Firstly, I am very grateful to my supervisor (Directeur de thèse), Paul Montgomery, whom I greatly respect, for supervising my research work over the years. Many thanks for all the advice, the encouragement, the availability and the hours he spent for reading and correcting the manuscript. I am also very grateful to my co-supervisor (Encadrant), Fabien Salzenstein, for his great availability for discussion whenever I had a problem or a question about my PhD research. Many thanks also for the help, the encouragement, and the contribution for the improvement of the manuscript.

Thanks are given to Audrey Leong-Hoï and Husneni Mukhtar for providing the measurements on real samples and to Stéphane Perrin for useful discussions, all from the IPP team. Finally, I will always remember all of the kindness of my colleagues in the IPP team and to all parties that cannot be mentioned one by one for their support during my study.

My appreciation is also extended to the institution where I work, the Metrological Resources Development Centre, which gave me permission to study in France. I am also grateful to the Indonesian Government (Ministry of Trade), which provided the scholarship for my studies. Several surface analysis techniques, such as the classical stylus probe, SEM (Scanning Electron Microscopy), AFM (Atomic Force Microscopy), confocal microscopy and interference microscopy, are important in the field of materials characterization, industrial metrology, and inspection [1], [START_REF] Bennett | Stylus profiling instrument for measuring statistical properties ofsmooth optical surfaces[END_REF], [START_REF] Hattori | Analysis of microscopic areas on wafer surfaces usingg STM/AFM[END_REF]. It is often useful to use several techniques to elucidate the various surface characteristics, especially since there is no single technique capable of providing all the information on the morphology of a surface. The technique of interference microscopy, which makes use of light interference as the optical probe, has the advantages of being non-destructive and fast [START_REF] Schmit | Performance advances in interferometric optical profilers for imaging and testing[END_REF], [START_REF] De | Principles of interference microscopy for the measurement of surface topography[END_REF], [START_REF] Pfeiffer | 3D nano surface profilometry by combining the photonic nanojet with interferometry[END_REF]. There are two families of techniques in interference microscopy: Phase Shifting Microscopy (PSM) and

TABLE OF FIGURES

LIST OF TABLES

Coherence Scanning Interferometry (CSI). PSM is a mathematical method of fringe interpolation based on the introduction of known phase shifts between the two arms of the interferometer. The technique is well suited for the analysis of small surface roughness (depth < 200 nm) and commonly has a nanometric axial resolution. The dynamic range of PSM is limited to λ/2 due to the periodicity of the interference fringes that induce a 2π ambiguity in the measured surface profiles.

Due to this limitation, the second family of techniques, i.e. CSI was developed to allow the measurement of much deeper surfaces. The advantages of this technique compared to the PSM are the larger vertical depth of field and the possibility of measuring height differences of several microns and more [START_REF] Caber | Interferometric profiler for rough surfaces[END_REF], [8]. Nowadays, the use of CSI, which is also common known as white light interferometry, is widely used in various fields, including surface metrology, materials characterization, and medical imaging. For instance in the field of metrology, white light interferometry has been applied in the characterisation of the fabrication of Micro Electro Mechanical Systems (MEMS) and other micro-component devices [START_REF] O'mahony | Characterization of micromechanical structures using white-light interferometry[END_REF], [START_REF] Grigg | Static and Dynamic Characterization of {MEMS} and {MOEMS} Devices Using Optical Interference Microscope[END_REF], [START_REF] Bosseboeuf | Application of microscopic interferometry techniques in the MEMS field[END_REF]. White light interferometry has also been applied for the measurement of surface roughness and microscopic structures in materials science and microelectronics [START_REF] Montgomery | Implementation of a fringe visibility based algorithm in coherence scanning interferometry for surface roughness measurement[END_REF], [START_REF] Blunt | White Light Interferometrya production worthy technique for measuring surface roughness on semiconductor wafers[END_REF]. In another field, Dubois et al. developed full-field optical coherence tomography (FF-OCT) based on white light interferometry for highresolution optical imaging of biological tissue [START_REF] Dubois | Three-dimensional cellular-level imaging using full-field optical coherence tomography[END_REF], [START_REF] Dubois | Ultrahigh-resolution full-field optical coherence tomography[END_REF], [START_REF] Dubois | High-resolution fullfield optical coherence tomography with a Linnik microscope[END_REF]. This technique achieves a better spatial resolution than conventional OCT, without using an expensive light source, achieving a resolution of axial × transverse = 0.7 µm × 0.9 µm. Conventional OCT is usually based on a fibre Michelson interferometer illuminated by a broad-bandwidth spatially coherent source (laser). The measurement requires scanning of a point probe in the transverse direction over the sample in order to obtain the cross sectional image. The spatial resolution of OCT in the axial direction is defined by the coherence length of the illumination source, typically 10-15 µm in the case of a super luminescent diode (SLD) [START_REF] Dubois | High-resolution fullfield optical coherence tomography with a Linnik microscope[END_REF], [START_REF] Hee | Handbook of optical coherence tomography[END_REF], [START_REF] Bott | Optical Coherence Tomography Principles and applications[END_REF]. By using ultra-short femto second laser technology, the spatial resolution of OCT has improved down to ∼ 1 µm. Nowadays, OCT can be applied at the cellular level [START_REF] Dubois | Three-dimensional cellular-level imaging using full-field optical coherence tomography[END_REF], [START_REF] Hee | Handbook of optical coherence tomography[END_REF].

A schematic layout of a simplified OCT system is illustrated in Fig. 1. The light from a low coherence source is coupled into a fiber-optic Michelson interferometer. This light is then split at a fiber coupler into the reference and the sample arm. The light retroreflected from the reference mirror and the light backscattered from the sample is recombined in the coupler and generates an interference pattern, which is detected by a single point detector (a photodiode) [START_REF] Hee | Handbook of optical coherence tomography[END_REF], [START_REF] Popescu | Optical coherence tomography: Fundamental principles, instrumental designs and biomedical applications[END_REF], [START_REF] Izatt | Theory of Optical Coherence Tomography[END_REF].

Fig. 1 Schematic of the OCT system [START_REF] Hee | Handbook of optical coherence tomography[END_REF] The use of white light interference fringes as an optical probe in microscopy has a long history. In 1665, Robert Hooke investigated the coloured fringe that can be observed in white light when the two glass plates are in contact [START_REF] Hariharan | Optical interferometry[END_REF]. This phenomenon can be called as the starting point of optical interferometry. Coloured fringe were then studied by Isaac Newton in 1717 as an interference pattern created by the reflection of light between two surfaces (known as "Newton's rings"). Since then, the development of white light interferometry has been extended right up to the present day. The method developed for the three dimensional measurement of surface topography using white light interferometry is generally based on the analysis of fringe contrast. In this method, vertical fringe scanning (either the object or reference mirror) is carried out, followed by measuring the fringe intensity at each point of the object surface. The fringe pattern is observable when the optical path difference between the reference mirror position and the object surface is smaller than the coherence length of the used light. The zero value of the optical path difference is related to the surface position of the object (sample).

Developments in digital signal processing now allow three dimensional measurement using white light interferometry rapidly and automatically. The first automated 3D measurement using white light interferometry was reported by Balasubramanian in 1980 [START_REF] Balasubramanian | Optical system for surface topography measurement[END_REF]. The variation in fringe contrast is recorded using a detector (CCD), point by point for each surface position. If the maximum fringe contrast level is observed, that point is determined as having a zero path difference that indicates the surface position. In 1987, Davidson reported the application of white light interferometry for integrated circuit inspection and metrology [START_REF] Davidson | An Application Of Interference Microscopy To Integrated Circuit Inspection And Metrology[END_REF]. In this application, the use of an automated mechanical piezoelectric stage for fringe scanning was applied. The fringe intensity was captured using a detector, and then the peak of the fringe contrast was detected for profiling the object (integrated circuits). In 1990, by extending the concept of the Davidson's work and focusing on the profilometry capabilities of white light interferometry, Lee and Starnd reported the advantage of white light interferometry over the conventional microscope in lateral resolution [START_REF] Lee | Profilometry with a coherence scanning microscope[END_REF]. In 1992, the work on three dimensional sensing using white light interferometry for rough objects was reported [START_REF] Dresel | Three-dimensional sensing of rough surfaces by coherence radar[END_REF].

In 1990, Chim and Kino introduced another way of fringe analysis by digital filtering of the interferogram (interference data). The fringe analysis used two different algorithms based on the Fourier Transform and Hilbert transform for extracting the fringe envelope [START_REF] Kino | Mirau correlation microscope[END_REF], [START_REF] Chim | Three-dimensional image realization in interference microscopy[END_REF]. In 1993, Caber used the demodulation technique adapted from communications theory for retrieving the fringe envelope [START_REF] Caber | Interferometric profiler for rough surfaces[END_REF], [START_REF] Caber | New interferometric profiler for smooth and rough surfaces[END_REF], [START_REF] Caber | Effect of detector noise on the positioning accuracy of an autofocus system[END_REF]. In 1993, Montgomery and Fillard applied the algorithm of PFSM (peak fringe scanning microscopy) [START_REF] Montgomery | Peak fringe scanning microscopy: submicron 3D measurement of semiconductor components[END_REF], [START_REF] Montgomery | Method and apparatus for interferometrically inspecting a surface of an object[END_REF], a very efficient algorithm which was then adapted for 3D measurement in real time [START_REF] Montgomery | Real time surface morphology analysis of semiconductor materials and devices using 4D interference microscopy[END_REF], [START_REF] Montgomery | High-speed, on-line 4D microscopy using continuously scanning white light interferometry with a high-speed camera and real-time FPGA image processing[END_REF]. In 1996, Larkin developed the Five Sample Adaptive (FSA) algorithm which is an efficient algorithm in computational time for retrieving the fringe contrast envelope [START_REF] Larkin | Efficient nonlinear algorithm for envelope detection in white light interferometry[END_REF]. This was then demonstrated to be a member of the wider family of very compact and efficient Teager-Kaiser Energy Operators by Salzenstein et al. (TKEO) [START_REF] Salzenstein | 2d discrete high order energy operators for surface profiling using white light interferometry[END_REF], [START_REF] Boudraa | Time-Frequency Signal Analysis and Processing: TEAGER-KAISER ENERGY OPERATORS IN TIME-FREQUENCY ANALYSIS[END_REF], [START_REF] Salzenstein | Teager-Kaiser energy and higher-order operators in white-light interference microscopy for surface shape measurement[END_REF]. The work demonstrated that the TKEO algorithms are robust and very competitive with the other types of fringe envelope detection algorithms, being very efficient in terms of computation time, and making them quicker than the other techniques when implemented on the appropriate hardware. The extended versions of this operator show that the method substantially provides the effective results in term of the surface measurement [START_REF] Salzenstein | Local frequency and envelope estimation by Teager-Kaiser energy operators in white-light scanning interferometry[END_REF]. Then, another method using the wavelet transform algorithm in order to retrieve the fringe envelope in white-light interferometry was applied by Sandoz in 1997 [START_REF] Sandoz | Wavelet transform as a processing tool in white-light interferometry[END_REF]. The advantage of the wavelet transform method implemented in white light interferometry is that it is very robust to noise.

Most of the above fringe envelope detection methods are implemented on the one dimensional fringe signal (1D approach). The work in [START_REF] Blunt | White Light Interferometrya production worthy technique for measuring surface roughness on semiconductor wafers[END_REF] used a 2D imaging processing method of the XZ images in an image stack for tomographic analysis of transparent layers. We also demonstrated in [START_REF] Montgomery | Implementation of a fringe visibility based algorithm in coherence scanning interferometry for surface roughness measurement[END_REF], [START_REF] Salzenstein | Two-dimensional continuous higher-order energy operators[END_REF] the ability of 2D approaches to compete with some classical methods used in the field of interferometry, in terms of robustness.

Whereas most methods only take into account the 1D data, it is advantageous to take into account the spatial neighborhood using multidimensional approaches (2D, 3D). The objective of this project was therefore to study and develop new n-D approaches using Teager Kaiser energy operators for fringe analysis in CSI. This study is one part of dealing with several challenges of fringe signal processing currently in CSI. The challenges include the precision required along Z (optical axis) and XY (lateral direction), speed of processing, quantitative and qualitative aspects, noise aspects, the offset component, n-D approaches, and a formalised protocol for processing. Another challenge is the degree of complexity of the surface, that can be classified going from the simplest to the more complex, by a smooth homogenous surface, a rough surface, a transparent layer with defects, a cell in water, and skin/tissue, as shown in Fig. 2. The first chapter discusses the state of the art of Coherence Scanning Interferometry as related to this project. We mainly discuss the general principles of CSI and the procedure of fringe analysis. We study and observe the structure of the fringe signal and the various approaches (Z-scan and XZ-scan techniques) for fringe analysis. The study of pre-processing for the offset component removal, microscope system, and description of the samples which we use in the work are also reported.

In the second chapter, we begin our study by evaluating the performance of envelope detection using the 1D Teager Kaiser Energy Operator, which is compared to other techniques. These methods consist of the Fourier Transform (TF), wavelet, the FSA (Five-Sample-Adaptive). We have developed a simulation program (on MATLAB) that allows the comparison of the performance of different methods using a synthetic fringe signal (a synthetic transparent layer). Then, we have studied the realization of the algorithms using real data, in this case on fringe images from a resin layer on Silicon.

In the third chapter, we implement the 2D approach using Teager Kaiser for fringe analysis. In this work, the study of the robustness of the 2D approach in CSI was carried out for the characterization of a transparent polymer film. The results demonstrate that the XZ fringe envelope extracted by the 2D Teager Kaiser approaches gives more satisfactory results than the 1D approach by revealing the internal structures and the rear surface of the transparent polymer film. The technique also results in improved details in the XZ images as well as more accurate measurements of the thickness of the polymer film.

In the fourth chapter, we present the study of the application of the 3D Teager Kaiser Energy Operator (3DTKEO), which is developed on the basis of the multidimensional Teager Kaiser energy operator. Through a simulation using a synthetic fringe signal, we have evaluated the robustness of the 3DTKEO's performance in fringe signal processing, which is compared to the 1D and 2D approaches. In addition, we have also used the algorithm on real data, in this case a step height standard (VLSI Standard Inc.) to evaluate the accuracy of the measurement. In addition, we have enriched the field of study by testing the algorithm on various other samples, such as graphene, DOE (Diffractive Optical Elements), resin on silicon, cable, and rock.

Chapter 1. COHERENCE SCANNING INTERFEROMETRY

In this chapter, we discuss several of the existing three dimensional surface profiling techniques, including Atomic Force Microscopy (AFM), confocal microscopy, and interference microscopy. Futhermore, we describe how the technique of Coherence Scanning Interferometry (CSI) generates an interferogram (fringe signal). We study and observe the structure of the fringe signal and the various approaches (Z-scan and XZ-scan techniques) for the fringe analysis. Then, we focus on the steps of the fringe analysis procedure which we perform in this work. The study of pre-processing for the offset component removal and determination of the surface structure of the sample which we use in the work are also reported.

3D SURFACE PROFILING

The measurement of 3D surface structures is an important field in materials characterization and industrial metrology. The existing techniques of 3D surface topography measurement consist of contact profilometers and optical profilometers. A stylus profiler is the oldest contact profiler, using a tactile probe to measure the surface profile [1], [START_REF] Bennett | Stylus profiling instrument for measuring statistical properties ofsmooth optical surfaces[END_REF]. The technique works by moving the object surface in relation to the stylus tip and sensing the height variations of the stylus tip to determine the surface height profile. In the 1980's, another type of scanning probe profilometer involving near field microsopy was developed [START_REF] Binnig | Scanning Tunneling Microscopy-from Birth to Adolescence (Nobel Lecture)[END_REF]. The first mode was Scanning Probe Microscopy (SPM), based on electron tunnelling, that works by moving a fine tip in close proximity to the sample surface, to within several nm to a few angstroms depending on the technique. The Atomic Force Microscope (AFM) is the most popular SPM, using Van der Waal's forces that can provide 3D images of surfaces generally at the nanometer scale. AFM is popular particularly for use in measuring non-conductive materials such as cells, bacteria, viruses and proteins [START_REF] Markiewicz | Atomic force microscopy probe tip visualization and improvement of images using a simple deconvolution procedure[END_REF], [START_REF] Tokumoto | Nanometer Modifications of Non-Conductive Materials Using Resist-Films by Atomic Force Microscopy[END_REF].

AFM has the advantages of nanometric resolution, but the images take in general several minutes to several tens of minutes or more to acquire. Optical profilers on the other hand have the advantage of allowing high speed surface profiling. The two most common optical profilers are confocal microscope and white light interferometery (Coherence Scanning Interferometry).

Confocal microscope. The confocal microscope is based on focusing principle, as illustrated in Fig. 3. that uses a pinhole to ensure only light at the point of focus on the sample surface can enter the detector. Another type, the chromatic confocal microscope was then developed which the vertical scanning is not required [1], [START_REF] Schmit | Performance advances in interferometric optical profilers for imaging and testing[END_REF], as shown in Fig. 3(c). Rather than using the vertical scanning, the technique uses an objective with an axial chromatic aberration that has a different focus position for each different wavelength corresponding to the height of the surface. A spectrometer in place of CCD as a detector, detects the wavelength value.

The measurement of the object focus position based on the spectrum measurement makes the measurement process much faster. The technique has the disadvantage of the optical sensitivity for the inhomogeneous materials. Similarly as the conventional confocal system, another disadvantage is the necessity of lateral scanning of the sample. Phase Shifting Microscopy. There are two main techniques used in interference microscopy: Phase Shifting Microscopy (PSM) and CSI [START_REF] Montgomery | Implementation of a fringe visibility based algorithm in coherence scanning interferometry for surface roughness measurement[END_REF], [8], [START_REF] Fillard | Near Field optics and nanoscopy[END_REF]. PSM is a mathematical method of fringe interpolation based on the introduction of known phase shifts between the two arms of the interferometer. These phase shifts vary the optical path difference (OPD) which result in several interferograms. The technique is well suited for the analysis of small surface roughness (depth < 200 nm) and commonly has a nanometric axial resolution. The PSM also has the capability to deliver results with low noise and high precision for smooth optical surfaces. However, the dynamic range of PSM is limited to λ/2 due to the periodicity of the interference fringes that induce a 2π ambiguity in the measured surface profiles. Due to this limitation, the second family of techniques, i.e. CSI was developed to allow the measurement of much deeper surfaces. The advantages of this technique compared to the PSM are the larger vertical depth of field and the possibility of measuring height differences of several microns and more [START_REF] Caber | Interferometric profiler for rough surfaces[END_REF], [8].

In PSM technique, a monochromatic or quasi-monochromatic light source is used for illumination on the sample such as with Köhler illumination. During the measurement, the optical path difference (OPD) is changed by taking three or more images using the camera. The phase of the interference signal is then analyzed at each point on the image using the PSM algorithm. The altitude which represents the surface profile is calculated based on the result of the phase measurement.

The following Eq.(1.1) [START_REF] Abderrazzaq | Développement de la microscopie interférométrique pour une meilleure analyse morphologique des couches minces e t épaisses des matériaux semiconducteurs et optiques[END_REF] express the intensity at a coordinate point (x,y) in an interference pattern:

(1.1)
where I o (x,y) is the irradiance, γ 0 is the fringe visibility (also called modulation or contrast), φ(x,y) is the signal phase, and α i is the phase shift. Based on the Eq. (1.1) mathematically, the three unknown parameters: I o (x,y), γ 0 and φ(x,y) can be calculated by using at least three interferograms. The precision of the measurement in PSM technique can be improved using higher numbers of interferograms.

In order to introduce the phase shifts in PSM, there are two basic modes, i.e. the discrete mode and continuous mode. For description of this phase shift in discrete and continuous mode, the algorithm of 4 steps of 120° is chosen as illustration.

Discrete mode. The discrete mode of this technique based on the discrete phase steps calculation is also known as the phase stepping microscopy. The illustration in Fig. 4 shows a phase determination from 4 discrete mode which is represented by the intensity of the interference fringes. The phase is then can be calculated using the following equation, which is usually displayed as a grayscale image of the phase:

(1.2) Fig. 4 Phase determination, from 4 discrete steps of 120° [START_REF] Abderrazzaq | Developpement de la microscopie interferometrique pour une meilleure analyse morphologique des couches minces et epaisses des materiaux semiconducteurs et optique[END_REF].

Continuous mode. The continuous mode of the phase change technique in PSM, also known as the technique of the phase integration. As illustrated in Fig. 5, the technique is based on a linear variation of the phase. The parameter T is the period time of the change of the phase, while the parameter τ, which depends on the number of interferograms, is a time of the acquisition of an image. Fig. 5 The technique of the change of phase [START_REF] Abderrazzaq | Developpement de la microscopie interferometrique pour une meilleure analyse morphologique des couches minces et epaisses des materiaux semiconducteurs et optique[END_REF].

In this technique several interferograms over the period T (assuming T is continuous) are recorded in order to determine the initial phase. On the other hand, one interferogram represents each integration time τ. Fig. 5 shows the illustration of phase determination in continuous mode corresponding to the example of Fig. 4. The integration time represents respectively the phase determination of 4 steps of 120°.

PSM algorithms

Several phase reconstruction algorithms have been developed. The phase measurement algorithms are based on the acquisition of a series of N interferograms (N ranging generally from 3 to 7) obtained with known phase shifts. The essential parameters in these measurement techniques are the mode of variation of the phase (discrete or continuous), the frequency of acquisition of the interference fringes and the number N of interferograms used to calculate the phase at a given instant.

a. Three step technique

If a phase shift of 2π is introduced between the object and reference beams, the phase φ(x,y) can be deduced from the intensities I 1 , I 2 , I 3 of the three interferograms recorded for the angular offsets π/4, 3π/4, 5π/4. The intensities measured are given by:

            10 20 30 , , 1 cos ( , ) 4 3 , , 1 cos ( , ) 4 5 ,
The resolution of these equations gives the expressions of phase and visibility at a point of coordinates (x,y) as follows:

                        32 12 22
1 2 2 3 0 ,, ( , ) ,, , , , , , 2,

b. Four step technique

The intensity of the interference fringes for phase shifts of 2π is given by: By a combination of these equations, the phase and visibility can be calculated with the following expressions:

                    10 20 30 40 , , 1 cos ( , ) , , 1 cos ( , ) 2 , , 1 cos ( , ) 3 ,
                        42 13 22 4 2 1 3 0 ,, ( , ) ,, , , , , , 2,

c. Five step technique

The intensity of the interference fringes for phase shifts of 2π is given by:

                          10 20 30 40 50 , , 1 cos ( , ) , , 1 cos ( , ) 2 , , 1 cos ( , ) , , 1 cos ( , ) 2 , , 1 cos ( ,

1.7)

By a combination of these equations, the phase and visibility can be calculated by the following expressions:

                                24 3 5 1 2 2 2 4 3 5 1 0 2 , , ( , ) 2 , , , 2 , , 2 , , , , 4,

I x y I x y x y arctg I x y I x y I x y I x y I x y I x y I x y I x y xy

I x y               (1.8)
As mentioned previously, the limited depth range of PSM led to the development of white light interferometry and a different technique altogether based on the fringe envelope detection to give unambiguous depth measurement over large depths. This is now presented in the next section.

GENERAL PRINCIPLE OF CSI

A typical layout of a CSI system is shown in Fig. 6(a) [START_REF] Bosseboeuf | Application of microscopic interferometry techniques in the MEMS field[END_REF]. The working principle of the CSI technique is based on the cross coherence analysis between the reference beam and the reflected object beam which come from a low coherence source using a beam splitter.

During the measurement of the sample surface, the reference beam is reflected from the reference mirror, whilst the reflected object beam is reflected from the sample [START_REF] Fallis | [END_REF]. The two light beams are then combined at the detector. The interference will occur when the optical path difference between the reference beam and the reflected object beam is close to zero. This is when the optical path length to the sample is nearly identical with the optical path length to the reference mirror. With the goal of finding the interference maximum, fringe scanning is carried out at each point on the sample surface, point by point. The fringe intensities, which vary according to the change in distance between the sample and the objective (in z axis), are captured by the detector (camera) generating the stored data signal, or interferogram. Fringe analysis is then applied in order to retrieve the peak of the fringe contrast envelope which indicates the axial position of the surface of the sample. Fig. 7 shows how the interference is constructed at each pixel, point by point in the detector (camera). Fig. 7 The interferogram construction on a surface using CSI (source: Guide to the measurement using CSI [START_REF] Leach | A National Measurement Good Practice Guide[END_REF]).

The advantage of white light over monochromatic light is its ability to avoid ambiguity in order to determine the fringe order due to white light having a low coherence length. On the other hand, many factors can affect the accuracy of the CSI measurements, including the camera performance, the control and linearity of the piezocontrolled vertical scanning, the metrology frame design, the environment, and the sample stability [START_REF] Leach | A National Measurement Good Practice Guide[END_REF].

FRINGE SIGNAL

The light intensity giving rise to the fringe signal, s(x,y,z), captured from a detector (CCD) as the optical path difference is varied through focus in a white light interferometer, has the following form [START_REF] Kino | Mirau correlation microscope[END_REF], [START_REF] Larkin | Efficient nonlinear algorithm for envelope detection in white light interferometry[END_REF]: Eq.(1.9) corresponds to the simple case of light propagation in air which is assumed to be equal to 1. In the case of the light passing through a transparent layer, a double signal is produced, one from each side of the layer (Fig. 8). In this case the refractive index is greater than one and wavelength dependent, thus inducing the phenomenon of dispersion for the signal from the other side of the layer. In this case the actual distance for the second signal is measured by Eq.(1.10). CSI measures the optical path difference along the optical axis Z. By measuring the distance  separating the peaks of the two envelopes, the actual distance d of the transparent layer at that point (X,Y) can be measured. If the sample displacement d along the optical axis (Z) and the refractive index n of the layer is known, the actual distance d is:

                00 0 4 , , , , , , .cos ,
  d n (1.10)
A completely description about this topic will be discussed in Chapter 2 and Chapter 3 in measurement the sample of resin on Si and Mylar polymer film.

Z-SCAN TECHNIQUE (ID)

The description of fringe analysis in CSI using the Z-scan technique is shown in Fig. 9.

By means of a single vertical scan of the sample, a stack of XYZ images is generated.

Signal processing is then used to obtain the fringe envelopes along Z in order to measure the positions of the peak of the fringe envelope which corresponds to the height of the surface at each pixel in the XY image [START_REF] Halter | Characterization of inhomogeneous colloidal layers using adapted coherence probe microscopy[END_REF], [START_REF] Montgomery | Tomographic analysis of medium thickness transparent layers using white light scanning interferometry and XZ fringe image processing[END_REF].

XZ-SCAN TECHNIQUE (2D)

Two dimensional fringe signal processing can be chosen using the raw XYZ data by operating on the XZY images as shown in Fig. 10. The technique is based on analyzing the XZ fringe images at a given point along the Y axis. The fringes are low pass filtered in 2D so as to find the fringe envelope. By processing a given XZ image, the cross sectional profile of a transparent sample can be obtained [START_REF] Montgomery | Tomographic analysis of medium thickness transparent layers using white light scanning interferometry and XZ fringe image processing[END_REF], also known as a B-scan in optical coherence tomography.

ANALYSIS OF WHITE LIGHT INTERFERENCE FRINGES

In general, the techniques of signal processing developed in this work consist of three main steps: pre-filtering, envelope detection and post-filtering (Fig. 11). The envelope detection is needed in order to obtain the fringe envelope of which the peak represents the surface position. Pre-filtering is used to remove the offset component and reduce the noise, while post-filtering is used to determine more precisely the measurement.

Fig. 11 The procedure of white light interference fringe analysis [START_REF] Gianto | Comparison of envelope detection techniques in coherence scanning interferometry[END_REF].

Pre-filtering

The robustness of the signal processing of fringe signals depends on the sensitivity to the different sources of signal noise and artifacts. Another problem lies in an additive offset component (background) which can appear in the fringe signals during the acquisition process, particularly over large scanning depths. In order to remove this offset component and the additive noise, it is important to filter out both of them before applying fringe envelope detection.

Envelope Detection

In the following we present the different techniques used to retrieve the fringe envelope in CSI.

Analytic Signal (Hilbert Transform)

In analytic signals, the Hilbert Transform is often used for the purpose of amplitude demodulation. If we consider a real signal s(t), then the analytic signal s A (t) is defined as:

          exp AH s t s t is t A t i t       (1.11)
Where the Hilbert transform of a signal is given by:

      11 H su s t s t du t t u         (1.12)
In the frequency domain, the analytic signal corresponds to:

      1 A S sign S         (1.13)
In particular, for obtaining the analytic signal s A (t), the negative frequency component of the signal is suppressed, which can be performed using a Fast Fourier Transform (FFT). The main steps of the Analytic Signal (Hilbert Transform) technique using a Fast Fourier Transform (FFT) calculation are outlined in Fig. 12. The FFT is applied to the signal in Fig. 12(a), producing the spectrum of the signal (Fig. 12(b)). The FFT coefficients that correspond to negative frequencies are then replaced with zeros (Fig.

12(c))

. Finally, the fringe envelope is extracted (Fig. 12(d)) by calculating the inverse FFT of the positive frequency packet in Fig. 12(c). By removing negative frequencies from the spectrum of real signal s(t), the signal provided by inverse FFT becomes complex as shown in Eq. (1.11), which gives directly the phase (t) and the amplitude

A(t). (a) (b) (c) (d)
Fig. 12 Fringe envelope detection process using Hilbert Transform technique [START_REF] Gianto | Comparison of envelope detection techniques in coherence scanning interferometry[END_REF].

Five Sample Adaptive (FSA) algorithm

The FSA technique is a fast and simple algorithm which has been commonly used in CSI to retrieve the fringe envelope [START_REF] Larkin | Efficient nonlinear algorithm for envelope detection in white light interferometry[END_REF]. The main idea of the technique is the application of phase shifting algorithms for white light interferogram demodulation and to use the fringe visibility (or modulation) to calculate the fringe envelope. In the case of the FSA technique, five interferograms are captured by a digital imaging system.

At each pixel location in the image (x,y), the value of the visibility(x,y) of the fringes at that point is calculated from the intensities of five sampling positions, I n-2 to I n+2 , along the optical axis z (Fig. 13):

    22 2 4 1 3 5 1 ( , ) 4 2 4 A x y I I I I I       (1.14)
where I 1 to I 5 are the interferogram intensities, giving exact values when the phase shifting between interferograms is 90. The amplitude A(x,y) can be calculated with

Teager Kaiser Energy Operator (TKEO)

The Teager Kaiser Energy Operator (TKEO) [START_REF] Kaiser | On a simple algorithm to calculate the energy of a signal[END_REF] is an operator that tracks the instantaneous energy of a signal. This non-linear energy operator and its 1D/2D discrete versions has found applications in various fields of signal and image processing due to its success in analysing and demodulating AM-FM signals with high resolution, simplicity, and efficiency [START_REF] Salzenstein | Local frequency and envelope estimation by Teager-Kaiser energy operators in white-light scanning interferometry[END_REF]. In its discrete version, only three samples are required at each time instant. In the continuous case, the Teager Kaiser Energy Operator is defined by:

              2 2 2 2 s z s z s z s z s z s z s z zz                (1.15)
where s(z) is the signal, s (z) and s (z) means the first derivative and the second derivative of s respectively. A discrete forward and backward approximation of the derivatives of Eq.(1.15) leads to the discrete TKEO [START_REF] Maragos | Speech nonlinearities, modulations, and energy operators[END_REF]:

        2 11 s n s n s n s n        (1.16)
Let us consider a mono-component continuous time AM-FM signal:

        cos s t a t t   (1.17)
where a(t) represents the spatially varying amplitude and (t) is the phase signal. The output of the Teager Kaiser Energy Operator applied to s(t), is given by

      2 s t a t t       , where   t 
is the instantaneous phase. Moreover, the Energy Separation Algorithm (ESA), estimates the instantaneous envelope a(t) and the instantaneous phase   t  as follows:

            ; s t s t a t t st st                    (1.18)
The main disadvantage of the Teager Kaiser Energy Operator lies in its sensitivity to noise, due to the successive derivations. In the context of noisy data, a filter such as the Savitzky-Golay approach, improves the robustness of TKEO.

Continuous Wavelet Transform (CWT)

The Wavelet transform is a time-frequency analysis technique which has been widely used in signal analysis and processing [START_REF] Peng | Application of the wavelet transform in machine condition monitoring and fault diagnostics: A review with bibliography[END_REF], [START_REF] Adeli | Analysis of EEG records in an epileptic patient using wavelet transform[END_REF], [START_REF] Swami | Speech enhancement by noise driven adaptation of perceptual scales and thresholds of continuous wavelet transform coefficients[END_REF]. Due to its ability to decompose locally the signal into different scale/frequencies, the CWT highlights the region of interest, where the local frequency corresponds to the carrier frequency of the CSI signal. Moreover, the CWT method seems to be robust to noise, providing accurate measurements of the surface.

The CWT transform function of a fringe signal s(z) can be expressed as:

    1/ 2 1 , zb W a b s z dz a a          (1.19)
where W(a,b) represents the CWT coefficient function, ψ(z) is a continuous function in time and frequency domain called the mother wavelet, a is the scale factor, and b is the shift factor. The well-known complex Morlet wavelet [START_REF] Tay | Demodulation of a single interferogram based on continuous wavelet transform and phase derivative[END_REF], [START_REF] Li | Statistical searching of deformation phases on wavelet transform maps of fringe patterns[END_REF], [START_REF] Wielgus | Amplitude demodulation of interferometric signals with a 2D Hilbert transform[END_REF], [START_REF] Salzenstein | Local frequency and envelope estimation by Teager-Kaiser energy operators in white-light scanning interferometry[END_REF], [START_REF] Suzuki | Low-coherence interferometry based on continuous wavelet transform[END_REF],

[64], [START_REF] Medicus | Fourier Transform and Wavelet Algorithms for Calculating Height in White Light Interferometry[END_REF] has been chosen as the mother wavelet since its properties have strong

Post-filtering

For noisy fringe signals, for instance as shown in Fig. 15(a) even though pre-filtering is used to remove the offset component and suppress the noise before the process of envelope detection, the noise can still appear in the resulting fringe signal, as shown in Fig. 15(b). This affects the process of envelope detection so that the fringe patterns appear in the resulting fringe envelope, as illustrated in Fig. 15(c). Because of this, a smoothing filter such as a cubic spline is needed to improve the envelope peak detection (Fig. 15(d)).

Then, the curve fitting method using Gaussian fitting is implemented in order to determine more precisely the envelope peak.

PERFORMANCE EVALUATION OF DIFFERENT OFFSET COMPONENT REMOVAL TECHNIQUES IN CSI

Generally, when we apply signal processing, the main aim is to remove part of the noise present in the signal or to eliminate different sources of variation (e.g. background) that are not related to the measured variable. The existence of a variable background offset to the signal can be caused by the effects of defocusing (3D PSF along Z), the Köhler illumination (conical illumination at each point on the sample), contrast variations across the sample (XY) and diffraction effects at step edges. In the case of fringe signal processing, it is important to filter out the background (offset component) before applying fringe envelope detection. The background variation is more observable on rougher surfaces or edges of structures and results from the response of the imaging objective to diffraction and out of focus effects. This generally consists of a slowly varying

The Offset Component removal technique

The following are most common offset component removal techniques in CSI:

a. Global Average

The global average technique makes use of the average of the signals in order to suppress the offset component. Global averaging removes the mean value from the vector x, which can be expressed as:

1 1 N i i i i x x x N    (1.21)

b. Local Average

The local averaging technique is a fast and simple algorithm which has been commonly used in signal processing to suppress the offset component. The value of the pre-filtered signal for a single sampling position x i is calculated from the intensities of three sampling positions, x i-1 to x i+1 , which can be expressed as:

  11 3 i i i ii x x x xx    (1.22)

c. Central Derivative

In signal processing, the derivative is a common method used to remove the background variation of a signal. The simplest form of numerical approximation of the derivative is the first order derivative based on the finite difference method, which has three form

      ' 2 f x h f x h fx h     ( 1

d. Empirical Mode Decomposition

After studying different techniques for supressing the background variation, as an alternative to the first derivative, we have introduced the use of the well-known decomposition technique known as Empirical Mode Decomposition (EMD). The EMD method has the ability to decompose a signal into oscillatory components, known as intrinsic mode functions (IMFs). The principle of EMD is the use of a repeated shifting process on the analyzed signal. For a given signal x(t), all extrema (local maxima and minima) are detected and connected by an interpolation technique, such as the cubic spline, that produces respectively the upper envelope e max (t) and the lower envelope e min (t). The average of the two envelopes is then computed, m(t) = (e max (t) + e min (t))/2 and the remaining detail d(t) = x(t)m(t), is extracted. The above procedure is repeated in order to obtain the first intrinsic mode functions (IMFs), d 1 (t) which satisfy two conditions:

1. The number of extrema (the sum of the maxima and minima) and the number of zero-crossings must either be equal or differ at most by one.

2. At any point of an IMF the mean value of the envelopes of the local maxima and minima is zero.

After the

1 st IMF is obtained, a residue r(t) = x(t) -d 1 (t)
    1 () N k k x t d t r t    (1.24)
The lower-order IMF's typically represents fast oscillation modes, while the higher order IMF's and the residue typically represent slow oscillation modes. Due to this characteristic, the EMD technique seems to be well adapted for suppressing the slowly varying offset. However, as with the central derivative technique, the EMD method used as a pre-filter can also fail in the presence of significant noise. For this reason, we have tested more robust filters such as the Savitzky-Golay or Wiener family in combination with the derivative and EMD methods.

In order to remove the offset and reduce the noise, three different algorithms have been studied in this work which combine the derivative/EMD technique and denoising filter (Savitzky-Golay/Wiener [START_REF] Boudraa | EMD-based Signal Noise Reduction[END_REF], [START_REF] Lim | Two-Dimensional Signal and Image Processing[END_REF], [START_REF] Tomaniac | [END_REF]). The corresponding algorithms are referred to with the following names: i) Pre-filter 1: combination of derivative and Wiener filter.

ii) Pre-filter 2: combination of derivative and Savitzky-Golay filter.

iii) Pre-filter 3: combination of EMD and Savitzky-Golay filter.

The denoising technique

The following gives a description of the denoising techniques used in this work:

a. Wiener Filter

Let us consider the signal corrupted by an independent zero-mean white Gaussian noise, which is modeled as [START_REF] Jin | ADAPTIVE WIENER FILTERING OF NOISY IMAGES AND IMAGE SEQUENCES[END_REF]:

    , ( , ) , y i j x i j n i j  (1.25)
where y(i,j) is the noisy signal, x(i,j) is the original signal and n(i,j) is a Gaussian noise.

The aim of the technique is to remove the noise and to obtain a linear estimate   ˆ, x i j which minimizes the mean squared error (MSE) [START_REF] Jin | ADAPTIVE WIENER FILTERING OF NOISY IMAGES AND IMAGE SEQUENCES[END_REF].

      2 ,1 1 ˆ,, N ij MSE x i j x i j N    (1.26)
where N is the number of elements in x(i,j). When x(i,j) and n(i,j) are stationary Gaussian processes the Wiener filter is the optimal filter. Specifically, when x(i,j) is also a white Gaussian process the Wiener filter has a very simple scalar form:

                2 22 , ˆ, , , , ,, x xx xn ij x i j y i j u i j u i j i j i j       (1.27)
where σ 2 , µ are the signal variances and means, respectively, and where we will normally assume the mean of the noise to be zero.

b. Savitzky-Golay Filter

The Savitzky-Golay filter is a smoothing technique for a signal that was proposed by Savitzky and Golay to increase the SNR (signal to noise ratio) without significantly distorting the signal. The filter is defined as a weighted moving average with weighting given as a polynomial of a specific degree [START_REF] Azami | An Improved Signal Segmentation Using Moving Average and Savitzky-Golay Filter[END_REF], [START_REF] Tomaniac | [END_REF]. The illustration of Savizky Golay filer is shown in Fig. 17.

Fig. 17 Illustration of the Savitzky Golay filter

The coefficients of a Savitzky-Golay filter, when applied to a signal, perform a polynomial P of the degree k, is fitted to points of the signal, where N describes the window size (neighboring points, including the point to be smoothed). The advantage of this filter is that it tends to keep features of the distribution such as those relative to the maxima and minima which are often flattened by other smoothing techniques such as MA (moving average).

Simulation results

In order to test these techniques by simulation, we generate different types of synthetic fringe signals corresponding to the two fringe signals coming from the top and bottom surfaces of a transparent layer. We then used these to test all pre-processing techniques, based locally on the general model expressed by Eq. (1.9) along the optical axis Z, with a 40 nm sampling step, for each lateral position X. The previously cited pre-processing techniques were then applied to the synthetic fringe signals, namely global averaging, local averaging, the central derivative and the EMD filter. The results are then shown for the constant offset (Fig. 19), linear offset (Fig. 20) and non-linear offset (Fig. 21).

Based on the results in Fig. 19, Fig. 20, and Fig. 21, it can be seen that the type of pre-processing technique required to suppress the offset component from the fringe signal depends on the nature of the signal and the computational time available and the precision required. The computation times for each of these processes are given in Table 1 using Matlab on a PC equipped with an Intel Xeon processor (2.33 GHz, 8 Go RAM). 1). However, as illustrated in Fig. 19

SURFACE MEASUREMENT ERRORS IN CSI

As with any measurement technique, while CSI has significant advantages for optical profiling, the measurements are also prone to different types of noise, artefacts and errors.

These are now fairly well known and have been described in the literature over the last decade, namely: ghost steps, the batwing effect, slope dependent errors, material effects and multiple scattering effects. These are now described.

a. Ghost Steps

When measuring a perfectly flat surface with CSI, depending on the algorithm and parameters used, step artefacts can be introduced into the measurement [START_REF] Lehmann | Fundamental aspects of resolution and precision in vertical scanning white-light interferometry[END_REF], [START_REF] Ghim | Complete fringe order determination in scanning white-light interferometry using a Fourier-based technique[END_REF]. This kind of error is commonly referred to as ghost steps. The simplest form is due to the sampling step when interpolation is not used. This results in the presence of steps corresponding to the scanning step between images. Another form of ghost step error is a type of fringe order error, which correspond to an error of the measured profile of  0 /2 [START_REF] Gao | Surface measurement errors using commercial scanning white light interferometers[END_REF]. The asymmetries in CSI, which arise for instance from imperfect beam splitter, can lead to the ghost step error in the measured profile [START_REF] Pförtner | Dispersion error in white-light Linnik interferometers and its implications for evaluation procedures[END_REF].

b. Batwing Effect

This measurement error is known as batwing effect due to the shape, rather like that of batwings, as illustrated in Fig. 22. The error appears in the measurement of step height, which is less than the coherence length of the light source used [START_REF] Harasaki | Fringe modulation skewing effect in white-light vertical scanning interferometry[END_REF], [START_REF] Harasaki | Improved vertical-scanning interferometry[END_REF]. The batwing effect generally appears for the white light vertical scanning technique due to the diffraction effect on the edges. For a typical white light interferometer, the fringe contrast envelope obtained, which is Gaussian, skews on the surface close to the edges leading to the error. The batwing effect must be considered when carrying out the measurement calibration using a step height standard. . Source: Guide to the Measurement using CSI [START_REF] Leach | A National Measurement Good Practice Guide[END_REF] Fig. 22 A square wave grating that shows the batwing effect at the step edges

c. Multiple Scattering

For the object with the depth profile such as V-grooves, the incident light is scattered multiple times due to the geometry of the object [START_REF] Gao | Surface measurement errors using commercial scanning white light interferometers[END_REF], [START_REF] Coupland | Measurement of steep surfaces using white light interferometry[END_REF]. The effect of multiple scattering is discussed in [START_REF] Gao | V-Groove Measurements using White Light Interferometry[END_REF] using a silicon V-groove specimen, as shown in Fig. 23.

Fig. 23 Profile measurement of V-groove using white light interferometry [START_REF] Gao | V-Groove Measurements using White Light Interferometry[END_REF] As illustrated in the figure, at the bottom of the groove, the measured profile (blue line)

and the known form of the groove (black line) only coincides at the apex, while the measured profile around the apex show the inverted form of the V-groove. Generally, in white light interferometry, multi scattering causes the over estimation for the surface roughness [START_REF] Gao | Surface measurement errors using commercial scanning white light interferometers[END_REF], which can be considered to be similar to small V-grooves.

d. Dispersive Effects in Dissimilar Materials

The optical properties of the object surface can also be a source of measurement error in white light interferometry [START_REF] Gao | Surface measurement errors using commercial scanning white light interferometers[END_REF], [START_REF] Palodhi | Absolute surface topography measurement with polarisation sensitive coherence scanning interferometry[END_REF], [START_REF] Park | Compensation of phase change on reflection in whitelight interferometry for step height measurement[END_REF]. The surface height measurement will be affected since different materials exhibit different phase changes on reflection [START_REF] Harasaki | Offset of coherent envelope position due to phase change on reflection[END_REF]. For instance, because of the complex refractive index of metallic surface, this will lead to shifting of the position peak of the coherence envelope function which results in the measurement error. In the article [START_REF] Harasaki | Offset of coherent envelope position due to phase change on reflection[END_REF], it is demonstrated that the largest phase dispersion on reflection is shown by metallic surfaces, shifting the peak position of the coherence envelope by 10 -40 nm.

MICROSCOPE SYSTEM

The equipment developed for testing the developed algorithm for fringe signal processing in this thesis consists of four adapted microscopes, a Leica DMR-X microscope (Fig. 24) [START_REF] Montgomery | Implementation of a fringe visibility based algorithm in coherence scanning interferometry for surface roughness measurement[END_REF], [START_REF] Montgomery | Tomographic analysis of medium thickness transparent layers using white light scanning interferometry and XZ fringe image processing[END_REF], [START_REF] Pecheva | White Light Scanning Interferometry Adapted for Large-Area Optical Analysis of Thick and Rough Hydroxyapatite Layers[END_REF], [START_REF]Étude des techniques de super-résolution latérale en nanoscopie et développement d'un système interférométrique nano-3D[END_REF].

equipped
The Leica DMR-X (Fig. 24) consists of the following five main elements [START_REF]Étude des techniques de super-résolution latérale en nanoscopie et développement d'un système interférométrique nano-3D[END_REF]:

a. The stand that supports and maintains various optical mechanisms and tubes.

b. The illumination source (halogen or tungsten filament lamp or LED).

c. The interference objective that makes it possible to view interference fringes superimposed on the image of the sample surface and to modify the degree of magnification.

d. The motorised sample platform for focusing and positioning the observed sample.

e. The observation head which allows both direct binocular observation by eye and image detection using a sensor such as a CCD camera to record the observed image.

Fig. 24 The modified Leica DMR-X interference microscope developed in the IPP team [START_REF]Étude des techniques de super-résolution latérale en nanoscopie et développement d'un système interférométrique nano-3D[END_REF].

STRUCTURE OF THE SAMPLES SURFACE

In order to evaluate the performance of the algorithms developed in this work, the following samples that have different structures were used. They were chosen because they vary from a simple smooth surface through to increasingly complex structures.

a. Single surface

The samples that have a single surface which are used in the thesis consist of two types of structures: (i) a sample having a smooth surface, such as a step height standard (SHS), with a step having a height of 1.7803 ± 0.011 µm [START_REF] Harasaki | Improved vertical-scanning interferometry[END_REF], [START_REF] Roy | White-light interference microscopy: minimization of spurious diffraction effects by geometric phase-shifting[END_REF] and (ii) a sample having a rough surface, such as a rock surface, i.e. limestone, in which the fringes are very noisy [START_REF] Mukhtar | Rock surface roughness measurement using CSI technique and analysis of surface characterization by qualitative and quantitative results[END_REF]. The sample of SHS is used to calibrate and test the precision of the measurement using the developed algorithm, while the rock sample is used to test the robustness of the algorithm on the noisy data of the rough sample (Chapter 4). The interferograms of the samples are shown in Fig. 25.

b. Transparent layer on substrate

The next sample consists of a layer of resin on silicon used for making diffractive optical elements (Fig. 26) [START_REF] Flury | Rapid prototyping of diffractive optical elements for high-power lasers using laser ablation lithography fabrication and coherence probe microscopy analysis[END_REF]. The description of acquisition data of resin on Si obtained by the microscope system is given in Appendix-1: Data of Sample. This sample is of interest because it produces two interferograms in each signal from the air/resin surface and resin/silicon interface, the distance between them indicating the optical thickness of the layer. Interference microscopy can thus also be used to non-destructively analyze a surface buried under a transparent layer. However, several factors can contribute to the degradation of the wavefront reflected from the interface, for instance the aberrations due to the presence of the transparent layer, the variations of the refractive index of the transparent layer, the shape and geometry of the transparent layer and interface, the edges of the layer etc. In Chapter 2, the transparent layer on substrate sample is used to evaluate the performance of certain algorithms to measure the thickness profile of the transparent layer and to analyze the layer/substrate interface. The transparent polymer film sample consists of a Mylar film used as an insulator in microelectronics (Fig. 27) [START_REF] Benatmane | 3D analysis of buried interfaces using interference microscopy[END_REF], [START_REF] Leong-Hoï | Detection of defects in a transparent polymer with high resolution tomography using white light scanning interferometry and noise reduction[END_REF]. The description of acquisition data of Mylar polymer film obtained by the microscope system is given in Appendix-1: Data of Sample. This sample is of interest because as well as the upper and lower reflecting surfaces, it is a bit more complex since it contains low contrast buried structures which are not easily observable using classical optical techniques. The sample is used to test the performance of the developed algorithms in detecting the rear surface position and the presence of the buried structures (Chapter 3).

RESOLUTION

The spatial resolution of an optical technique refers to the smallest value that can be measured along the spatial dimension. In other words, the spatial resolution is the minimum distance that can be measured between two points of space. In this section, we will discuss the spatial resolutions of optical microscopes, i.e. lateral and axial resolution.

Lateral resolution

The optical microscope is limited by the diffraction of light, which leads to the spatial resolution of a microscope [START_REF] Den Dekker | [END_REF], [START_REF] Pichon | Enregistrement optique haute densité : étude physique et physicochimique du phénomène de Super-Résolution To cite this version[END_REF]. In 1873, Ernst Abbe defined the resolution limit of an optical microscope as the minimum lateral distance between two objects allowing them to be observed separately. This distance is given by = λ / (2 ), where λ is the wavelength of the light and NA is the numerical aperture of the objective. The NA as applied to a microscope objective is a measure of the ability to gather light and resolve fine specimen detail at a fixed object distance. The higher the numerical aperture of the total system, the better the resolution.

Two-point resolution

The two-point resolution criterion corresponds to the ability of an imaging system (an eye intensity of an Airy spot corresponds to the PSF (Point Spread Function) of the system.

The two-point resolution criteria are based on the observation of PSF recovery from two point sources of equal intensity. Several the types of this criteria for example are the Rayleigh, Abbe, and Sparrow criteria. The distance between the two spots from the point sources is given by the following relation:

K RK NA   (1.28)
with the numerical aperture of the objective microscope A = ⨯ sin θ where θ is the half angle of the cone of light that can enter the objective and is the refractive index of the medium between the objective and the object, λ is the wavelength of the light source and K is a constant, between 0.47 and 1, according to the criterion and the illumination type of the light source used [START_REF] De Lega | Lateral resolution and instrument transfer function as criteria for selecting surface metrology instruments[END_REF]. According to the Rayleigh criterion, the distance of the maxima of the PSF is an essential value in the determination of the resolution and it is therefore important that the maxima are well separated from each other. This implies that the resolution is all the better as the diameter of the Airy spot is small. So another way to determine the resolving power of an optical instrument is to measure the PSF's FWHM (Full Width at Half Maximum) [START_REF] Heintzmann | Breaking the resolution limit in light microscopy[END_REF], as shown in Fig. 28.

Table 2 The lateral resolution criteria based on the illumination type

Criteria

Fig. 28 The illustration of 3D PSF

Axial resolution

The axial resolution of a microscope corresponds to the depth resolution and depends on the temporal coherence length of the illumination source and the numerical aperture of the lenses used. As for lateral resolution, the axial resolution can be determined from the system of PSF (point spread function) by measuring its width at half-height along the optical axis [START_REF] Dubois | Three-dimensional cellular-level imaging using full-field optical coherence tomography[END_REF] and can be defined by the following relationship: where n is the refractive index of the medium, λ is the central wavelength of the source and Δλ is the total spectral width at mid-height of the source (FWHM) [START_REF] Dubois | High-resolution fullfield optical coherence tomography with a Linnik microscope[END_REF]. The temporal coherence length of the source is given by = / Δλ where is the speed of propagation of the wave. Therefore, the use of an illumination source with a wide spectrum, having a low coherence length, can improve the axial resolution of the system. When objectives have large numerical apertures, the axial resolution of the system depends mainly on the NA of the objectives [START_REF] Dubois | High-resolution fullfield optical coherence tomography with a Linnik microscope[END_REF] and can be approximated by the following relation:

  0.44 1 cos axial R n     (1.30)
with α the half-angle of the opening cone and λ the central wavelength of illumination.

RÉSUMÉ DU CHAPITRE 1

Dans ce chapitre, nous décrivons comment la technique de l'interférométrie à balayage de L'intensité lumineuse donnant lieu au signal de frange, s (x, y, z), capté à partir d'un détecteur (CCD) lorsque la différence de trajet optique varie par focalisation dans un interféromètre de lumière blanche, a la forme suivante:

                00 0 4 , , , , , , .cos ,

Chapter 2. COMPARISON OF PRE-FILTERING AND ENVELOPE DETECTION TECHNIQUES

In this chapter, we describe the Teager-Kaiser Energy Operator which we use as the proposed multidimensional algorithm (1D, 2D, 3D) for envelope detection in CSI. Then the study of the performance comparison of the different pre-filtering and envelope detection techniques in fringe signal processing is described. For comparison, results

using the algorithms are presented on the synthetic fringe signal and on real data, i.e. the resin on silicon sample. In addition, a study of the selection of the mother wavelet for the Continuous Wavelet Transform is presented in its use for obtaining the fringe envelope.

TEAGER KAISER ENERGY OPERATOR

Signal energy

The estimation of signal energy is commonly used to characterize a signal. In traditional signal processing, the energy of a signal is defined as the square of the signal magnitude, or the envelope of the squared signal magnitude or the integral of the squared signal magnitude.

Let us consider a signal x(t). With a time period 2T, the signal x(t) has the energy E which is defined as follows:

  2 T tT E x t dt    (2.1)
while the total energy is defined by

  2 , t E x t dt     (2.2)
Note that the total energy of the signal in Eq.(2.2) is not the same as the instantaneous energy which is what we need to seek. Then the instantaneous amplitude can be found for fringe analysis. In the case of simple harmonic motion, i.e. fundamental sinusoidal oscillation, this expression shows that the energy required to produce a simple sinusoidal signal is a function of the frequency and amplitude. This observation was used by Kaiser to define the Teager Kaiser-Energy Operator [START_REF] Kaiser | On a simple algorithm to calculate the energy of a signal[END_REF].

Teager Kaiser Energy Operator

The above observation of the signal energy in light is discussed in the article by Kaiser [START_REF] Kaiser | On a simple algorithm to calculate the energy of a signal[END_REF]. Here, Kaiser derives the equation of energy for a simple oscillator from the basic physics of motion.

The energy is presented as being directly corresponding to the square of the product of the pulsation frequency and amplitude. The second order differential equation in the case of using the basic physics of motion for a simple spring and mass oscillator is used as the beginning of the development of the operator, as illustrated in the following equation:

2 2 0 d x k x dt m  (2.
3)

The Eq.( 2.3) represents the law of motion of a system consisting of a mass m that is suspended from a spring with force constant k > 0. The system can be regarded as a model of a simple mechanical acoustical system, in which the object oscillates, generating the pressure wave in the neighbouring medium.

The solution of Eq.(2.3) is the simple harmonic motion, which can be defined as

x(t) = A cos (t + )
where A corresponds to the oscillation amplitude,  represents the oscillation frequency, that is given by k m   , and  is the arbitrary initial phase.

The simple harmonic motion has the energy E that is defined as the total energy coming from the kinetic energy of the mass and the potential energy in the spring, as follows: Based on this equation, we can observe that the energy of the simple harmonic system is proportional to the squared of the product of the frequency ω and the amplitude A, and varies as function of time t, while the x 2 (t) only takes into account the potential energy of the signal source.

Discrete Teager Kaiser Energy Operator

Let us consider a discrete-time signal x n representing the motion of the oscillator body,

given by:

  cos n x A n     (2.

6)

Where A corresponds to the oscillation amplitude, Ω corresponds to the frequency and is given by Ω = 2πf/F s , where f represents the oscillation frequency, F s represents the sampling frequency, while the parameter φ corresponds to the arbitrary initial phase.

In Eq.(2.6), there are three parameters A, Ω, and φ, with unknown values. This means that mathematically, it is necessary to construct three different equations based on the formula in Eq.(2.6) in order to obtain the values of these parameters from three samples of the signal x n .

          1 1 cos cos 1 cos 1 n n n x A n x A n x A n                 (2.7)
Through the trigonometric formulas, then we obtain:

    2 2 2 2 11 cos sin nn x x A A        (2.8)   2 2 2 11 sin n n n A x x x     (2.9)
The solution to Eq.(2.9) with respect to Ω is specific and singular provided that Ω is restricted to be less than π/2 and has a positive value. The approximated approach of this equation can be carried out by noting that if Ω is small, sin (Ω) ≈ Ω. If we limit the value of Ω to Ω < π/4 = 0.7854, i.e. f /f s < 1/8 then the relative error is always below 11 %.

Thus the discrete version of Teager Kaiser-Energy Operator can be defined as the following equation:

        2 11 d x n x n x n x n        (2.

10)

Based on the Eq.(2.10), the instantaneous energy  d of a signal x(n) can be estimated by substituting x in the formula of the TK energy operator. This operator is nonlinear and has the following properties [START_REF] Boudraa | Time-Frequency Signal Analysis and Processing: TEAGER-KAISER ENERGY OPERATORS IN TIME-FREQUENCY ANALYSIS[END_REF], [START_REF] Maragos | On amplitude and frequency demodulation using energy operators[END_REF]:

•  d [•] is independent of the initial phase φ 0 . •  d [x(n)]=  d [x(-n)]. •  d [x(n)
] is robust to the division by zero error even when the signal passes through zero, as no division operation is required.

•  d [x(n)
] is capable of responding rapidly to changes in both frequency and amplitude (instantaneous operator).

Continuous Teager Kaiser Energy Operator

The discrete version of the Teager-Kaiser Energy Operator has been defined in the previous section. In the continuous time domain, the Teager-Kaiser Energy Operator is then defined as:

    2
                  2 2 2 2 2 2 22 sin cos cos sin cos c x t A t A t A t A t t A                   (2.12)
As illustrated in Eq.(2.12), the energy of the signal x(t) = A cos (ωt) is defined by the squared product of the frequency ω and the amplitude A. In fringe analysis, the main disadvantage of the Teager-Kaiser Energy Operator is its sensitivity to the signal noise that appears during the acquisition process. In the application of a noisy fringe signal, a denoising filter, such as the Savitzky-Golay filter can be applied in combination with the operator to make the envelope detection more robust to noise.

Performance Comparison Of Different Mother Wavelet

In Continuous Wavelet Transform Algorithm On Fringe Signal Processing

Application wavelet analysis in fringe signal processing

The wavelet transform is a relatively robust mathematical tool for signal analysis and processing. In optical measurement, wavelet analysis has been widely used for fringe signal processing. In 1997, Sandoz applied the wavelet transform to the analysis of white light interferometry [START_REF] Sandoz | Wavelet transform as a processing tool in white-light interferometry[END_REF]. In this application, the accurate phase measurements are obtained from simple correlation computations which are used to accurately identify the local fringe peak, which indicates the surface height of a sample. Meanwhile, similar work was reported in 1997 [START_REF] Recknagel | Measurement and analysis of microtopography using wavelet methods[END_REF] which also applied wavelets for the analysis of white light interferometry. In this application, the fringe envelope of a white light interferogram corresponds to the absolute modulus of the maximum correlation coefficient. The fringe analysis based on a local fringe peak can provide a higher resolution than current methods that are is based on a global envelope peak. Moreover, the wavelet analysis seems to be robust to noise [START_REF] Peng | Application of the wavelet transform in machine condition monitoring and fault diagnostics: A review with bibliography[END_REF], [START_REF] Huang | Comparison of Fourier transform, windowed Fourier transform, and wavelet transform methods for phase extraction from a single fringe pattern in fringe projection profilometry[END_REF], providing accurate measurements of the surface.

Mother wavelet selection

The success of the CWT for retrieving the fringe envelope depends on the selection of a mother wavelet [START_REF] Sokolovsky | Analysis of interferograms of multi-layered biological samples obtained from full field optical coherence tomography systems[END_REF]. In order to obtain satisfactory results, the mother wavelet should be of the same form as the signal. The fringe signal which will be analysed is characterized by a high frequency sinusoid that is modulated by a Gaussian function. As regards the characterization, wavelet families that visually have a similar shape to that of the fringe signal are the Morlet, complex Morlet, Gaussian, complex Gaussian, and Mexican hat [START_REF] Jiang | Multi-frequency fringe projection profilometry based on wavelet transform[END_REF], [START_REF] Kocahan | Determination of phase from the ridge of CWT using generalized Morse wavelet[END_REF], [START_REF] Chopra | Choice of mother wavelets in CWT spectral decomposition[END_REF] as shown in Fig. 29. In this study, all these wavelets were tested and performed using a synthetic fringe signal.

The synthetic signal we used in this simulation to test all the wavelets is based locally on the general model expressed by equation (1.9) along the optical axis Z, with a 40 nm sampling step, for each lateral position X. The resulting XZ image is shown in Fig.

Simulation results

For further observation of the use of the complex Gaussian and complex Morlet as mother wavelets in the CWT algorithm, we carried out the simulation using the procedure described in section 2.4.1, as shown in Fig. 35. In the simulations, we use the EMD and Table 3 summarizes the results of the performance comparison of the complex Morlet and complex Gaussian in the CWT algorithm on fringe signal processing. The table shows the mean absolute error (mae) of surface extraction obtained using the CWT algorithm (complex Gaussian and complex Morlet wavelet), as illustrated in Fig. 32(b) and Fig. 33(b). The value of mean absolute error shows the precision of the measurement results using the fringe signal processing used. Comparing the results in the table, both the CWT using the complex Gaussian and complex Morlet wavelet are similar in terms of performance in the casees of the noiseless signal (σ=0%) and a noisy signal (σ=10% and σ=20%), providing a maximum difference of mae = 0.24 nm. Based on these results, it can be observed that the use of the complex Gaussian wavelet in the CWT algorithm is as competitive as the complex Morlet wavelet as the mother wavelet in fringe signal be used for the performance comparison. An additive non-linear offset and Gaussian noise are added onto the synthetic data (20%) which means that the root mean square (rms) of the added Gaussian noise is 20% of the maximum amplitude of the second layer fringe signal. Regarding the removal of the offset, two procedures are considered: a) the classical centered derivative, which has been chosen for its stability [START_REF] Yang | Two-step phase-shifting fringe projection profilometry[END_REF], and b) as an alternative method, the EMD which helps to remove the low frequency component given by the higher order IMF's and the residue [START_REF] Flandrin | Detrending and denoising with empirical mode decompositions[END_REF]. On the other hand, to process the noisy data, the Savitzky-Golay filter is compared with the classic Wiener filter, using a local window around each sample. Based on empirical data, the window length of 5 seems to optimize the performance for this approach (with a 40 nm sampling step). Concerning the Savitzky-Golay filter, we have applied it along the lateral axis using a third order polynomial. Based, on our data, this parameter seems to ensure the better robustness to noise. The simulation results of performance comparison of pre-filter 1, pre-filter 2, and pre-filter 3, are shown in Fig. 34(b,c,d). It can be observed in Fig. 34, that the three algorithms provide competitive results for the offset removal and noise reduction. However, they have different capabilities for maintaining the fringe signal intensities.

Comparisons are made between the fringe signal intensities of the pre-filtered (red line, Fig. 34) and the ideal signals i.e. without the offset and noise (blue line, Fig. 34). As illustrated in Fig. 34(b), the resulting amplitude of the fringe signal using pre-filter 1, is clearly lower than the ideal synthetic fringe signal (without the offset or the noise). This means that a degradation of the intensities occurs during the pre-filtering process.

Similar results are also obtained using pre-filter 2 (Fig. 34(c)), although it maintains better the fringe signal intensities of the second layer. This means that the use of the Savitzky-Golay filter is better at maintaining the amplitude of the fringe signal than the wiener filter in the pre-filtering step. A decrease in the intensity may seriously degrade the performance of the algorithm in order to extract the fringe envelope, particularly in the area where the SNR is low. Meanwhile, as illustrated in Fig. 34(d), the pre-filter 3 is able to remove the offset and reduce the noise, while almost maintaining the amplitude of the fringe signal. Based on the simulation, we observe that pre-filter 3 using a combination of the EMD and Savitzky-Golay provides a better result than the others in removing the offset, suppressing the noise and particularly in maintaining the fringe amplitude.

COMPARISON OF ENVELOPE DETECTION TECHNIQUES

Comparison Procedure

In this section, results are presented of simulations performed using a Matlab program in order to compare the performance of the envelope detection techniques. The simulation procedure is illustrated in Fig. 35. In the simulation, we used a synthetic fringe signal corresponding to the known profile of the reference surface (h ref ). Then we perform the fringe analysis procedure on the synthetic fringe signal. This consist of a first step of prefiltering for removing the offset component and suppressing the noise, a second step of fringe contrast envelope detection and a third step of post-filtering using smoothing and Gaussian fitting for determining more precise the envelope peak position, which indicates to the surface height (h). Finally, we calculate the error of the surface height estimation (h) in respect to the reference surface (h ref ). A smaller value of the error rate (mae) represents a higher accuracy. Fig. 35 The comparison procedure to evaluate the performance of the different algorithms [START_REF] Gianto | Comparison of envelope detection techniques in coherence scanning interferometry[END_REF].

Synthetic Samples

The synthetic signal we used in this simulation to test all the cited algorithms is based locally on the general model expressed by Eq.(1.9) along the optical axis Z, with a 40 nm sampling step, for each lateral position X. The resulting XZ image is shown in Fig. 36.

The two synthetic images used represent flat and wavy transparent layers on a substrate, giving two fringe signals along Z corresponding to the air/layer surface and layer/substrate interface. In this case, an additive non-linear offset and Gaussian noise are added onto the synthetic data (10% and 20%). The Gaussian noise value of 10% means that the root mean square (rms) of the added Gaussian noise is 10% of the maximum amplitude of the second layer fringe signal. Considering the synthetic fringe signal, the rms of 10% is equal to 19 dB, while the rms of 20% is equal to 13 dB.

Simulation Results

Fig. 37 and Fig. 38 show an example of the results of fringe analysis for the flat and wavy transparent layers as shown in Fig. 36 using the procedure as described in section 2.4.1.

In detail, Fig. 37(a) and Fig. 38(a) show the pre-filtered signal in two dimensions (XZ image) after carrying out the pre-filtering process. In these figures, it can be seen that the pre-filter which we used has successfully removed the offset component and suppresses the noise (compare with the XZ image in Fig. 36). After the pre-filtering process, then we carry out the envelope detection on the pre-filtered signal, obtaining the fringe contrast envelope in two dimensional (XZ image), as shown in Fig. 37 interface) as the final result obtained by the fringe analysis using the procedure as described in section 2.4.1. In these figures, it can be seen that the measured profiles (red and green line) coincide with the reference surface (blue line) of the synthetic transparent layers which we used in the simulation. For quantitative evaluation, then the error rate (mean absolute error) of the measured profile is calculated in respect to the surface reference, as shown in Table 4, Table 5 and Table 6. In this simulation, we carry out the procedure as described in the section 2.4.1 in order to compare the performance of fringe analysis using a combination of different prefiltering as described in the section 2.3 and different envelope detection as described in the section 1.6.2. The results of performance comparison using this procedure are summarized in the following Table 4 -Table 6. 5 and Table 6 correspond respectively to the pre-filter 1, 2 and 3, the related algorithms being respectively denoted by A, B, C, D (Table 4), E, F, G, H (Table 5) and I, J, K, L (Table 6). For each table, the envelope technique associated with each procedure is indicated. For instance: A, E, and I are related to the Hilbert Transform; B, F, and J are related to the FSA, C, G, and K are related to the TKEO, while D, H, and L use CWT). The same post-processing mentioned in section 1.5.3, has been applied. For each procedure, the simulations have been calculated with different levels of noise (). The performance of each procedure has been quantitatively evaluated by the measurement of the calculation time and the error rate (mean absolute error) between the reference surface and the estimated one (for both surface layers). Based on the mae in the case of a noiseless signal (σ = 0%), all algorithms perform similarly providing an average value of mae = 0 nm for the flat transparent layer and average value of mae = 1 nm for the wavy transparent layer. In the presence of noisy data (σ =10% or σ =20%), the mae values significantly differ from each other. More precisely the mae values of E, F, G, and H in Table 5 are lower than the mae values of algorithms A, B, C, and D in Table 4, which indicates that the use of the pre-filter 2 provides a better performance than the pre-filter 1 in terms of precision. Thus, the Savitzky-Golay filter is more effective than the Wiener filter regarding the noise. However, the combination of EMD and the Savitzky-Golay filter (pre-filter 3) provides the best performance for removing the offset and reducing the noise which improves the precision of surface measurement, as shown by the smaller value of mae in Table 6.

In regard to the main step in the procedure of fringe analysis, i.e. envelope detection process, it can be observed that the TKEO algorithm (called "K") is a competitive algorithm in terms of robustness and computational time compared with the other techniques. TKEO provides better surface extraction than the Hilbert Transform (algorithm called "I") and FSA (algorithm called "J"). In the case of the flat transparent layer, TKEO provides the most precise results for the 1 st surface extraction and close to the performance of CWT (called "L") for the buried interface (2 nd surface) extraction, having the smallest mean absolute error value as shown in Table 3. For the wavy transparent layer, the most precise measurement is provided by CWT, while the performance of TKEO is closer to CWT than the others. However, TKEO has the advantage of being more compact computationally so TKEO is far better in terms of computational time than CWT. For instance, TKEO taking 6.5 s while CWT taking 30.6 s in the context of a noisy fringe signal with σ = 20%. On the other hand, we have also evaluated the performance of post-processing that we used in our procedure of fringe analysis. Table 7 shows the performance comparison between the fringe analysis with and without interpolation on Gaussian fitting. Based on these results, we have demonstrated that the use of interpolation with Gaussian fitting successfully improves the axial sensitivity of the profile measurement. The results provide a significant improvement in terms of precision of surface extraction, as represented by the smaller value of mae in Table 7, compared without the use of interpolation with Gaussian fitting as post-processing. The value of mae obtained for fringe analysis using interpolation with Gaussian fitting being 2 to 3 times smaller, corresponds to a higher precision.

RESULTS USING MEASUREMENTS ON RESIN LAYER ON SILICON

The performance of the algorithms using real data, i.e. a fringe image XZ measured on a resin layer on Si with (570 x 111) pixels, is shown in Fig. 39(a). This real data was taken on the Leitz-Linnik interference microscope with the following optical parameters: For the results of the measurements on this sample, the performance is evaluated by measuring the total (or peak-valley) roughness R t and arithmetic roughness R a of the profiles from each algorithm. The maximum height of the profile R t , is computed as follows [START_REF]Standard[END_REF]:

max min t R z z  (2.13)
Where z max (resp. z min ) represents the highest (resp. the lowest) peak of the surface profile. R a is the average deviation of the roughness profile height z j from a mean line z over the evaluation length N [103]. The smaller is the value of this roughness parameter, the better is the performance of the algorithm, in terms of sensitivity. The results comparing the different algorithms are given in Table 8.

Based on the results in Table 8, for the single surfaces (ROI-1 and ROI-3), all the algorithms show excellent performance, as illustrated by the very small value of R t = 18.0-27.0 nm and R a = 2.8-6.2 nm (which is below the sampling step of 90 nm). Then for the transparent surface regions (ROI-2 and ROI-4), the algorithms seem to provide comparable results according to the value of the roughness parameters R t and R a , where the maximum difference is 9 nm. The different envelope techniques provide similar performance for the single substrate surface as well as for the transparent layer, since the fringe image is slightly noisy. In accordance with the results of the performance comparison based on synthetic images, the performance of the different envelope detection techniques depends on the SNR in the fringe image, while they are similar for the noiseless fringe images. The best performance is provided by HT (Fig. 42(a)) and TKEO (Fig. 42(d) with nearly similar values of 2.7 and 2.6 respectively. The value for FSA (Fig. 42(b)) is slightly worse, at 1.9, but nonetheless capable of discriminating the two surfaces. The parameter ∆I derived from the CWT approach, (see Fig. 42(c)) cannot be calculated since it fails to distinguish between the envelopes.

In this work, the CWT algorithm actually used is not able to distinguish between two successive fringe envelopes, due to the non-optimal use of the CWT. Further work involving the testing of the trade-off between space and frequency to find the optimal value would most likely have made it possible to allow this distinction, but the wavelets technique was considered not to be central to the actual work in hand. For each point (X,Y), the identification of the position of the first envelope peak (Fig. 43) gives the posiiton of the surface (Z 1 ) and the peak of the second envelope gives the positon along Z of the interface (Z 2 ). If this procedure is carried out for each point, the (XZ) profile of the corresponding points on the surface and the buried interface can be made. A correction to the position of the interface profile needs to be performed due to the value of the refractive index of the resist layer (n = 1.644) being greater than unity, the corrected value of Z being:

The altitude (Z axis) measurement of resin on SI

   1 2 1 Z Z Z Z n (2.16)

Approximate empirical expression of the refractive index

The Cauchy formulas are valid for media whose absorption bands are all in the ultraviolet. The resin verifies this condition which favored this choice of formulas:

     2 4 B C n A (2.

17)

Resin is a transparent material in the visible region: it is therefore far from the absorption bands, which allows on the one hand, to consider that its refractive index is real and on the other hand, to make a limited development based on the wavelength.

Fig. 44 The dispersion curve corresponding to the resin layer

In these areas of transparency, the refractive index follows a function that decreases with wavelength (Fig. 44), or, which amounts to the same, a function that increases with frequency. Such a law of dispersion is called the law of normal dispersion. Media used in optics, which are obviously transparent in the visible, have larger indices of refraction in the blue than in the red: n blue > n red .

The dispersion curve (Fig. 44) and the Cauchy equation (Eq.2.17) describe how the refractive index of the photosensitive resin changes as a function of the wavelength of the light incident on the film. The Cauchy equation is given by:

  24 BC nA      (2.18)
where λ corresponds to the wavelength, n corresponds to the refractive index, while A, B, C represent the coefficients. By performing the fitting using the data provided in Fig. 44, the coefficient values for resin can be obtained:

B = 1.5935 C = 1.8854 x10 6 D = 4.1211 x10 10
Thus, the refractive index of resin for λ = 610 nm is calculated as n = 1.644. By using this refactive index value and the Eq.(2.16), the altitude (Z axis) of the sample of resin on Si can be obtained, as illustrated by Fig. 45.

Fig. 45 The altitude Z (axis) of resin on Si

CONCLUSION

A study has been made on the comparison of different pre-processing and different envelope detection techniques in CSI using synthetic images and real measurements of different types of thin transparent films, resulting in the following conclusions:

 The combination of EMD and Savitzky-Golay filters provides the best performance for suppressing noise and the offset component while maintaining the original signal intensity.

 The CWT and TKEO are both competitive for providing surface extraction:

-TKEO is slightly better in terms of computational time.

-CWT is slightly better for the wavy transparent layer, but TKEO is quicker in terms of computational time.

-In this work, TKEO is more competitive than CWT for resolving two close layers although this may be due to a non optimal use of the CWT used. The ability of CWT to allow different space frequency trade-offs that provides the opportunity to prioritize either space resolution or frequency resolution could lead to distinguishing between closely spaced layers.

 The use of an interpolation method with Gaussian fitting as the post-processing has been successfully used to improve the the axial sensitivity of the measurement, with the error rate (mae) being from 2 to 3 times smaller than if it was not used, corresponding to a higher precision.

RÉSUMÉ DU CHAPITRE 2

Dans

Chapter 3. 2D FRINGE PROCESSING IN CSI

In this chapter, we study the performance of 2D fringe processing in CSI developed based on the two dimensional Teager Kaiser Energy Operator [START_REF] Salzenstein | A new class of multidimensional Teager-Kaiser and higher order operators based on directional derivatives[END_REF] on a transparent film sample containing defects. A robust fringe analysis is needed for improved characterization of the transparent polymer film, i.e. a Mylar, which is complex and contains buried layers with low contrast. The algorithms are compared to the classical two dimensional Hilbert Transform [START_REF] Wielgus | Amplitude demodulation of interferometric signals with a 2D Hilbert transform[END_REF], [START_REF] Wielgus | Evaluation of amplitude encoded fringe patterns using the bidimensional empirical mode decomposition and the 2D Hilbert transform generalizations[END_REF], [START_REF] Bovik | Handbook of image and video processing[END_REF], [START_REF] Hutchison | Image Analysis and Recognition[END_REF] and the 1D Fringe Processing using the well-known FSA algorithm. In addition, we also report the study of the developed algorithm for the detection of the layer number in multiple layers using Gaussian curve fitting.

But first we discuss the notion of the independence of neighboring pixels, which is an assumption made in the CSI technique that leads to the ability to measure larger steps than in PSM. But in reality, as we shall see, there is a certain dependence between pixels due to the experimental conditions, which motivates the use of 2D and 3D processing.

The dependence of neighbouring pixels in PSM processing

As mentioned in chapter 1, the technique of PSM is limited to the measurement of small surface roughness or sample surfaces that are assumed to present height differences of less than /2 between neighbouring image points, as illustrated in Fig. 46(a). In the case of this surface type, representing a small step, PSM can be used to measure the height of each point independently of its neighbours.

Fig. 46 The limitation of the PSM technique in height measurement

But in the case of a sample surface having a deeper smooth and continuous surface, for instance in the case of a spherical surface such as a microlens (Fig. 47), it is well known that the 2π phase discontinuities can be corrected as shown in Fig. 48. The use of phase unwrapping in the PSM technique therefore takes into account the neighbourhood values of pixels. This means that measurements using the PSM technique in this case are not independent of their neighbours.

The dependence of neighbouring pixels in CSI processing

In the history of the development of interference microscopy, white light interferometry (CSI) was therefore proposed as a solution to measuring large step heights.

Making the assumption that CSI allows the measurement of the absolute height of a surface at each point independently of its neighbours by localizing each point along Z in relation to the scanning performed by the piezoelectric actuator, in principle any step height can be measured. This is the reason why we can categorize this signal processing as 1D fringe signal processing.

However, in practice, in an experimental system, the measurement at a given pixel is not absolutely independent of its neighbours for several reasons. This means that there is a link between neighbouring pixels and that the measurement of the height of a surface at each point is not absolutely independently of its neighbours.

Fig. 49 The illustration of airy spot on the pixels

The neighbourhood link is also present in the measurement of certain step heights, as shown in Fig. 50 where there is over and undershoot near to the step instead of a square step when the height is close to the coherence length of the light illumination [START_REF] Harasaki | Fringe modulation skewing effect in white-light vertical scanning interferometry[END_REF], [START_REF] Harasaki | Improved vertical-scanning interferometry[END_REF]. This is known as the "batwing" effect on the edge of the step caused by the mixing of signals from the top and bottom of the step, demonstrating again the dependence of neighboring pixels.

Fig. 50 The batwing effect on the measurement of a step A third reason is that for most surfaces where there are no holes present, it can be assumed to be continuous laterally, whatever the roughness. The measurement of the surface roughness is always an approximation of the real surface topography, limited by the various parameters of the measurement system, so that the assumption needs to be made that a neighbouring point is linked to its neighbour.

In the present work, we therefore propose performing 2D and 3D fringe processing that takes into account the spatial neighbourhood, so that the independence of measurements between neighboring pixels is therefore lost, at least on a small scale of several pixels. Taking into account diffraction, step effects and the assumption that the surface is locally continuous justifies the use of 2D/3D algorithms and their assumptions (that neighbouring pixels are linked) and the estimation of the local slope.

2D FRINGE ENVELOPE DETECTION

Analytic signal-Hilbert Transform (HT2D)

The one dimensional form of the Hilbert transform of a signal is given by:

      11 H su s x s x du x x u         (3.1)
This Hilbert transform is often used in analytic signals and is useful for the purpose of amplitude demodulation [START_REF] Larkin | Natural demodulation of twodimensional fringe patterns I General background of the spiral phase quadrature transform[END_REF], [START_REF] Potamianos | A comparison of the energy operator and the Hilbert transform approach to signal and speech demodulation[END_REF]. In particular, for obtaining the analytic signal, the negative frequency component of the signal is suppressed [START_REF] Gabor | Theory of Communication[END_REF]. The analytic signal s A (x) is defined as:

          exp AH s x s x is x A x i x       (3.2)
In the frequency domain, the analytic signal corresponds to:

      1 A S sign S         (3.3)
Based on Equation ( 17), it can be seen that we can obtain the instantaneous amplitude (envelope) of a real signal s(t) from the absolute value of s A (t). Meanwhile for the 2D signal, the technique is extended by a basic approach of the 2D Hilbert Transform H{s(x,y)}:

         22 , 11 ,, H s u v s x y s x y dudv xy x u x v         (3.4)
which leads to the 2D analytic signal s A (x,y), which is given by:

      , , , AH s x y s x y is x y  (3.5)
In the frequency domain, the 2D analytic signal s A (x,y) corresponds to:

        , 1 , A S i sign sign S          (3.6)
Notice with this equation, that we can understand that Eq.

    , Re , A A s x y i s x y       (3.7)

  ( , ) ,

A A x y s x y  (3.8)

Teager Kaiser Energy Operator (TKEO2D)

For local energy tracking measurement, the Teager-Kaiser Energy Operator (TKEO) is well known in digital signal processing, particularly on oscillatory signals due to the fact that the operator is simple to implement and is computationally very efficient. In the one dimensional and continuous version of the Teager Kaiser Energy Operator, the energy of a signal x(t) is defined by the following equation [START_REF] Salzenstein | A new class of multidimensional Teager-Kaiser and higher order operators based on directional derivatives[END_REF]:

        2 (1) (0) (2) s t s t s t s t       (3.9)
When applied to a signal s(t) = a(t) cos (ωt), the TKEO formula yields: a(t) 2 ω(t) 2 . It is assumed that the instantaneous amplitude a(t) and the instantaneous frequency ω(t) do not vary too much or too fast with respect to the average value of ω(t). Given the simplicity of the TKEO (and the extended versions of TKEO) and the broad applicability of the AM-FM model in signal processing and communication systems, this operator leads to the Energy Separation Algorithm (ESA) defined by [START_REF] Quatieri | Energy Separation in Signal Modulations with Application to Speech Analysis[END_REF]:

            ; x t x t t a t xt xt                  (3.10)
The TK operator, which is usually called as TKEO limited to the second order, has been extended into higher-order differential operators (DEOs) [START_REF] Maragos | Higher order differential energy operators[END_REF]:

          (1) ( 1) (0)
( ) kk k x t x t x t x t x t      (3.11)
Practically, the operators (DEOs) have been implemented to the 2D case of fringe signals in the article [START_REF] Salzenstein | Two-dimensional continuous higher-order energy operators[END_REF], [START_REF] Boudraa | 2D Continuous Higher Order Energy Operators[END_REF] in order to improve the fringe envelope extraction to detect the surface height of the sample.

The generalization of the higher order differential operators (DEOs) is as follows [START_REF] Salzenstein | energy operators[END_REF]:

          ( ) ( ) ( ) ( ) , , , p q m l p q m l x t x t x t x t x t     (3.12)
with integers p+q = m+l, (p,q) ≠ (m,l). For (p=1, q=1, m=0, l=2), the generalized operator is reduced to the TK operator. Finally, all these operators can be extended to the multidimensional case as k th -order tensors H (H stands for higher) [START_REF] Salzenstein | A new class of multidimensional Teager-Kaiser and higher order operators based on directional derivatives[END_REF]:

  ,, d d d d u du du du du k p m TT m l p q H m l p q s s s s s                    (3.13)
with k=p+q=m+l, (m,l)=(p,q). Vector u = (x 1

x 2 … x n ) is n-dimensional, s(u) is a
multidimensional signal, and ⊗ means the Kronecker product. The second-order operator (k=2) provides the directional TKEO, which extends one dimensional classical TKEO using directional derivatives along any vector v as follows:

          2 2 2,v 2 2 u = v u v u u u vv T T ss s s s                  (3.14)
The Eq. (3.14) can be applied for the AM-FM signal demodulation. For any ndimensional local AM-FM signal defined by:

        u u cos u sA   (3.15)
We have the following approximation [START_REF] Salzenstein | A new class of multidimensional Teager-Kaiser and higher order operators based on directional derivatives[END_REF]:

            2 2 22 2,v u u u u w u v v T s A A            (3.16)
w is the local frequency w = (w 1 ,w 2 ,...,w n ) T , which is the gradient of the phase , where   w u w   , when the local amplitude A(u) does not vary too fast compared to the carrier [START_REF] Chonavel | A new class of multi-dimensional Teager-Kaiser and higher order operators based on directional derivatives[END_REF]. The vector v is a normalized vector v = (v 1 , v 2 ,...,v n ) T . The choose of the directional derivative has been discussed in article [START_REF] Salzenstein | Multi-dimensional higher order differential operators derived from the Teager -Kaiser energy-tracking function[END_REF].

It is then possible to compute the instantaneous envelope:

          22 2,v 2,v 2 2 2 4,v 2,v uu u uu v ss A ss                     (3.17)

THE ROBUSTNESS EVALUATION OF 2DTKEO AND 2DHT

To evaluate the robustness of 2DTKEO [START_REF] Salzenstein | A new class of multidimensional Teager-Kaiser and higher order operators based on directional derivatives[END_REF] and 2DHT [START_REF] Wielgus | Amplitude demodulation of interferometric signals with a 2D Hilbert transform[END_REF], [START_REF] Wielgus | Evaluation of amplitude encoded fringe patterns using the bidimensional empirical mode decomposition and the 2D Hilbert transform generalizations[END_REF], [START_REF] Bovik | Handbook of image and video processing[END_REF], [START_REF] Hutchison | Image Analysis and Recognition[END_REF], it has been applied respectively to a noiseless signal (σ=0%) and noisy signals (σ=10% and σ=20%) using a synthetic signal as shown in Fig. 51. In order to improve the axial sensitivity and precision regarding the peak detection, cubic spline smoothing followed by a Gaussian interpolation has been used, which is adapted to the physical model of the envelope. Fig. 52 and Fig. 53 show the results obtained. Comparing the results in Table 9, both the Z-scan technique (FSA) and XZ-scan technique (2DTKEO and 2DHT) are similar in terms of performance in the case of the noiseless signal (σ=0%) providing an average value of mae = 2.1 nm. Meanwhile for a noisy signal (σ=10% and σ=20%), the XZ-scan technique is generally more robust than the FSA. Moreover, it can also be observed that the use of 2DTKEO is as competitive as classical 2DHT in terms of performance for a noisy signal. The error rate value (mae) of surface measurement extracted by both of 2DTKEO and the 2DHT is nearly similar for a noisy signal (σ = 10% and σ = 20%). The reason that 2DTKEO is more robust than the FSA algorithm is due to the use of a 2D gradient smooth by two-dimensional Gaussian filter.

DETECTION OF THE LAYERS NUMBER ON A TRANSPARENT LAYERS

One of the benefits of CSI is the ability to separate multiple reflections from a transparent layer structure. Fig. 54 shows the scheme of the interferogram construction on a transparent layer using CSI. As illustrated in Fig. 54, we found that there are three clearly identifiable modulation envelopes corresponding to surface reflections from the transparent layer boundaries (top surface, rear surface, and structure). In the case of a transparent layer as shown in the figure, the interferogram has two conditions: (i) an interferogram has two modulation envelopes corresponding to the top and rear surfaces;

(ii) an interferogram has more than two modulation envelopes due to the presence of the structures. In this case of the sample with multiple layers, an adaptive algorithm in fringe signal processing is required to identify the layers number on an interferogram.

Fig. 54 Scheme of the interferogram construction on a transparent layer using CSI [START_REF]Étude des techniques de super-résolution latérale en nanoscopie et développement d'un système interférométrique nano-3D[END_REF] In this work, in order to find the layer number on an interferogram, we perform threshold filtering using Gaussian curve fitting on the fringe envelope, which is applied after the step of envelope detection. The Gaussian curve fitting is performed on the number of points around the identifiable local maximum on the fringe envelope during the peak detection process. Through this Gaussian curve fitting, we can obtain the characteristics of each of the Gaussian functions, such as its height, position, and width.

Fig. 55(a) shows how this procedure has been tested on synthetic data which contains multiple Gaussian functions. As illustrated in Fig. 55(a), by using this procedure, we obtain the four identifiable Gaussian functions as the candidates. In order to identify the number of Gaussian function which indicates the different layers of the sample, we make use of the height and width obtained using Gaussian curve fitting to filter the Gaussian functions which are not suitable for the characteristics of the Gaussian functions that we seek to define. For example, we seek the Gaussian function with a width > 10 and a height (amplitude) > 2. By defining this threshold value of the height and width, then we obtain only two Gaussian functions which are suitable (red colour), as shown in Fig. As illustrated by the interferogram in Fig. 56, we have generated the synthetic fringe signal corresponding to multiple layers (top surface, rear surface, and some structures). We perform the algorithm for the detection of the layer number as postprocessing after applying the process of envelope detection. Fig. 57 shows the fringe envelope of the synthetic fringe signal of the transparent layers. As illustrated in the figure, an adaptive algorithm is required as post-processing to identify whether an interferogram has two, three, or four modulated envelopes corresponding to the different positions of the layer/surface. In this simulation, the width of the Gaussian has been identified for our constructed synthetic fringe signal. We can adjust the minimum width threshold value corresponding to 0.75 times the identified width of the Gaussian based on the coherence length of the fringe signal. We did not choose the exact identified width threshold due to the fact that the fringe contrast envelope (Gaussian) which we seek could be distorted by the noise.

Meanwhile, for the amplitude threshold, we adjust it by 3* the standard deviation of the noise. This amplitude threshold is obtained empirically and corresponds to the fringe amplitude being higher than 3* the standard deviation of the noise. The standard deviation of the noise is estimated on the region of the interferogram which does not have a fringe pattern (the background region). By using the adjusted threshold, this makes it Fig. 59 Comparison of the measured structures with the reference structures For the results of the measurements on this sample, the performance is also evaluated by measuring the local total (or peak-valley) roughness R t and arithmetic roughness R a of the profiles after removing the slopes. By using Eq.(2.13) and Eq.(2.14), it seems that the total roughness R t for the top surface, internal structures (1-4), and rear surface has the same value, i.e. 5.33 nm, while the arithmetic roughness R a also has the same value, i.e. 1.21 nm. The small value of roughness parameter indicates that the measured profiles has represent well the homogeneous reference structures.

b. Real data (transparent polymer film)

The performance of the algorithm was tested using real data, i.e. a transparent polymer film, of which the interferogram is shown in Fig. 60 and the envelope is illustrated in Fig. 61. The description of acquisition data of the measurement of a transparent polymer film obtained by the microscope system is given in Appendix-1: Data of Sample. As shown in Fig. 61, it is apparent that there are three clearly identifiable envelopes in the interferogram which indicate the top surface, rear surface, and some internal structures. In order to apply our algorithm to this data, it was necessary to adjust the threshold value. As with the synthetic data, the amplitude threshold is adjusted by 3* the standard deviation of the noise, which is estimated on the region of interferogram without fringes.

Meanwhile, the width threshold is adjusted based on the coherence length of the fringe signal. The width (variation) w of Gaussian function considering to the fringe envelope of the signal can then be calculated based on the following equation [117], [START_REF] Perrin | Development and characterization of an optical coherence tomography micro system[END_REF]:

2 0 2 ln 2 8ln 2 2.3548 c wL           (3.18)
Where  0 is the mean of wavelength from the light source, while  is the spectral bandwidth of the camera that will limit  (which is smaller than the wider spectral bandwidth of the black body emission of the light source) for the illumination actually used in the microscope. (amplitude and width of Gaussian). As illustrated in Fig. 62(a), the algorithm succeeded in finding the layer number of the transparent polymer film, hence making it possible to identify the position of the top surface, the rear surface, and the presence of internal structures. However, it is apparent that there is still an artefact that appears during the processing of surface detection, as shown with the blue line in Fig. 62(a). In order to suppress the artefact, the neighbourhood number is added to the identifiable surface as a threshold parameter in the algorithm. This filters out the identifiable surface that has less than three surface neighbours to remain within the size of the Airy disc so as not to lose lateral information. The result is shown in Fig. 62(b).

RESULTS OF MEASUREMENTS ON MYLAR POLYMER FILM USING Z-SCAN AND XZ-SCAN TECHNIQUE

The fringe signals from the Mylar polymer film (Fig. 63) were obtained using the adapted Leitz-Linnik interference microscope. The description of the acquisition data of the measurement of the Mylar polymer film obtained by the microscope system is given in Appendix-1: Data of Sample. The method using the averaging technique with dark and flat corrections was performed in order to reduce the noise [START_REF] Leong-Hoï | Detection of defects in a transparent polymer with high resolution tomography using white light scanning interferometry and noise reduction[END_REF]. The previously described

Z-scan technique (1D Fringe Processing) results

Performing the Z-scan FSA approach on the fringe signal of the Mylar polymer film leads to the fringe envelope image shown in Fig. 64(a). The fringe envelope is shown as a logarithmic transformation in order to facilitate the identification of the presence of the structures and the rear surface. This XZ image is sufficient to provide the information regarding the structures and the rear surface.

XZ-scan technique (2D Fringe Processing) results

In order to obtain better results on the Mylar polymer film, the XZ-scan technique was performed with the previously mentioned post-processing techniques.

Measurement of thickness of Mylar polymer film

The surface extraction using the XZ-scan technique can be used for the thickness measurement of the Mylar polymer film, known to be between 3 µm and 3.6 µm. Table 10 and Fig. 70 show the mean values of the thickness for a given Region of Interest (ROI). The ROI is taken every lateral span of 10 µm along the lateral axis X. The thickness of the Mylar polymer film at the point X can be measured by calculating the distance ∆ separating the surface position along the optical axis Z [START_REF] Abderrazzaq | Developpement de la microscopie interferometrique pour une meilleure analyse morphologique des couches minces et epaisses des materiaux semiconducteurs et optique[END_REF], [START_REF] Benatmane | Buried interface characterization by interference microscopy[END_REF]. By using the value of refractive index (n) of the Mylar polymer film which is known (1.63 for λ = 700 nm), the thickness d is: Thus, the refractive index of Mylar polymer film for λ eff = 720 nm is calculated as follows:

  d n (3.19)
 

Error Approximation of Thickness Measurement of Mylar Film

The error approximation of the thickness measurement is given by: The main sources of error are:

-The variation of the refractive index n 1 of the medium, -The variation in the measurement of ∆.

a. Error n 1 (δ n )

The main contribution of refractive index error comes from the light source used (white light), δ λ :

    1 1 1 2 2 n nn     (3.21)
If we take into account the spectral response of the CCD sensor, that extends from λ 1 = 350 nm to λ 2 = 1100 nm, the value of this error can be estimated by:

    350 nm 1100 nm 2 n nn    (3.22)
The refractive index of Mylar polymer film for λ eff = 720 nm is calculated as follows:

  Meanwhile the refractive index for white light over the range of λ 1 = 350 nm to λ 1 = 1100 nm, is given by:   By using Equation [START_REF] Montgomery | Method and apparatus for interferometrically inspecting a surface of an object[END_REF], we obtain the error approximation of the thickness contributed by the refractive index error, to be 0.0565.

b. Error ∆ (δ ∆ )

Inaccuracies in the position of the central fringe and systematic errors cause an error that affects the values of the scanning steps. The error contribution of δ ∆ comes from the imperfection of the piezoelectric stepper calibration (δ e ) and the resolution of its plate displacement (δ p ). Using the values of δ e = 5 nm and δ p = 1 nm for the piezo stepper used, the error value of δ ∆ is given by:  

CONCLUSION

The study of the robustness of 2D fringe processing in CSI has been presented for the characterization of a transparent Mylar polymer film. We have demonstrated the ability of 2D approaches to compete with some classical methods (1D approaches) used in the field of interferometry, in terms of robustness. These results demonstrate that the XZ fringe envelope extracted by the 2D fringe processing provides more satisfactory results than the 1D fringe processing in revealing the internal structures and the rear surface, which is contained in Mylar polymer film. The technique also allows an improvement in the details in the XZ images as well as more accurate measurements of the thickness of the polymer film. In this work, we make the hypothesis that the XYZ image stack of the fringe signals generated by CSI follows the model of a three dimensional AM-FM signal. It is obviously intrinsically difficult to represent the 3D signal. We consider this three dimensional AM-FM of XYZ fringe signal as a reconstruction of the projection of the XZ slices and YZ slices. For example, we have the 3D AM-FM signal of the fringe signal as follows: The following relations are the different operators developed by Maragos and Bovik, Boudraa, and Larkin for 2D signal s(x,y). These operators are denoted respectively by Φ B , Φ C and Φ D [START_REF] Salzenstein | Two-dimensional continuous higher-order energy operators[END_REF], [START_REF] Salzenstein | A new class of multidimensional Teager-Kaiser and higher order operators based on directional derivatives[END_REF].

RÉSUMÉ DU CHAPITRE 3

          00 , , , , , .cos
       
                                                                                               
                                         
In for the case of a 3D signal, the operators can be expressed by an energy tensor as follows [START_REF] Salzenstein | A new class of multidimensional Teager-Kaiser and higher order operators based on directional derivatives[END_REF], [START_REF] Salzenstein | Multi-dimensional higher order differential operators derived from the Teager -Kaiser energy-tracking function[END_REF]:

          2 2 2 2 2 2 2 2 2 2 2 2 22 , , , , , , , , ...
                                                                                                      2 2 2 2
                                        (4.8)

Discrete 3D Teager Kaiser Energy Operator

In this section, we recall the mathematical model of discrete 3D Teager Kaiser Energy Operator, which was introduced in the article [START_REF] Boudraa | Teager-Kaiser energy methods for signal and image analysis: A review[END_REF]. As far as we know, no one has applied this discrete model to fringe signal processing, nor compared it with the performance of the algorithm of the continuous 3D

                                      2 11 22 33 12 21 13 31 23 32 2 2 2 ,, , , 1, , 1, , , , , 1, , 1, , , , , 1 , , 1 2 , , , , , , , , 2 , , , , , , , , 2 , , , d x y xy
x z xz yz s x y z s i j k s i j k s i j k s i j k s i j k s i j k s i j k s i j k s i j k s i j k s i j k s i j k s i j k s i j k s i j k s i j k s i j k

s i j k s i j                                           , , , , , yz k
s i j k s i j k  (4.9)

Where s x (i,j,k), s y (i,j,k) and s z (i,j,k) represent the first order derivative along x (resp. y and z). The parameter s xy (i,j,k) represents the second order derivatives according to x and y (resp. s xz (i,j,k) and s yz (i,j,k) ). The first part of (4.8) corresponds to the 2D discrete energy operator described in by Maragos and Bovik [START_REF] Maragos | Image demodulation using[END_REF] and is identical to discrete energy operator developed in [START_REF] Yu | A novel nonlinear filter for image enhancement[END_REF] for digital image edge detection. Consider a 3-dimensional an AM-FM signal defined by:

      , , , , cos where ;
;

x y z x x e x x e x x e s i j k A i j k i j k x x x                (4.10)
The variable x e (resp. y e and z e ) represents the sampling period along the x-axis (resp. the y-axis and z-axis). The assumption that the amplitude A(i,j,k) varies more slowly than the carrier signal cos (Ω x i + Ω y j + Ω z k), (locally constant), yields immediately:

                                      1, , 1
i j k s i j k s i j k A i j k i j k s i j k s i j k s i j k A i j k i j k s i j k s i j k s i j k A i j k i j k s i j k s i j k s i j k A i j k                                                              sin cos 1, , 1
i j k s i j k s i j k s i j k A i j k i j k s i j k s i j k s i j k A i j k i j k                               (4.11)
Finally, a discrete version of the C2TKEO, where the amplitude A(i,j,k) is locally constant, is given by:

                                              22 2 2 2 2 2 , , , , 1, , 1, , , , , 1, , 1, , , , , 1 , , 1 2 , , , , , , , , 2 , , , , , , , , 2 , , , , , , , , , , sin ,

x y z s i j k s i j k s i j k s i j k s i j k s i j k s i j k s i j k s i j k s i j k s i j k s i j k s i j k s i j k s i j k s i j k s i j k s i j k s i j k s i j k s i j k

A i j k A i j                                       2
k A i j k A i j k A i j k A i j k A i j k                   (4.12)
The same discrete gradient approximations and recurrence formula lead to:

            44 4 2 , , , , sin , , sin , , sin
, , sin sin sin

x x y y z z x y z s x y z s i j k s i j k s i j k A i j k                   (4.13)
Thus ( 2) and ( 3) provide an estimation of the envelope. Note that using the formula ( 2

                           2
s i j k A i j k s i j k A i j k s i j k A i j k s i j k s i j k A i j k si                                                               2 2 2 2 2 2 , , ,
j k s i j k A i j k s i j k s i j k A i j k                          (4.14)
The previous equations provide flexible ways to detect the carrier frequencies and the envelope, such as:

    2 2
,, sin ,,

x x s i j k s i j k        (4.15)               2
i j k                                         (4.16)                   2 2 2 22 22 ,, sin sin sin ,, , , 4 , , , , z
z x z z x z x z s i j k sign s i j k s i j k s s s s sign s i j k s i j k                                 (4.17)       2 2 4
,, ,, ,,

s i j k A i j k s i j k        (4.18)

PROCEDURE OF 3D FRINGE PROCESSING

In general, the techniques of 3D fringe processing developed in this work consist of three main steps: (i) pre-processing for offset removal and suppressing the noise, (ii) envelope detection and (iii) post-processing in order to determine the measurement results of surface height more accurately. In order to describe more clearly this procedure of 3D fringe signal processing, we use the measurements on the sample of Graphene, consisting of an image stack XYZ of 99 x 338 x 50 pixels. The description of the acquisition data of measurements on the Graphene obtained by the microscope system is given in Appendix-1: Data of Sample. In the case of noisy fringe signals, the fringe patterns can still appear in the resulting fringe envelope even though a pre-filtering process has been used before applying the fringe envelope detection. For this reason, a smoothing filter such as a cubic spline is used to suppress the fringe patterns on the envelope, hence allowing an improvement in the accuracy of the envelope peak detection. The peak position obtained is then used as an initial value along with the neighbourhood at the peak for the next process, i.e. Gaussian fitting. The curve fitting method using Gaussian fitting is implemented in order to determine more precisely the envelope peak and to improve the axial sensitivity. ) has the advantage of better measurements in the terms of precision. This can be observed by the smaller mean absolute error value. The smaller is the value of this mean absolute error, the better is the performance of the algorithm, in terms of precision.

EVALUATION OF THE MEASUSUREMENT ACCURACY USING STEP HEIGHT STANDARD

For further evaluation of the developed algorithm of continuous 3DTKEO, the measurement accuracy is calibrated using a calibration standard, i.

PERFORMANCE COMPARISON OF 3D FRINGE PROCESSING USING CONTINUOUS 3DTKEO AND DISCRETE 3DTKEO

In the previous section, we have described the developed mathematical model of discrete 3DTKEO in order to analyse directly the stack image of the fringe signal. It is obvious that we hope this discrete approach of 3DTKEO will give the advantageous in terms of computational time. In this section, we report the study of the application of the discrete 3DTKEO on fringe signal processing in a comparison with the results from the continuous 3DTKEO. For this purpose, we use a real data sample, i.e. measurements made on the Cable, which has the fringe signals, as illustrated in Fig. 106. As illustrated in Fig. 106, it seems that before applying the envelope detection, the pre-processing step is needed in order to remove the offset and to suppress the noise. Fig. 107 shows the pre-filtered fringe image and its fringe profile in 1D after the preprocessing step.

                                            22 2 2 , , , , 1, , 1, , , , , 1, , 1, , , , , 1 , , 1 2 , , , , , , , , 2 , , , , , , , , 2 , , , , , , , , d x y xy
x z xz y z yz s x y z s i j k s i j k s i j k s i j k s i j k s i j k s i j k s i j k s i j k s i j k s i j k s i j k s i j k s i j k s i j k s i j k s i j k s i j k s i j k s i j k s i j k

                    
Pour l'évaluation de l'algorithme développé de 3DTKEO continu en termes de précision, la mesure 3D est étalonnée en utilisant une norme d'étalonnage, c'est-à-dire une norme de hauteur de pas (SHS). Ces normes sont couramment utilisées pour l'étalonnage de mesure de profileur mécanique ou optique, y compris la microscopie interférentielle.

Pour la traçabilité de la norme, il est certifié par l'Institut National de Métrologie. La caractéristique de SHS que nous avons utilisée pour cet étalonnage est le paramètre de hauteur de pas. Dans ce travail, nous avons utilisé une norme de hauteur de pas qui est fabriquée par VLSI Standard Inc, consistant en 25 mm x 25 mm x 3 mm, et qui a une valeur de hauteur de pas de 1,7803 ± 0,011 μm. Sur la base des résultats de l'étalonnage utilisant notre traitement de signal de frange 3D développé, nous observons que le taux d'erreur de notre mesure sur un étalon est toujours en dessous de la limite de tolérance de de l'erreur SHS (< 0.011 μm

GENERAL CONCLUSION

In this thesis, we have presented the study of the use of Teager Kaiser Signal processing for fringe analysis in CSI. The main aim of the research project was to develop new n-D approaches (1D, 2D, 3D) which are suitable for improved characterization of more complex surfaces and transparent layers using white light interferometry.

We begin our study by evaluating the performance of envelope detection using the 1D Teager Kaiser Energy Operator, which is compared to other techniques. These methods consist of the Fourier Transform (TF), wavelet and the FSA (Five-Sample-Adaptive) algorithm. We have developed a simulation program (in MATLAB) that allows the comparison of the performance of the different methods using a synthetic fringe signal (a synthetic transparent layer). Further comparison was then carried out using real data, i.e. the fringe image of a resin layer on Silicon. In addition, we also evaluated the performance of different types of pre-processing in order to remove the offset component and suppress the noise in the fringe analysis. This study yields the following conclusions:

 The combination of EMD and Savitzky-Golay filters provides the best performance for suppressing the noise and the offset component while maintaining the original signal intensity.

 CWT and TKEO are both competitive for providing surface extraction:

-TKEO is slightly better in terms of computational time.

-CWT is slightly better for the wavy transparent layer, but TKEO is quicker in terms of computational time.

-TKEO is more competitive than CWT for resolving between two close layers.

After a study of 1D fringe signal processing, we implement the 2D approach using Teager Kaiser for fringe analysis. The study of the robustness of 2D fringe processing in CSI has been carried out for the characterization of a transparent Mylar polymer film. We have demonstrated the ability of 2D approaches to compete with some classical methods (1D approaches) used in the field of interferometry, in terms of robustness. These results demonstrate that the XZ fringe envelope extracted by the XZ-scan technique provides more satisfactory results than the Z-scan approach in revealing the internal structures and the rear surface. The technique also allows an improvement in the details in the XZ images as well as more accurate measurements of the thickness of the polymer film.

CONCLUSION GÉNÉRALE

Dans cette thèse, nous avons présenté l'étude de l'utilisation du traitement du signal de Teager Kaiser pour l'analyse des franges dans CSI. L'objectif principal du projet de recherche était de développer de nouvelles approches n-D (1D, 2D, 3D), qui conviennent à la caractérisation améliorée de surfaces plus complexes et de couches transparentes utilisant l'interférométrie à lumière blanche.

Nous commençons notre étude en évaluant la performance de la détection -TKEO est légèrement meilleur en termes de temps de calcul.

-CWT est légèrement meilleur pour la couche transparente ondulée, mais TKEO est plus rapide en termes de temps de calcul.

-TKEO est plus compétitif que CWT pour résoudre entre deux couches proches.

APPENDIX-2: MULTI-SCALE ROUGHNESS MEASUREMENT OF CEMENTITIOUS MATERIALS USING WINDOW RESIZING ANALYSIS

The work during this PhD also involved a contribution to the development of the "window re-sizing" technique [START_REF] Apedo | Cement paste surface roughness analysis using coherence scanning interferometry and confocal microscopy[END_REF], [START_REF] Apedo | Geometrical roughness analysis of cement paste surfaces using coherence scanning interferometry and confocal microscopy[END_REF], [START_REF] Montgomery | Multi-scale roughness measurement of cementitious materials using different optical profilers and window resizing analysis[END_REF] for comparing the results of measurements performed by two different profiling systems (interference microscopy and confocal chromatic microscopy). This work was carried out in the context of collaboration with the GCE team at ICube.

Development of "window resizing" technique

The usual roughness amplitude parameters are measured, namely H mm , the peak-valley roughness, the difference in height between the highest and lowest measured points, R a , the arithmetic average of the absolute values of the roughness and R q , the root mean square (RMS) value of the roughness.

Sliding window method used

In order to compare data from both techniques, the average value of these roughness parameters (H mm , R a and R q ) is calculated at different scales from the data in the form of an XYZ matrix using sliding windows XY of increasing size [126], [START_REF] Kopp | Fluctuations of the dynamic fracture energy values related to the amount of created fracture surface[END_REF]. The (x,y,z) coordinates of the measured points are considered on a regular grid as defined in Fig. 114 where the crossings of the grid lines correspond to the positions of pixels in the image data. The points are positioned every dx and dy along the x and y axes respectively (see Fig. 114) and dx = dy. The height, z i,j , measured at a given pixel corresponds to the i th column and j th row.

A sliding window or cell is defined as a set of points forming a square with a size δ, an integer. A cell centred on a point (i,j), contains all of the points from (x i,j -δdx) to (x i,j + δdx) and from (y i,j -δdx) to (y i,j + δdx). For example, for δ = 1, the cell contains 9 points; for δ = 2, the cell contains 25 points, etc. The value of δ is increased up to the point where there are less than four cells for averaging the different roughness parameters. Each point is centred on a cell having the length of a side equal to 2δdx (Fig. 114). The points that are too close to the edges where it is not possible to centre a cell of the correct size are eliminated. The number of columns and rows used are i max and j max respectviely.

The values of (z max -z min ) are calculated for each cell, and then averaged over all of the cells, where:

(z max )i,j = Max(z n,m ),n ∈ [i -δ,i + δ],m ∈ [j -δ,j + δ] (z min )i,j = Min(z n,m ),n ∈ [i -δ,i + δ],m ∈ [j -δ,j + δ]

(1.1)

The average value of H mm is then given by:

(1.2)

The average value of R a is given by:

(1.3)

The average value of R q is given by:

(1.4)

Where:

(1.5)

with relatively smooth facets, the difference between the lateral resolutions of both techniques does not affect much the results. For the measurement of large smooth grains,

Fig. 1

 1 Fig. 1 Schematic of the OCT system [17] ............................................................................

Fig. 2

 2 Fig. 2 Challenges of fringe signal processing in CSI ...........................................................

Fig. 3

 3 Fig. 3 Schematic of confocal microscope: showing the scanning (a) in the focal plane of the objective and (b) out of focus. (c) schematic of chromatic confocal microscopy [44] ..

Fig. 4 Fig. 5 Fig. 8

 458 Fig. 4 Phase determination, from 4 discrete steps of 120°[47]. ......................................... Fig. 5 The technique of the change of phase[47]. .............................................................. Fig. 6 (a) A schematic layout of a CSI system (Leitz-Linnik interference microscope) and (b) typical signal from a single surface [49]. ..................................................................... Fig. 7 The interferogram construction on a surface using CSI (source: Guide to the measurement using CSI [50]). ........................................................................................... Fig. 8 Double signal of white light interference fringes from a transparent layer (synthetic signal). ................................................................................................................................

Fig. 9 Z

 9 Fig. 9 Z scan technique. .....................................................................................................

Fig. 10 XZ

 10 Fig. 10 XZ scan technique [49].......................................................................................... Fig. 11 The procedure of white light interference fringe analysis [53]. ............................

Fig. 12 Fig. 13 FSA

 1213 Fig. 12 Fringe envelope detection process using Hilbert Transform technique [53]. ........ Fig. 13 FSA Algorithm ...................................................................................................... Fig. 14 The modulus of coefficient CWT of the fringe signal [53]. .................................. Fig. 15 The use of post-filtering in CSI: (a) fringe signal, (b) prefiltered signal, and its envelope obtained (c) without post-filtering and (d) using post-filtering. ......................... Fig. 16 An example of background variation on the fringe signal (resin on Si)................ Fig. 17 Illustration of the Savitzky Golay filter .................................................................

Fig. 18

 18 Fig. 18 Synthetic fringe signals with: (a) constant, (b) linear, (c) non-linear offset components. .......................................................................................................................

Fig. 19

 19 Fig. 19 Prefiltered signal obtained using: (a) Global average, (b) Local average, (c) Central derivative, (d) EMD, applied to the synthetic fringe signal with a constant offset component. ......................................................................................................................... Fig. 20 Prefiltered signal obtained using: (a) Global average, (b) Local average, (c) Central derivative, (d) EMD, applied to the synthetic fringe signal with a linear offset component. .........................................................................................................................

Fig. 24

 24 Fig. 24 The modified Leica DMR-X interference microscope developed in the IPP team [83]. .................................................................................................................................... Fig. 25 XZ images showing the interferograms from (a) a Step Height Standard and (b) a rock surface [85]. ............................................................................................................... Fig. 26 An XZ image showing the interferogram on a transparent layer on a substrate consisting of a resin layer on a silicon substrate ................................................................

Fig. 27

 27 Fig. 27 An XZ image showing the interferogram of the transparent polymer film sample. ............................................................................................................................................ Fig. 28 The illustration of 3D PSF ..................................................................................... Fig. 29 Wavelet familes: (a) Morlet, (b) complex Morlet, (c) Gaussian, (d) complex Gaussian, and (e) Mexican hat ...........................................................................................

Fig. 30

 30 Fig. 30 Synthetic fringe signal in (a) XZ image and (1D) .................................................

Fig. 31

 31 Fig. 31 Fringe envelope obtained using wavelet familes: (a) Morlet, (b) complex Morlet, (c) Gaussian, (d) complex Gaussian, and (e) Mexican hat ................................................

Fig. 32

 32 Fig. 32 Signal processing analysis for a wavy synthetic transparent surface: (a) output of pre-filtering using the EMD-SGolay filter; (b) surface profile, (c) 2D and (d) 1D fringe envelope obtained by CWT(complex Gaussian). .............................................................. Fig. 33 Signal processing analysis for a wavy synthetic transparent surface: (a) output of pre-filtering using the EMD-SGolay filter; (b) surface profile, (c) 2D and (d) 1D fringe envelope obtained by CWT (complex Morlet). ................................................................. Fig. 34 (a) Synthetic signal with noise σ =20% and prefiltered signal resulting from: (b) pre-filter 1; (c) pre-filter 2; (d) pre-filter 3. ........................................................................

Fig. 35

 35 Fig. 35 The comparison procedure to evaluate the performance of the different algorithms[53]. ...................................................................................................................

Fig. 36 Fig. 38

 3638 Fig. 36 Synthetic fringe signal with a non-linear offset and 10% Gaussian noise (XZ image) 256 x 256 pixel on a (a) flat transparent layer and (b) wavy transparent layer. .... Fig. 37 Signal processing analysis for synthetic flat transparent surface: (a) output of prefiltering using EMD-SGolay filter; (b) surface profile, (c) XZ image of envelope and (d) 1D fringe envelope obtained by TKEO. ............................................................................

Fig. 40 2D

 40 Fig. 40 2D envelope peak detection obtained by: (a) HT, (b) FSA, (c) CWT, (d) TKEO.

Fig. 41

 41 Fig. 41 Region of interest (ROI). ....................................................................................... Fig. 42 1D fringe envelopes obtained by: (a) HT, (b) FSA, (c) CWT, and (d) TKEO at x = 11.41 µm near the resin edge where the two surfaces are closer together. ........................ Fig. 43 Typical fringe signals from a sample of a resin layer on Silicon .......................... Fig. 44 The dispersion curve corresponding to the resin layer .......................................... Fig. 45 The altitude Z (axis) of resin on Si ........................................................................ Fig. 46 The limitation of the PSM technique in height measurement ............................... Fig. 47 The measurement of a spherical surface using the PSM technique resulting in phase discontinuities .......................................................................................................... Fig. 48 The technique of phase unwrapping in PSM to measure deeper structures .......... Fig. 49 The illustration of airy spot on the pixels ..............................................................

Fig. 50

 50 Fig. 50 The batwing effect on the measurement of a step ................................................. Fig. 51 (a) Synthetic XZ fringe image 256 x 256 pixel and (b) profile of fringe signal along the optical axis Z. .....................................................................................................

Fig. 52

 52 Fig. 52 Analysis of a noisy synthetic fringe signal (σ=20%): (a) 2D and (b) 1D fringe envelope obtained by Z-scan technique (FSA), (c) surface profile. .................................. Fig. 53 Analysis of a noisy synthetic fringe signal (σ=20%): (a) 2D and (b) 1D fringe envelope obtained by XZ-scan technique (2DTKEO), (c) surface profile. ....................... Fig. 54 Scheme of the interferogram construction on a transparent layer using CSI [83] Fig. 55 Identify the number of Gaussian function using curve fitting ............................... Fig. 56 (a) Synthetic XZ fringe image 256 x 256 pixel and (b) profile of fringe signal along the optical axis Z of transparent multilayer. ............................................................

Fig. 57

 57 Fig. 57 Fringe envelope of a synthetic transparent layers .................................................

Fig. 58 (

 58 Fig. 58 (a) Surface detection on the synthetic transparent layers and (b) detected surface profile ................................................................................................................................. Fig. 59 Comparison of the measured structures with the reference structures ..................

Fig. 60

 60 Fig. 60 Interferogram of a transparent polymer film ......................................................... Fig. 61 Fringe envelope of the interferogram from a transparent polymer film ................

Fig. 71

 71 Fig. 71 Thickness of Mylar polymer film. ......................................................................... Fig. 72 The technique of 3D fringe processing using 3DTKEO .....................................

Fig. 73

 73 Fig. 73 Stack image XYZ of fringe signals s(x,y,z) ......................................................... Fig. 74 The stack image XYZ of fringe signal corresponds as reconstruction of the projection XZ slices and YZ slices .................................................................................. Fig. 75 The fringe envelope and the local frequency/orientation .................................... Fig. 76 3D surface profile, XY surface profile z 0 (x,y), and the line surface profile ....... Fig. 77 Fringe signals of Graphene .................................................................................. Fig. 78 Pre-filtered fringe signals of Graphene after pre-processing step ....................... Fig. 79 The volume of amplitude A(x,y,z) of fringe signals obtained using 3D fringe envelope detection ........................................................................................................... Fig. 80 Improvement results of 3D envelope detection using post-processing ...............

Fig. 81

 81 Fig. 81 Surface extraction of a sample of Graphene obtained using 3D fringe signal processing ........................................................................................................................

Fig. 82

 82 Fig. 82 Synthetic surface (XYZ image) 256 x 256 x 25 pixel and it's fringe signals: (a), (c), (e) flat transparent layer and (b), (d), (f) wavy transparent layer. ............................. Fig. 83 Signal processing analysis for synthetic flat transparent surface ........................

Fig. 84

 84 Fig. 84 Signal processing analysis for synthetic wavy transparent surface .....................

Fig. 86 1D

 86 Fig. 86 1D Fringe envelope obtained using (a) 1DTKEO, (b) 2DTKEO, (c) 3DTKEO .

Fig. 87

 87 Fig. 87 Camera image (XY slice) of the step-height standard (VLSI Standard Inc.) ...... Fig. 88 Fringe signal processing of Step Height Standards (SHS) by the modified Leitz-Linnik microscope ........................................................................................................... Fig. 89 Surface extracted of the Step Height Standards (SHS) obtained using 3D fringe signal processing on measurements from the modified Leitz-Linnik microscope .......... Fig. 90 Line profile of the Step Height Standards (SHS) obtained using 3D fringe signal processing on measurements with the modified Leitz-Linnik microscope .....................

Fig. 91

 91 Fig. 91 Fringe signal processing of the Step Height Standards (SHS) from measurements on the new Fogale microscope ("Michelin") ................................................................... Fig. 92 Surface extracted of the Step Height Standard (SHS) obtained using 3D fringe signal processing from measurements with the new Fogale microscope ("Michelin") ...

Fig. 93

 93 Fig. 93 Line profile of the Step Height Standards (SHS) obtained using 3D fringe signal processing from measurements with the new Fogale microscope ("Michelin") .............

Fig. 94

 94 Fig. 94 Fringe signal processing of the Step Height Standard (SHS) from measurements with the "immersion" Fogale microscope ........................................................................ Fig. 95 Surface extracted of the Step Height Standard (SHS) obtained using 3D fringe signal processing on measurements with the "immersion" Fogale microscope .............. Fig. 96 Line profile of the Step Height Standard (SHS) obtained using 3D fringe signal processing on measurements with the "immersion" Fogale microscope ......................... Fig. 97 The measured step height obtained using: (a) Leitz-Linnik microscope; (b) "Michelin" Fogale Microscope; (c) "immersion" Fogale microscope ............................. Fig. 98 Fringe signals of DOE (Diffractive Optical Elements) ....................................... Fig. 99 Surface extraction of a sample of DOE (Diffractive Optical Elements) obtained using 3D fringe signal processing ....................................................................................

Fig. 100

 100 Fig. 100 Fringe signals of Resin on Silicon ....................................................................

Fig. 101 (

 101 Fig. 101 (a) Surface extraction of a sample of Resin on Silicon obtained using 3D fringe signal processing and (b) the line profile .........................................................................

Fig. 102

 102 Fig. 102 Fringe signals from measurements on a Cable .................................................. Fig. 103 Surface extraction of a sample of Cable obtained using 3D fringe signal processing ........................................................................................................................ Fig. 104 Fringe signals of measurements on a Rock surface ...........................................

Fig. 108 XZFig. 110 3D

 108110 Fig. 108 XZ fringe envelope obtained using (a) Continuous 3DTKEO and (b) Discrete 3DTKEO .......................................................................................................................... Fig. 109 Comparison fringe envelope profile in 1D obtained using (a) Continuous 3DTKEO and (b) Discrete 3DTKEO ...............................................................................Fig. 110 3D Surface extraction obtained using (a) Continuous 3DTKEO and (b) Discrete 3DTKEO .......................................................................................................................... Fig. 111 Image of surface profile obtained using (a) Continuous 3DTKEO and (b) Discrete 3DTKEO ............................................................................................................ Fig. 112 Line profile comparison of Cable surface obtained using (a) Continuous 3DTKEO and (b) Discrete 3DTKEO ...............................................................................Fig. 113 Deviation value of surface extraction obtained using (a) Continuous 3DTKEO and (b) Discrete 3DTKEO ...............................................................................................Fig. 114 Coordinate system used showing basic cell for δ = 1 (in this case giving 3x3 = 9 points) ..............................................................................................................................Fig. 115 Measurement of surface roughness of unpolished samples using SCM and CSI .......................................................................................................................................... Fig. 116 Variation of different roughness values as a function of window size δdx for unpolished cement ...........................................................................................................

  ABBREVIATIONS

Fig. 2

 2 Fig. 2 Challenges of fringe signal processing in CSI

Fig. 3 (

 3 a) shows the sample in the focal plane of the objective, while the sample moving out of the focus is shown in Fig. 3(b). The complete image is formed through point by point scanning of the illuminating and detecting pinhole on the spot over the sample surface. The image will not be formed if the sample moves out of focus, as shown in Fig. 3(b).

Fig. 3

 3 Fig. 3 Schematic of confocal microscope: showing the scanning (a) in the focal plane of the objective and (b) out of focus. (c) schematic of chromatic confocal microscopy[START_REF] Mukhtar | Development of compensated immersion 3D optical profiler based on interferometry[END_REF]

Fig. 6 (

 6 Fig.6(a) A schematic layout of a CSI system (Leitz-Linnik interference microscope) and (b) typical signal from a single surface[START_REF] Gianto | Study of Robustness of 2D Fringe Processing in Coherence Scanning Interferometry for the Characterization of a Transparent Polymer Film[END_REF].

9 )

 9 The function s corresponds to the intensity signal at a given point of the sample surface (x,y), where z represents a vertical scanning position in relation to the surface. The parameter z is referred to as k, with k the index-scanning step and  the value of scanning step. The height of the surface z 0 (x,y) is a spatial function that depends on lateral coordinates x and y. The quantity a(x,y,z) is an offset intensity related to the reference and object beam intensities, b(x,y) is the fringe contrast, g(z) is the fringe envelope function related to the spectral profile of the white light source, and λ 0 is the mean wavelength of the light source. The phase offset related to the phase change on reflection is represented by α(x,y). The fringe signal in Fig.6(b) shows a synthetic fringe signal from a single surface that has been generated based on the structure of white light interferences in Eq.(1.9).

Fig. 8

 8 Fig. 8 Double signal of white light interference fringes from a transparent layer (synthetic signal).

Fig. 9 Z

 9 Fig. 9 Z scan technique. Z-scan technique allows the initial manual investigation of the nature and quality of the fringe signal obtained which is generated by CSI. A priori information of the nature of the fringe signal is useful for the checking the presence of artefacts, for instance in a complex layer.

Fig. 10 XZ

 10 Fig. 10 XZ scan technique [49].

  Fig. 13 FSA Algorithm

Fig. 14

 14 Fig. 14 The modulus of coefficient CWT of the fringe signal [53]. In order to compute each coefficient W(a,b), the fringe signal is convolved with a set of complex Morlet wavelets. As a function of scale factors a and shift b, W(a,b) can be represented by a scalogram, as shown in Fig. 14. The fringe envelope corresponds to the absolute modulus of the maximum correlation coefficient as shown by the red line in the scalogram.

Fig. 15

 15 Fig. 15 The use of post-filtering in CSI: (a) fringe signal, (b) prefiltered signal, and its envelope obtained (c) without post-filtering and (d) using post-filtering.

  Fig.[START_REF] Dubois | High-resolution fullfield optical coherence tomography with a Linnik microscope[END_REF] An example of background variation on the fringe signal (resin on Si)

  types, i.e. forward difference, backward difference, and central difference. By removing the values of the signal where their adjacent/neighbourhood values are almost equal, the first derivative effectively removes the lower frequencies of the signal, i.e. the offset component. Among the three types of the first derivative, the central difference gives the most accurate approximation, and hence it is commonly used for suppressing the trend and the offset in white light fringe analysis. The central difference formula for the first derivative of f(x) which is based on the values of the function at the points f(x+h) and f(xh), can be expressed as:

. 23 )

 23 This Eq.(1.23) expresses that the first derivative of f(x) at a point x is derived by subtraction of its immediate neighboring points (x+h) and (x-h). It removes the values of two successive points of f(x) which are the same and leaves the values which are different. When it is performed on an entire signal, the first derivative f'(x) effectively removes the intensity background and suppresses the lower frequency signal, although it emphasizes the higher frequency signal which tends to accentuate the noise. The first derivative therefore works well for noiseless signals whose derivation values can be calculated precisely and is not good for high frequency suppression.

Fig. 18

 18 Fig. 18 Synthetic fringe signals with: (a) constant, (b) linear, (c) non-linear offset components.

Fig. 19 Fig. 20 Fig. 21

 192021 Fig. 19 Prefiltered signal obtained using: (a) Global average, (b) Local average, (c) Central derivative, (d) EMD, applied to the synthetic fringe signal with a constant offset component.

  Fig. 21(b), the resulting amplitude of the fringe signal is clearly lower than the ideal synthetic fringe signal (without the offset). This means that a degradation of the intensities occurs during the pre-processing process. Similar results are also obtained using the central derivative technique, as illustrated in Fig. 19(c), Fig. 20(c), and Fig. 21(c), although taking from 3 to 4 times longer than the global and local averaging. The resulting amplitude of the fringe signal is also clearly lower than the ideal synthetic fringe signal, although the decrease is not as great as that for the local averaging. A decrease in the intensity may seriously degrade the performance of the algorithm in order to extract the fringe envelope, particularly in areas where the SNR is low. The best results are provided by the EMD technique, as illustrated in Fig. 19(d), Fig. 20(d), and Fig. 21(d) which is clearly able to remove the offset, while almost maintaining the amplitude of the fringe signal. But this good result is at the expense of the computation time which takes from 6 to 31 times the time required by the three other techniques.

  Fig. 25 XZ images showing the interferograms from (a) a Step Height Standard and (b) a rock surface [85].

Fig. 26

 26 Fig. 26 An XZ image showing the interferogram on a transparent layer on a substrate consisting of a resin layer on a silicon substrate

Fig. 27

 27 Fig. 27 An XZ image showing the interferogram of the transparent polymer film sample.

  or an optical instrument) to distinguish two point sources of equal luminosity and separated by a minimum distance. When two distinct points are very close, they will have a spots (Airy spots) which overlap and if they are not sufficiently separated, may therefore no longer be distinguished from each other. These point sources are represented by two well separated diffraction patterns, or Airy disks. Experimentally, the normalized lateral axis x ( m)

  Abbe 0.5 A R NA  

  cohérence (CSI) génère un interférogramme (signal des franges). Nous étudions et observons la structure du signal des franges et les différentes approches (techniques Zscan et XZ-scan) pour l'analyse des franges. Ensuite, nous nous concentrons sur les étapes de la procédure d'analyse des franges que nous effectuons dans ce travail. L'étude du prétraitement pour enlever les composants "offset" et la détermination de la structure de surface de l'échantillon que nous utilisons dans le travail sont également rapportés.Le principe de la technique CSI est basé sur l'analyse de cohérence croisée de deux faisceaux lumineux de faible cohérence, le faisceau objet étant réfléchi par l'échantillon, tandis que le faisceau de référence est réfléchi par un miroir de référence. Les deux faisceaux lumineux sont ensuite combinés au niveau du détecteur. L'interférence se produit lorsque la différence de chemin optique entre le faisceau de référence et le faisceau d'objet réfléchi est proche de zéro. C'est-à-dire lorsque la longueur du trajet optique vers l'échantillon est presque identique à la longueur du trajet optique par rapport au miroir de référence. Dans le but de trouver le maximum d'interférence, un balayage de frange est effectué sur chaque point de la surface de l'échantillon, point par point. Les intensités de contraste des franges qui tiennent compte du changement de distance entre l'échantillon et l'objectif (en axe z) sont captées par le détecteur (caméra) générant le signal de frange, également connu sous le nom d'interférogramme. L'analyse des franges est alors nécessaire pour récupérer le pic d'enveloppe de contraste de frange qui indique la position de surface de l'échantillon.

  fonction s correspond au signal d'intensité à un point donné de la surface de l'échantillon (x,y), où z représente une position de balayage vertical par rapport à la surface. La quantité a(x,y,z) est une intensité de décalage liée aux intensités de faisceau de référence et d'objet, b(x,y) est le contraste de frange, g(z) est la fonction d'enveloppe de frange liée au profil spectral du source de lumière blanche, et λ 0 est la longueur d'onde moyenne de la source lumineuse. Le déphasage lié au changement de phase à la réflexion est représenté par α(x,y). En général, les techniques de traitement du signal développées dans ce travail se composent de trois étapes principales: un pré-filtrage, la détection d'enveloppe et un postfiltrage. La détection d'enveloppe est nécessaire pour obtenir l'enveloppe de franges dont le pic représente la position de surface. Le pré-filtrage est utilisé pour supprimer le composant de décalage et réduire le bruit, tandis que le post-filtrage est utilisé pour déterminer plus précisément la mesure. Dans ce chapitre, nous présentons les différentes techniques utilisées pour extraire l'enveloppe de frange dans la CSI, à savoir la Transformée de Hilbert (HT), l'algorithme Adaptatif à cinq échantillons (FSA), la Transformée en ondelettes continue (CWT) et l'Opérateur de Teager Kaiser-Energy (TKEO). Par contre, nous avons également rapporté notre étude de la comparaison des performances de différents pré-traitements afin d'éliminer la composante de décalage du signal de frange (moyenne globale, moyenne locale, dérivée centrale, décomposition en mode empirique). L'équipement développé pour exécuter l'algorithme pour le traitement des signaux de frange dans cette thèse consiste en deux microscopes adaptés, un microscope Leica DMR-X équipé d'objectifs Mirau (x10, x40) et Michelson (x5) et un microscope Leitz-Linnik équipé d'objectifs de x50 (NA = 0,85). L'échantillon est monté sur une table de balayage vertical piézo-contrôlée pour le balayage en Z. L'actionneur piézoélectrique est commandé en boucle fermée avec un capteur de position capacitif, ayant une sensibilité de position de 1 nm. Une lampe à halogène est utilisée dans le Leica DMR-X et une lampe à incandescence est utilisée pour l'éclairage du Leitz-Linnik avec une lumière blanche (310 nm -1100 nm). L'acquisition d'image est réalisée avec un CCD Prosilica numérique ayant 1360x1024 pixels et un Aval1000 Basler ayant 1024x1024 pixels. Afin d'évaluer la performance des algorithmes développés dans ce travail, les échantillons suivants qui ont différentes structures ont été utilisés. Ils ont été choisis parce qu'ils varient d'une simple surface lisse à des structures de plus en plus complexes, y compris une surface unique, une couche transparente sur le substrat et un film de polymère transparent.

4 )

 4 By substituting the equation of x = A cos (ωt + φ) and its derivative, dx x dt  , we can obtain the equation of energy:

30 .Fig. 29 Fig. 31 Fringe

 302931 Fig. 29 Wavelet familes: (a) Morlet, (b) complex Morlet, (c) Gaussian, (d) complex Gaussian, and (e) Mexican hat

Fig. 33

 33 Fig. 33 Signal processing analysis for a wavy synthetic transparent surface: (a) output of pre-filtering using the EMD-SGolay filter; (b) surface profile, (c) 2D and (d) 1D fringe envelope obtained by CWT (complex Morlet).

Fig. 34

 34 Fig. 34 (a) Synthetic signal with noise σ =20% and prefiltered signal resulting from: (b) pre-filter 1; (c) pre-filter 2; (d) pre-filter 3.

Fig. 36

 36 Fig. 36 Synthetic fringe signal with a non-linear offset and 10% Gaussian noise (XZ image) 256 x 256 pixel on a (a) flat transparent layer and (b) wavy transparent layer.

  (b) and Fig. 38(b). The clearer observation of the pre-filtered signal and the fringe contrast envelope is illustrated in Fig. 37(d) and Fig. 38(d) in one dimension (the optical axis Z).

Fig. 37 Fig. 38

 3738 Fig. 37 Signal processing analysis for synthetic flat transparent surface: (a) output of prefiltering using EMD-SGolay filter; (b) surface profile, (c) XZ image of envelope and (d) 1D fringe envelope obtained by TKEO.

  objective x50, numerical aperture 0.85, pixel size of 0.113 µm, and effective average wavelength of 720 nm. The step height of the piezo scanner in order to scan the sample along the optical axis was 90 nm over a dynamic range of 10 µm. A description of the acquisition data of the resin on Si sample obtained by the microscope system is given in Appendix-1: Data of Sample.

Fig. 39 Fig. 40 2D

 3940 Fig. 39 Real fringe measurements on the resin on silicon sample: (a) raw XZ image data and (b) output of pre-filtering of (a) using the EMD-SGolay filter.

  Fig. 41 Region of interest (ROI).

However, let usFig. 42 1D

 42 Fig. 42 1D fringe envelopes obtained by: (a) HT, (b) FSA, (c) CWT, and (d) TKEO at x = 11.41 µm near the resin edge where the two surfaces are closer together.

Fig. 43

 43 Fig. 43 Typical fringe signals from a sample of a resin layer on Silicon

  ce chapitre, nous décrivons l'opérateur d'énergie Teager-Kaiser que nous utilisons comme algorithme multidimensionnel proposé (1D, 2D, 3D) pour la détection d'enveloppe dans la CSI. Ensuite, l'étude de la comparaison des performances des différentes techniques de pré-filtrage et de détection d'enveloppe dans le traitement des signaux de franges est décrite. A titre de comparaison, des résultats utilisant les algorithmes sont présentés sur le signal de frange synthétique et sur des données réelles, c'est-à-dire l'échantillon de résine sur silicium. En outre, une étude de sélection de l'ondelette mère sur la transformée en ondelettes continue est présentée dans son utilisation pour l'obtention de l'enveloppe de franges. Afin de supprimer le décalage et réduire le bruit, nous comparons dans ce chapitre trois algorithmes différents qui combinent la technique dérivée / EMD et le filtre de débruitage (Savitzky-Golay / Wiener). Les algorithmes correspondants sont référencés avec les noms suivants:  Pré-filtre 1: combinaison de dérivée et filtre de Wiener,  Pré-filtre 2: combinaison de dérivée et filtre Savitzky-Golay,  Pré-filtre 3: combinaison d'EMD et filtre Savitzky-Golay. Pour l'évaluation des performances de différents pré-filtres ci-dessus, nous effectuons la simulation en les effectuant sur un signal de franges synthétiques, avec un pas d'échantillonnage de 40 nm. Un décalage non linéaire additif et un bruit gaussien sont ajoutés aux données synthétiques (20%), ce qui signifie que le carré moyen (rms) du bruit gaussien ajouté est égal à 20% de l'amplitude maximale du signal de frange de la seconde couche. Sur la base de la simulation, nous observons que le préfiltre 3 utilisant la combinaison d'EMD et Savitzky-Golay fournit le meilleur résultat que d'autres afin de supprimer le décalage et le bruit, en particulier dans le maintien de l'amplitude des franges. Une autre simulation a été réalisée afin d'évaluer notre procédure d'analyse des franges, en utilisant une combinaison de pré-filtrage différent et de détection d'enveloppe différente. En ce qui concerne l'étape principale de notre procédure, à savoir la détection de l'enveloppe des franges, nous observons que le TKEO est un algorithme compétitif en termes de robustesse et de temps de calcul. TKEO fournit une meilleure extraction de surface que la transformée de Hilbert et FSA. Dans le cas d'une couche transparente plate, TKEO fournit les résultats les plus précis pour l'extraction de la surface (1 e surface) et plus proche de la performance de CWT pour l'extraction de l'interface enterrée (2 e surface), donnant la valeur la plus petite d'erreur absolue moyenne. Pour la couche transparente ondulée, la mesure la plus précise est fournie par CWT, tandis que la performance de TKEO est la plus proche de CWT que les autres. Cependant, TKEO a l'avantage d'être plus compact en termes computationnels et ainsi bien meilleur en termes de temps de calcul que le CWT. Par exemple, TKEO prend 6,5 s tandis que CWT prend 30,6 s dans le contexte d'un signal de frange bruyant avec σ = 20%.D'autre part, nous avons également évalué les performances de post-traitement que nous avons utilisées dans notre procédure d'analyse des franges. Sur la base des résultats, nous avons démontré que l'utilisation de l'interpolation sur l'ajustement gaussien améliore la sensibilité axiale de la mesure du profil. Les résultats donnent une amélioration significative en termes de précision de l'extraction de surface, comme représenté par la plus petite valeur de mae, en tant que critère de comparaison, lorsque nous n'avons pas effectué l'interpolation sur l'ajustement gaussien comme post-traitement. La valeur de mae de l'analyse de frange en utilisant une interpolation sur un ajustement gaussien est 2 à 3 fois plus petit, ce qui correspond à une plus grande précision.Ensuite, les algorithmes sont réalisés sur des données réelles, c'est-à-dire d'une couche de résine sur Si. L'interférogramme venant de l'échantillon montre clairement la présence de grandes variations du décalage le long de l'axe optique près des bords de la couche de résine en raison d'effets optiques. Afin d'éliminer ce fond sur l'image des franges, le pré-filtrage est réalisé en utilisant la technique EMD en combinaison avec le filtre Savitzky-Golay. Ensuite, les différentes techniques de détection d'enveloppe ont été réalisées pour obtenir l'enveloppe de frange des signaux de franges pré-filtrés. Basé sur les résultats, on peut observer que la performance de chaque technique est différente, notamment en ce qui concerne les détails proches des bords de la couche de résine, particulièrement difficiles à mesurer. Dans la région des bords de la couche de résine, les meilleures performances sont fournies par HT et TKEO que par les autres pour distinguer clairement entre deux couches proches.

Fig. 47

 47 Fig.[START_REF] Abderrazzaq | Developpement de la microscopie interferometrique pour une meilleure analyse morphologique des couches minces et epaisses des materiaux semiconducteurs et optique[END_REF] The measurement of a spherical surface using the PSM technique resulting in phase discontinuities

  photodiodes on the camera sensor. The image in Fig.49illustrates the central Airy spot of a point source at infinity being correctly sampled by the camera according to the Nyquist criterion at twice the signal spatial frequency, a typical condition for microscope imaging.

( 3 . 6 )

 36 could be obtained by multiplying the transformed image and the appropriate mask in the frequency domain. If the 2D analytical signal    has been constructed, the phase and the amplitude of the signal s(x,y) could be obtained based on the following equation[START_REF] Bovik | Handbook of image and video processing[END_REF]:

Fig. 51 (Fig. 52 Fig. 53

 515253 Fig. 51 (a) Synthetic XZ fringe image 256 x 256 pixel and (b) profile of fringe signal along the optical axis Z.

  55(b). The number of Gaussian functions (envelopes) on an interferogram indicates the number of layers in the sample.

Fig. 55 56 .

 5556 Fig. 55 Identify the number of Gaussian function using curve fitting

Fig. 57

 57 Fig. 57 Fringe envelope of a synthetic transparent layers

Fig. 60

 60 Fig. 60 Interferogram of a transparent polymer film

Fig. 62

 62 Fig. 62 Result of surface detection using the neighborhoods number as the threshold

  Fig. 63 (a) XZ fringe image of a Mylar polymer and (b) the fringe signal along the optical axis Z.

Fig. 64 (Fig. 64 Fig. 65

 646465 Fig. 64 Logarithmic transformation 1 of fringe envelope obtained by (a) FSA, (b) postprocessing result using cubic spline smoothing.

Fig. 66 (Fig. 66 Fig. 67 Fig. 68

 66666768 Fig. 66 Logarithmic transformation of fringe envelope obtained by (a) 2DTKEO, (b) its post-processing result using cubic spline smoothing.

Fig. 69 B = 1 . 6073 C = 9 .7817 x10 3 D = 6 .

 6916073936 Fig. 69 The refractive index of the Mylar Fig. 69 shows the refractive index of the Mylar polymer film. Based on this figure, it can be seen that its value vary as a function of the wavelength. In order to calculate the index value of n(λ) of the Mylar polymer film, we applied the Cauchy equation, which shows the empirical relationship between the wavelength and the refractive index. The Cauchy equation is given by:

  The inhomogeneity of the refractive index of the Mylar is a factor that can accentuate the degradation of the wavefront reflected by the interface, which introduces an additional measurement error. It was assumed in our study that the index is homogeneous and constant across the depth of the layer. In reality, several variations are possible: random, linear or quadratic. Each variation has a different effect on the value of the measurement.

Fig. 70

 70 Fig. 70 Thickness measurements of Mylar polymer film.

Fig. 71

 71 Fig. 71 Thickness of Mylar polymer film.

2 ) 4 )

 24 Fig. 74 The stack image XYZ of fringe signal corresponds as reconstruction of the projection XZ slices and YZ slices Based on Eq.(4.2) and as illustrated in Fig. 74, fixing a slice x 0 yields the 2D AM-FM signal (YZ slice), which is represented by the following equation:

Fig. 77

 77 Fig. 77 Fringe signals of Graphene

Fig. 80

 80 Fig. 80 Improvement results of 3D envelope detection using post-processing

Fig. 81

 81 Fig. 81 Surface extraction of a sample of Graphene obtained using 3D fringe signal processing

Fig. 82

 82 Fig. 82 Synthetic surface (XYZ image) 256 x 256 x 25 pixel and it's fringe signals: (a), (c), (e) flat transparent layer and (b), (d), (f) wavy transparent layer.

Fig. 83 andFig. 83 Fig. 84 Fig. 86 1D

 83838486 Fig. 83 and Fig. 84 show the example of the results of signal processing for the flat and wavy transparent layers. Fig. 83(a) and Fig. 84(a) show the fringe envelope obtained, while Fig. 83(b) and Fig. 84(b) show the surface extracted by performing signal processing using 3D fringe processing. Meanwhile, Fig. 85 shows the results of surface

  e. Step Height Standard (SHS). These standards are common used for measurement calibration of mechanical or optical profilers, including interference microscopy. For the traceability of the standard, it is certified by a National Metrology Institute (NMI). The feature of SHS that we used for this calibration is the step height parameter. In this work, we used a step height standard fabricated by VLSI Standard Inc, consisting of a 25 mm x 25 mm x 3 mm and has step height value of 1.7803 ± 0.011 µm. Fig. 87 shows the camera image of the step-height standard (VLSI Standard Inc.).

Fig. 87 Fig. 93

 8793 Fig. 87 Camera image (XY slice) of the step-height standard (VLSI Standard Inc.)

Fig. 98 FringeFig. 101 (Fig. 102 FringeFig. 104 Fringe

 98101102104 Fig. 98 Fringe signals of DOE (Diffractive Optical Elements)

Fig. 106 (

 106 Fig. 106 (a) Fringe image of a Cable and (b) its fringe profile in 1D

Fig. 107

 107 Fig. 107 Prefiltered fringe image of a measurements from the Cable after the preprocessing step and (b) its prefiltered fringe profile in 1D

Fig. 108 XZFig. 110 3DFig. 111 Fig. 112

 108110111112 Fig. 108 XZ fringe envelope obtained using (a) Continuous 3DTKEO and (b) Discrete 3DTKEO

Fig. 113 2 d

 1132 Fig. 113 Deviation value 2 of surface extraction obtained using (a) Continuous 3DTKEO and (b) Discrete 3DTKEO



  d'enveloppe en utilisant l'opérateur 1D Teager Kaiser Energy, qui est comparé à d'autres techniques. Ces méthodes consistent en la Transformée de Fourier (TF), ondelettes, la FSA (Five-SampleAdaptive), Opérateur d'énergie de Teager Kaiser (TKEO). Nous avons développé un programme de simulation de franges blanches (sur MATLAB) qui permet de comparer les résultats de mesures synthétiques (une couche transparente) effectués par différentes techniques de traitement de signal. D'autre comparaison a ensuite été réalisée en utilisant des données réelles, c'est-à-dire l'image de frange de la couche de résine sur Silicium. En outre, nous avons également évalué les performances de différents types de pré-traitement afin de supprimer la composante de décalage et supprimer le bruit dans l'analyse des franges. Cette étude conduit les conclusions suivantes: La combinaison des filtres EMD et Savitzky-Golay permet d'obtenir les meilleures performances pour supprimer le bruit et supprimer le composant de décalage tout en maintenant l'intensité du signal original.  CWT et TKEO sont compétitifs pour fournir l'extraction de surface:

  Après une étude du traitement du signal frange 1D, nous implémentons l'approche 2D en utilisant Teager Kaiser pour l'analyse des franges. L'étude de la robustesse du traitement des franges 2D en CSI a également été réalisée pour la caractérisation d'un film de polymère Mylar transparent. Nous avons démontré la capacité des approches 2D Teager Kaiser à concurrencer certaines méthodes classiques (approches 1D) utilisées dans le domaine de l'interférométrie, en termes de robustesse. Ces résultats démontrent que l'enveloppe de frange XZ extraite par les approches 2D Teager Kaiser donne des résultats plus satisfaisants que l'approche 1D en révélant les structures internes et la surface arrière.La technique permet également une amélioration des détails dans les images XZ ainsi que des mesures plus précises de l'épaisseur du film polymère.La dernière partie de ce travail de thèse rapporte une application de 3D Teager Kaiser Energy Operator (3DTKEO), basée sur l'opérateur d'énergie TK multidimensionnel. pour l'analyse des franges dans CSI. Grâce à une simulation utilisant un signal de frange synthétique, nous avons démontré la compétitivité de la performance de 3DTKEO par rapport aux approches 1D et 2D. De plus, nous avons testé l'algorithme sur des données réelles, c'est-à-dire une norme de hauteur de pas (VLSI Standard Inc.) pour l'étalonnage de mesure afin d'évaluer la précision. Sur la base de ces résultats, nous avons observé que le taux d'erreur des mesures sur une norme d'étalonnage est encore inférieur à la limite de tolérance de l'erreur de norme de hauteur de pas (<0,011 μm). De plus, nous avons testé l'algorithme sur différents échantillons, tels que le graphène, les DOE (Diffractive Optical Elements), la résine sur silicium, le câble et la roche. L'algorithme fonctionne très bien et fournit des profils de surface satisfaisants des échantillons. D'autre part, nous avons également développé et appliqué la version discrète de 3DTKEO afin d'analyser directement le signal de frange 3D. En comparant cela avec le 3DTKEO continu, nous avons trouvé que l'approche discrète de 3DTKEO donne l'avantage d'un temps de calcul plus court, tandis que le 3DTKEO continu est plus robuste au bruit que le 3DTKEO discret. 072 µm Dynamic range = 23,000 µm

Fig. 114

 114 Fig. 114 Coordinate system used showing basic cell for δ = 1 (in this case giving 3x3 = 9 points)

  Fig. 116 Variation of different roughness values as a function of window size δdx for unpolished cement

Table 1

 1 Computation times for the different pre-processing techniques applied to the synthetic fringe signals ..............................................................................................

Table 2

 2 The lateral resolution criteria based on the illumination type .......................

Table 3

 3 Mean absolute error (mae) of surface extraction obtained using the CWT algorithm (complex Gaussian and complex Morlet wavelet) .......................................

Table 4

 4 Experiment result using pre-filter 1 and different envelope detection techniques (nm) ...............................................................................................................

Table 5

 5 Experiment result using pre-filter 2 and different envelope detection techniques (nm) ...............................................................................................................

Table 6

 6 Experiment result using pre-filter 3 and different envelope detection techniques (nm) ...............................................................................................................

Table 7

 7 Improvement of axial sensitivity of measurement obtained using Gaussian

Table 1 Computation times for the different pre-processing techniques applied to the synthetic fringe signals

 1

		Global Average	Local Average	Central Derivative	EMD
	Time computation (ms)	6.49	3.76	17.96	115.68
	The global averaging technique only works well for fringe signals with an offset
	which is relatively constant, but not for the linear and nonlinear offsets. The Local
	averaging technique is able to maintain the symmetry of the fringe pattern and has the
	fastest computation time (Table

the measured profile the known form of the groove profile

	microscope equipped with x50 (NA = 0.85) objectives (Fig. 6(a)), a modified Fogale
	microscope equipped with x40 immersion objectives used in water ("immersion" system)
	and a modified Fogale microscope equipped with a Nikon x50 Mirau objective
	("Michelin" system). The sample is mounted on a piezo-controlled vertical scanning table
	(the PIFOC scanner from PI) for Z-scanning. The piezo actuator is controlled in a closed
	loop with a capacitive position sensor, having a position sensitivity of 1 nm. For the
	illumination, a halogen lamp is used in the Leica DMR-X, an incandescent lamp in the
	Leitz-Linnik (310 nm -1100 nm) and white LED's in the Fogale "immersion" and
	"Michelin" systems. Image acquisition is performed with either a digital Prosilica CCD
	(type EC1380) having 1360x1024 pixels and a Basler Ava1000 having 1024x1024 pixels

with Mirau (x10, x40) and Michelson (x5) objectives, a Leitz-Linnik and a Firewire connection or a Sony 3CCD (type DXC-390P) and standard color acquisition board. Acquisition and processing is carried out on a PC equipped with an Intel Xeon processor (2.4 GHz, 8 Go RAM). The measurement system is controlled with software that has been developed in LabView on Windows 7

(64-bit)

Table 4 Experiment result using pre-filter 1 and different envelope detection techniques (nm)

 4 1 st surf. 2 nd surf. 1 st surf. 2 nd surf. 1 st surf. 2 nd surf. 1 st surf. 2 nd surf.

		A		B		C		D
	Pre-			Suppress the offset: 1 st order central derivative
	filt.			Suppress the noise: Wiener filter
	Env. Det.	HT		FSA		TKEO	CWT
	Post-	Smooth the amplitude envelope using cubic smoothing spline
	filt.	Correct the peak curve using Gaussian estimation and interpolation
				mean absolute error (nm)
	noise() flat transparent layer
	0%	0.00	0.05	0.00	0.05	0.00	0.05	0.01	0.05
	10%	3.07	7.27	4.21	7.09	3.08	7.11	2.44	6.65
	20%	5.35	14.52	7.55	14.40	5.19	14.58	4.29	13.31
	time	2.6 s		1.6 s		3.2 s		27.3 s
				wavy transparent layer
	0%	1.12	2.17	1.18	3.95	1.15	2.31	1.14	1.39
	10%	5.58	10.20	7.56	11.42	5.49	11.31	4.57	8.15
	20%	10.54	19.48	15.23	21.90	10.25	18.73	8.39	15.71
	time	2.6 s		1.6 s		3.2 s		27.3 s

Table 4 ,

 4 Table

Table 5 Experiment result using pre-filter 2 and different envelope detection techniques (nm)

 5

		E		F		G		H
	Pre-			Suppress the offset: 1 st order central derivative
	filt.			Suppress the noise: Savitzky-Golay filter
	Env. Det.	HT		FSA		TKEO	CWT
	Post-	Smooth the amplitude envelope using cubic smoothing spline
	filt.	Correct the peak curve using Gaussian estimation and interpolation
				mean absolute error (nm)
	noise() 1 st surf. 2 nd surf. 1 st surf. 2 nd surf. 1 st surf. 2 nd surf. 1 st surf. 2 nd surf.
				flat transparent layer
	0%	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
	10%	0.59	4.31	0.92	6.05	0.59	4.19	2.51	4.23
	20%	1.67	9.50	2.24	13.01	1.60	8.72	2.50	7.27
	time	2.6 s		1.6 s		3.2 s		27.3 s
				wavy transparent layer
	0%	1.02	1.03	1.02	1.02	1.03	1.06	1.02	1.04
	10%	3.96	8.68	5.05	12.00	3.79	8.75	3.91	7.53
	20%	7.03	17.81	10.06	28.03	7.03	16.74	6.08	14.33
	time	2.6 s		1.6 s		3.2 s		27.3 s

Table 6 Experiment result using pre-filter 3 and different envelope detection techniques (nm)

 6 1 st surf. 2 nd surf. 1 st surf. 2 nd surf. 1 st surf. 2 nd surf. 1 st surf. 2 nd surf.

		I		J		K		L
	Pre-			Suppress the offset: EMD
	filt.			Suppress the noise: Savitzky-Golay filter
	Env. Det.	HT		FSA		TKEO	CWT
	Post-		Smooth the amplitude envelope using cubic smoothing spline
	filt.	Correct the peak curve using Gaussian estimation and interpolation
				mean absolute error (nm)
	noise() flat transparent layer
	0%	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
	10%	0.56	4.25	0.80	5.06	0.51	4.01	1.05	3.73
	20%	1.66	8.35	1.96	10.45	1.58	8.32	1.65	7.28
	time	5.9 s		4.9 s		6.5 s		30.6 s
				wavy transparent layer
	0%	1..07	1.06	1.02	1.02	1.05	1.03	1.02	1.05
	10%	3.71	8.62	4.26	10.41	3.61	8.57	3.46	7.26
	20%	6.89	17.07	9.04	23.18	6.54	16..35	5.84	14.02
	time	5.9 s		4.9 s		6.5 s		30.6 s

Table 7 Improvement of axial sensitivity of measurement obtained using Gaussian fitting (nm)

 7

	Pre-			Suppress the offset: EMD
	filt.			Suppress the noise: Savitzky-Golay filter
	Env. Det.		HT	FSA		TKEO	CWT
	Post-filt.		Smooth the amplitude envelope using cubic smoothing spline
				mean absolute error (nm)
	noise() 1 st surf.	2 nd surf.	1 st surf.	2 nd surf.	1 st surf.	2 nd surf.	1 st surf.	2 nd surf.
			flat transparent layer -without interpolation on Gaussian fitting
	0%	10.06	10.06	10.06	10.06	10.06	10.06	10.06	10.06
	10%	10.07	10.68	10.07	11.07	10.07	10.62	10.07	10.53
	20%	10.12	12.90	10.17	14.28	10.11	12.34	10.16	12.14
			flat transparent layer -using interpolation on Gaussian fitting
	0%	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
	10%	0.56	4.25	0.80	5.06	0.51	4.01	1.05	3.73
	20%	1.66	8.35	1.96	10.45	1.58	8.32	1.65	7.28

Table 8 Experiment result using pre-filter 3 and different envelope detection techniques (nm)

 8

		ROI-1	ROI-2	ROI-3	ROI-4
	axis/length (0 -9 µm)	(20 -45 µm)	(55 -64 µm)	(20 -45 µm)
			HT
	Rt (nm)	18.0	99.0	27.0	63.0
	Ra (nm)	4.8	16.5	5.7	8.8
			FSA
	Rt (nm)	18.0	108.0	27.0	63.0
	Ra (nm)	4.6	15.8	5.3	8.6
			CWT
	Rt (nm)	18.0	108.0	27.0	63.0
	Ra (nm)	2.9	16.7	6.2	8.2
			TKEO
	Rt (nm)	18.0	108.0	27.0	63.0
	Ra (nm)	2.8	17.3	5.9	8.3

Table 9 Mean absolute error (mae) of surface extraction corresponding to synthetic fringe image

 9

		noise=0%	noise=10%	noise=20%
	Methods	Surf1 Surf2 Surf1 Surf2 Surf1 Surf2
			mean absolute error (nm)
	1DFSA	2.1	2.1	6.0	27.9	11.5	57.3
	2DTKEO	2.1	2.1	3.2	12.3	5.1	25.2
	2DHT	2.1	2.1	3.2	13.0	5.2	28.5

Table 10 Mean value of mylar polymer thickness Mylar thickness (µm) for each of ROI 0-10 10-20 20-30 30-40 40-50 50-60 60-70 70-80 80-90

 10

	3.48	3.44	3.36	3.27	3.32	3.24	3.28	2.98	2.90

  Basé sur du profil de surface obtenu en utilisant le traitement du signal frange 1D (FSA), nous avons observé que la technique fournit des résultats satisfaisants pour mesurer la surface supérieure, mais qu'elle n'est pas suffisante pour la mesure précise des profondeurs des structures internes ou des surface arrière. Afin d'obtenir de meilleurs résultats sur le film polymère Mylar, la technique XZ-scan (2DTKEO) a été réalisée. Les résultats démontrent que l'enveloppe de frange XZ extraite par le traitement du signal de frange 2D fournit des résultats plus satisfaisants que le traitement du signal frange 1D en révélant les structures internes et la surface arrière, qui est contenue dans un film de polymère Mylar. La technique permet également une amélioration des détails dans les images XZ ainsi que des mesures plus précises de l'épaisseur du film polymère.

	de couches dans un échantillon multicouche en utilisant l'ajustement de la courbe
	gaussienne.
	Pour évaluer la robustesse de 2DTKEO, comparée à la 1DFSA, nous avons effectué
	des simulations, qui étaient appliquées respectivement à un signal sans bruit (σ = 0%) et à
	des signaux bruyants (σ = 10% et σ = 20%) en utilisant un signal synthétique. Afin
	d'améliorer la sensibilité axiale et la précision de la détection de crête, un lissage spline
	cubique suivi d'une interpolation gaussienne a été utilisé, qui est adapté au modèle
	physique de l'enveloppe. Basé sur des profils de surface des résultats, la comparaison des
	performances qualitatives peut être observée entre la technique XZ-scan utilisant
	2DTKEO et la technique Z-scan utilisant l'algorithme FSA. On peut observer que la
	technique XZ-scan (2DTKEO) fournit une meilleure extraction de surface que la FSA à la
	fois sur les surfaces supérieure et arrière. De plus, le taux d'erreur absolu moyen (mae) a
	été calculé par rapport à la surface de référence. En comparant les résultats de l'évaluation
	quantitative, la technique Z-scan (FSA) et la technique XZ-scan (2DTKEO) sont
	similaires en termes de performance dans le cas du signal sans bruit (σ = 0%) fournissant
	une valeur moyenne de mae = 2,1 nm. La raison de 2DTKEO devrait être plus robuste
	que la FSA est l'utilisation d'une dérivée 2D par le biais du filtre gaussien, en présence de
	surfaces lisses. Pendant ce temps, pour un signal bruyant (σ = 10% et σ = 20%), la
	technique XZ-scan utilisant 2DTKEO est généralement plus robuste que la FSA, ce qui
	est montré par le taux d'erreur (mae) 2 fois plus petit, correspondant au une plus grande
	précision.
	Ensuite, nous évaluons ces deux algorithmes sur des données réelles, c'est-à-dire un
	film de Mylar qui a une couche multiple constituée de la surface supérieure, des
	structures internes et de la surface arrière. Les signaux de franges provenant du film de
	Dans ce chapitre, nous étudions les performances du traitement des franges 2D en CSI polymère Mylar ont été obtenus en utilisant le microscope interférentiel Leitz-Linnik
	développé basé sur deux dimensions-Teager Kaiser Energy Operator sur un échantillon adapté. La méthode utilisant la technique de moyennage avec des corrections champ
	multicouche. Une analyse de franges robuste est nécessaire pour une caractérisation sombre et plate a été réalisée afin de réduire le bruit. Les franges de Mylar correspondant
	améliorée du film de polymère transparent, c'est-à-dire un film de Mylar, qui est à la surface supérieure ont un contraste élevé, contrairement aux franges sur la surface
	complexe et contient des structures internes et une couche enterrées avec un faible arrière, en raison de l'imagerie dans l'air. Même si la technique de moyennage a augmenté
	contraste. Les algorithmes sont comparés à la transformée de Hilbert bidimensionnelle le SNR, le bruit peut encore apparaître dans les signaux de franges résultants du film.
	classique et au traitement frange 1D en utilisant l'algorithme FSA bien connu. En outre,
	nous rapportons également l'étude de l'algorithme développé pour la détection du nombre

  ) represents the sampling period along the x-axis (resp. the y-axis and z-axis). The discrete counterpart of the operator Φ 2 denoted by 2 d  applied to the signal s(u) is obtained in the same way, by performing the sum of the absolute values of the matrix elements, as follows:

									Teager Kaiser Energy Operator. In a
	3D context, let us give a local discrete AM-FM signal s(i,j,k) = A(i,j,k) cos (Ω	x i + Ω	y j +
	Ω	z k) where Ω	x = ω	x x e , Ω	y = ω	y y e , Ω	z = ω	z z	e . The variable x e (resp. y e and z

e

  | + |sin Ω z | equals one. According to this remark, there are particular values of (ω x ,ω y ,ω z ) and (xe,ye,ze) for which we expect more efficient demodulations. Finally, applying the transforms to the partial derivatives of an AM signal leads to the following results:

) or (3), one can directly demodulate a local AM signal, provided that the factor |sin Ω x | + |sin Ω y

Table 11 Performance Comparison of Envelope Detection using 1D TKEO, 2D TKEO and 3D TKEO (nm)

 11

	Pre-			Suppress the offset: EMD
	filt.		Suppress the noise: Savitzky-Golay filter
	Env. Det.	Z-Scan Technique (1D Fringe Processing using 1DTKEO)	XZ-Scan Technique (2D Fringe Processing using 2DTKEO)	XYZ-Scan Technique (3D Fringe Processing using 3DTKEO)
	Post-		Smooth the amplitude envelope using cubic smoothing spline
	filt.		Correct the peak curve using Gaussian estimation and interpolation
				mean absolute error (nm)
	noise()	1 st surf.	2 nd surf.	1 st surf.	2 nd surf.	1 st surf.	2 nd surf.
				flat transparent layer
	10%	0.71	4.55	0.52	3.94	0.33	1.47
	20%	2.05	8.75	1.39	8.19	0.41	3.49
				wavy transparent layer
	10%	3.51	8.63	3.10	6.41	1.87	2.91
	20%	6.08	18.17	5.51	13.06	2.42	4.83

Table 11

 11 summarizes the results of the performance comparison of fringe signal processing using 1D TKEO, 2D TKEO and 3D TKEO. Based on the table, we observe that the fringe signal processing which takes into account the spatial neighbourhood (2D, 3D

Table 12 Calibration results of 3D fringe signal processing (3D TKEO) using Step Height Standards (SHS)

 12

	Microscope type	Step Height Standard (µm)	Measurement results (µm)	Error (µm)
	Leitz-Linnik Mic.		1.79	0.009
	"Michelin" Fogale Mic.	1.7803 ± 0.011	1.778	0.002
	"Immersion" Fogale Mic.		1.776	0.004

Table 12

 12

	and Fig. 97 show the calibration results of 3D fringe signal processing
	(3D TKEO) using the Step Height Standard (SHS). Based on the table, we observe that
	the measurement error generated by the Leitz-Linnik microscope, the new Fogale
	(Michelin) Microscope, and the "immersion" Fogale microscope using 3D fringe signal
	processing is still below the tolerance limit of the Step Height Standard (SHS) error,

having a value of 0.011 µm. The results show that the 3D fringe signal processing (3D TKEO) algorithm developed is compatible with the 3D microscopes in the IPP team and give accurate results.

  ). D'autre part, nous avons également développé et appliqué la version discrète de 3DTKEO afin d'analyser directement le signal de frange 3D. L'intérêt espéré de cette approche discrète de 3DTKEO est un avantage en termes de temps de calcul. Dans ce chapitre, nous rapportons l'étude de l'application de 3DTKEO discrète sur le traitement des signaux de franges, qui est comparée à la 3DTKEO continue. A titre de comparaison, nous utilisons des données réelles, c'est à dire un câble, comme échantillon. Nous commençons par l'observation de la topographie de l'enveloppe des franges obtenue en utilisant le 3DTKEO continue et le 3DTKEO discret. Les enveloppes de franges obtenues en utilisant les deux algorithmes fournissent des résultats satisfaisants qui ont un contraste élevé, les enveloppes de frange étant clairement visibles sur l'image. En regardant de plus prêt, l'enveloppe de franges obtenue en utilisant le 3DTKEO discret a un meilleur contraste que ce du 3DTKEO continu. L'amplitude d'enveloppe de frange a deux fois l'amplitude de l'enveloppe de frange obtenue en utilisant le 3DTKEO continu. Ensuite, avec une observation plus approfondie, nous avons constaté la présence de petits artéfacts (presque inobservables) qui apparaissent dans l'image obtenue en utilisant le 3DTKEO discret. Par contre, aucun artefact apparaît sur l'image obtenue en utilisant le 3DTKEO continue.Ensuite, nous observons les profils de surface obtenus par les deux algorithmes.Basé sur de ceux-ci, nous avons observé que la caractéristique / structure du profil de surface obtenue en utilisant le 3DTKEO Continu et Discret sont relativement similaires.Puis nous avons comparé les résultats quantitativement en calculant la déviation des profils de surface. La valeur moyenne de la déviation pour ce profil de ligne est de 9,9 nm (avec un pas d'échantillonnage de 90 nm). En ce qui concerne le temps de calcul du traitement du signal des franges 3D, le 3DTKEO discret a l'avantage d'être plus compact en termes de calcul. Par exemple, en appliquant les algorithmes sur des mesures d'un échantillon de câble, le 3DTKEO discret prend 16,53 s alors que le 3DTKEO Continu prend 246,19 s pour un signal de frange avec une taille de données de 251701100 pixels. Le 3DTKEO discret est bien meilleur en termes de temps de calcul que ce du 3DTKEO continu.

ACKNOWLEDGEMENTS

This PhD research would not have been possible without the help, the support, and the contribution of many people and institutions.

The results in Fig. 31 show the fringe envelopes obtained using wavelet families: (a)

Morlet, (b) complex Morlet, (c) Gaussian, (d) complex Gaussian, and (e) Mexican hat. In this simulation, we adjust the scale frequency of 0.1 in the CWT algorithm, as refered to in article [START_REF] Mingzhou | Development of fringe analysis techniques in white light interferometry for micro-component measurement[END_REF]. Based on these figures, it can be seen that only one of the wavelet families is suitable, i.e. the complex Morlet, which provides a well-defined fringe envelope, as shown in Fig. 31(b). The other wavelets: Morlet, Gaussian, Mexican hat provide a significantly noisier fringe envelope, as shown in Fig. 31(a-c-e). The fringe envelope obtained using these mother wavelets contain a fringe pattern. Meanwhile, the fringe envelope obtained using a complex Gaussian provides a well-defined envelope without containing a significant fringe pattern, as shown in Fig. 31(d). However, the envelope amplitude increases by almost twice as much as the original fringe envelope. Further evaluation is required to see whether the increase in amplitude produces an effect on the height surface information contained in the peak of the fringe envelope. In this work, we therefore tested the use of the complex Gaussian as a mother wavelet in the CWT algorithm in order to obtain the envelope for fringe analysis, which is then compared to the complex Morlet.

Savitsky-Golay filters in the prefiltering step in order to remove the offset component and to suppress the noise. In the post-processing step, we only use the Gaussian estimation and interpolation to correct the peak curve, without smoothing due to the fringe envelope obtained using the CWT algorithms (complex Gaussian and complex Morlet) that already provides a well-defined fringe envelope without containing significant noise and fringe pattern. Fig. 32 and Fig. 33 show examples of the results of signal processing for the wavy transparent layers obtained using the CWT algorithms (complex Gaussian and complex Morlet). processing. This means that even though the fringe envelope obtained using the CWT (Gaussian complex morlet) has the envelope amplitude that is twice the original envelope, this algorithm is still able to maintain the height surface information in its fringe envelope peak.

COMPARISON OF PRE-FILTERING TECHNIQUES

In order to remove the offset and reduce the noise, in this section we compare three different algorithms which combine the derivative/EMD technique and denoising filter (Savitzky-Golay/Wiener). The corresponding algorithms are referred to with the following names:

 Pre-filter 1: combination of derivative and Wiener filter,  Pre-filter 2: combination of derivative and Savitzky-Golay filter,  Pre-filter 3: combination of EMD and Savitzky-Golay filter.

For the performance evaluation of the above different pre-filters, we carry out the simulation by performing them on a synthetic fringe signal, with a 40 nm sampling step, as shown in Fig. 34(a). It can be noted that the scanning step height chosen in interference microscopy during the data acquisition is a fixed step both for PSM and the FSA algorithm (π/2, linked to the effective wavelength). The reason for this is that the phase used for the algorithm is fixed at π/2. Even though this step can be variable for other TKEO algorithms, since the same fixed scanning step height is used for FSA, it will also Besides that, we also used another parameter, i.e. the ratio of number of the detected surface and number of the reference surface. Based on the Fig. 59, it seems that the ratio value is 100%, which means that our algorithm is able to detect correctly the entire assembly of reference structures.

Chapter 4. 3D FRINGE PROCESSING IN CSI

In the previous chapter 2 and 3, we report the study of 1D fringe signal processing (Zscan technique) and 2D fringe signal processing (XZ-scan technique) in CSI. We have demonstrated the ability of 2D approaches to compete with some classical methods (1D approaches) used in the field of interferometry, in terms of robustness. In addition, whereas most methods only take into account the 1D data, it would seem advantageous to take into account the spatial neighbourhood using multi-dimensional approaches (2D, 3D).

In this chapter, we present the study of the application of the 3D Teager Kaiser Energy Operator (3DTKEO), which is developed based on the multi-dimensional energy operator (2009, Salzenstein and Boudraa) [START_REF] Salzenstein | A new class of multidimensional Teager-Kaiser and higher order operators based on directional derivatives[END_REF], [START_REF] Salzenstein | Multi-dimensional higher order differential operators derived from the Teager -Kaiser energy-tracking function[END_REF]. Through a simulation using a synthetic fringe signal, we evaluated the robustness of the 3DTKEO performance in fringe signal processing, which is compared to the 1D and 2D approaches. Moreover, we also used the algorithm on real data, i.e. a step height standard (VLSI Standard Inc.) in order to evaluate the measurement accuracy. In addition, we enrich the field of study by testing the algorithm on measurements from different samples: Graphene, DOE (Diffractive Optical Elements), Resin on Silicon, Cable and Rock.

XYZ-SCAN TECHNIQUE (3D)

Fig. 72 The technique of 3D fringe processing using 3DTKEO Three dimensional fringe signal processing can be chosen using the raw XYZ data by operating on the XZY images as shown in Fig. 72. The technique is defined as an "XYZscan" which consists in analyzing the XYZ fringe signals (three dimensional signals). By processing a given XYZ fringe signals, we can obtain the 3D fringe envelope directly.

3D TEAGER KAISER ENERGY OPERATOR

In this work, we have applied the developed algorithm for continuous 3D Teager Kaiser Energy Operator based on the multi-dimensional energy operator [START_REF] Salzenstein | A new class of multidimensional Teager-Kaiser and higher order operators based on directional derivatives[END_REF], [START_REF] Salzenstein | Multi-dimensional higher order differential operators derived from the Teager -Kaiser energy-tracking function[END_REF], for fringe signal analysis using white light interferometry. In addition, we have also introduced the mathematical model and its application of the discrete 3D Teager Kaiser Energy Operator in the same field of fringe signal processing in CSI.

We now introduce the general image processing model of 3D Teager Kaiser Energy

Operator for the three dimensions of AM-FM signal f(x,y,z) as follows: Based on Fig. 86, the qualitative performance comparison for obtaining the fringe envelope can be observed for the different types of fringe signal processing. As illustrated in the figure, it can be observed that the fringe envelope obtained using 2D and 3D fringe processing (taking into account the spatial neighbourhood) is better than 1D fringe processing. The shape of the envelope obtained using 2D and 3D fringe signal processing

The final part of this thesis work reports an application of the 3D-Teager Kaiser Energy Operator (3DTKEO), based on the multi-dimensional energy operator for fringe analysis in CSI. Through a simulation using a synthetic fringe signal, we have demonstrated the competitiveness of the 3DTKEO performance, compared with the 1D and 2D approaches. Furthermore, we tested the algorithm on real data, i.e. a step height standard (VLSI Standard Inc.) for measurement calibration in order to evaluate the accuracy. Based on these results, we observed that the error rate of the measurements on a calibration standard is still below the tolerance limit of the Step Height Standards (SHS) error (< 0.011 µm). In addition, we have tested the algorithm on different samples, such as Graphene, DOE (Diffractive Optical Elements), Resin on Silicon, Cable and Rock. The algorithm works very well and provides satisfactory surface profiles of the samples. On the other hand, we have also developed and applied the discrete version of 3DTKEO in order to analyse directly the 3D fringe signal. Comparing this with the continuous 3DTKEO, we found that the discrete approach of 3DTKEO gives the advantage of shorter computational time, while the continuous 3DTKEO is more robust to noise than the discrete 3DTKEO.

LIST OF PUBLICATIONS AND COMMUNICATIONS

International peer-reviewed journals With:

(1.6)

And < . >i,j denotes the average of the quantity (.) within the cell centered at the point (i,j).

Results of roughness measurements

The results of the roughness measurements of the unpolished samples using SCM and