his first Ph.D. student working in the field of terahertz science and technology. His patient guidance, enthusiasm for the research and encouraging support have led me to go through all the tough times in the Ph.D. pursuit and to grow as a researcher in this wonderful world of terahertz. I am also grateful for the excellent example he has provided as a successful physicist, professor, and mentor. I would like to thank Dr. Alexandre Locquet for his support and for all the discussions and suggestions he provided for my research work. Answering his questions was challenging sometimes, but I have to admit that it was helpful for a better understanding of the key to the problems. I would like to express my gratitude to Prof. Nico F. Declecrq for his guidance on ultrasonics. Comparative studies between terahertz and ultrasonic imaging play an important role in this dissertation. Doing experiments in the LUNE lab was a precious experience for me, because it enabled me take advantage of my ultrasonics background and further improve my skills.

In regards to the investigation of fiber-reinforced composites, I thank Pascal Pomerade, a Ph.D. student in ENSAM, for performing fatigue testing and introducing various damages and defects into the composite samples. Pascal shared his knowledge of materials and mechanics, which helped me understand the damage mechanisms and verify the imaging results. For the study of artworks, I would like to acknowledge our collaborator Marcello Melis, the CEO of Profilcolore srl in Italy. We worked together on the multispectral imaging of art paintings, and I very much appreciated his enthusiasm and the kindness for providing various artworks as samples.

For this dissertation I would like to express my gratitude to the committee memv bers: Prof. Waymond R. Scott, Prof. Justin Romberg, and Prof. Paul L. Voss for their time, interest, and insightful comments.

My time at Georgia Tech was made enjoyable in large part due to many friends that became a part of my Ph.D. life. Many thanks to my dear friends, Dr. Xin Li, Dr. Fan Xu, Yang Wu, Xiaolong Wu, and Chien-Yuan Chang for sharing all my happiness and sadness no matter for the life or for the research, and offering help and support whenever needed.

Lastly, I would like to thank my parents for all their unconditional love and encouragement. I would like to thank them for raising me with a love of science and always supporting me in all my pursuits. vi In modern industry, fiber-reinforced composites, which combine the merits of high strength and low weight, offer an alternative to conventional structural materials and are widely used for various applications including aerospace, automotive, renewable energy and marine. As a result, the wide applicability of fiber-reinforced composites has created the correspondingly need for advanced NDE techniques for inspection and failure detection during manufacturing and maintenance. Numerous NDE techniques capable of characterizing damages and defects in fiber-reinforced composites have been developed, such as ultrasound, shearography, IR thermography, and X-ray radiography, to name a few. However, until now, only the ultrasonic C-scan has found widespread implementation in industry, because of simplicity of analysis and its effectiveness in geometrically locating damages and defects. Ultrasonic C-scans can provide a good trade-off between material penetration and measurement resolution, and ultrasonic C-scans in pulse-echo mode can also provide qualitative information in depth for thick fiber-reinforced composite samples. However, some of the difficulties associated with ultrasonic C-scans for the NDE of fiber-reinforced composites have been highlighted: (1) negligible quantitative information in depth can be obtained in thin samples with small thickness-to-wavelength ratio due to the relatively large time duration of ultrasonic pulse signal; (2) because of the attenuation of ultrasonic waves in fiber-reinforced composites (especially in glass fiber-reinforced composites), the operating frequency cannot be sufficiently high (usually less than 10 MHz), thus limiting the resolution; and (3) liquid coupling may be required. Although contactless ultrasonic techniques using laser and air-coupled transducers have been proposed, problem

(1) and (2) remain. Therefore, alternative nondestructive, noncontact, and nonionizing techniques with high resolution are still needed for inspection of fiber-reinforced composites.

In the steel industry, NDE techniques for monitoring and characterizing coatings on steel are essential to verify protection of the steel surface from corrosion during service. Coating failure can have many causes and manifestations. Especially, when a region of a coating system becomes detached from its substrate, the term 'adhesion failure' is commonly used. Delamination and blistering are two important types of coating failure in which compromised adhesion is strongly implicated. Since the exact cause and nature of coating failure is still in dispute, various NDE techniques have been explored to study damage mechanisms in coatings, such as electrochemical impedance spectroscopy (EIS), scanning electrochemical microscopy, acoustic emission, and thermography. However, not all of the NDE techniques mentioned above can provide quantitative information in depth, nor may they all be capable of monitoring the condition of adhesion. Scanning acoustic microscopy (SAM) and laser-ultrasonics can provide depth-specific information with enough resolution for characterizing coating systems; however, high attenuation in polymer materials limits the penetration depth of the ultrasonic waves. Therefore, alternative NDE techniques with relatively high resolution are still sought for quantitative evaluation of polymer-coated steel.

Great demand for advanced NDE techniques also appears in the field of cultural heritage conservation science, as preliminary noninvasive investigations are necessary before starting any conservation procedure on valuable artworks, in order to assist conservators in their decision-making process. The use of imaging techniques in the art conservation field ranges from well-established methodologies, such as high-resolution digital photography, visible raking light, UV fluorescence, X-ray radiography, IR reflectography, ect., to cutting-edge applications, such as multi-and hyper-spectral imaging techniques. In principle, all the above techniques could be used in order to reach the desired level of knowledge on the object under analysis. Nevertheless, most of these techniques lack penetration ability and can be only used to study surface or near-surface features. X-ray radiography is capable of penetration and can be used for investigating hidden features or defects. However, X-ray radiography cannot penetrate layers of high density pigments and cannot detect organic materials. X-ray radiation is ionizing, which is unsafe for humans. Moreover, all these techniques are lack of depth resolution and convey little about the structure of the artworks. THz imaging was first applied to the NDE of fiber-reinforced composites in 2006 [1] and has been explored extensively since. THz imaging enables to measure and determine the amplitude and the phase of a THz pulse transmitted through a sample.

Changes in the amplitude and the phase of the THz field are related to the complex refractive index of the material. A spectroscopic approach can be implemented as the THz spectrum can be achieved based on the Fourier analysis. Hence, both parts of the frequency dependent complex refractive index can be obtained, which are the key parameters of materials. The basic approach of material parameter esti-mation with THz waves involves constructing a theoretical model that incorporates the etalon effect and solving the inverse problem using a numerical algorithm to find complex refractive index values that fit the measured data to the model [2]. This approach is designed for estimating the material parameters of isotropic and homogeneous materials and can also be modified to the reflection configuration [3]. However, fiber-reinforced materials are anisotropic and inhomogeneous. For fiber-reinforced composites, the fiber fabrics are considered as the inclusion and the polymer matrix are considered as the host material. When the inclusions are small enough compared to the THz wavelength, effective medium approximations [4] can be used to equate the dielectric properties of individual components to the dielectric properties of their composite material. Naito et al. measured the dielectric properties of plain woven fabric glass fiber-reinforced epoxy matrix and eight-harness-stain fabric glass fiber-reinforced polyimide matrix composites in the THz range [5].

Besides the dielectric properties, fiber content and fiber orientation are also key parameters as both of them determine the mechanical performance of composites. Composite materials also suffer from burn damages due to high and periodic thermal stresses. Burn damage usually arises in the surface of composites where thermal gradients are the highest. Stoik et al. showed that THz imaging can detect burn damage in glass fiber composites generated by heating the part at 440 • C for several minutes [START_REF] Stoik | Nondestructive evaluation of aircraft composites using reflective terahertz time domain spectroscopy[END_REF]. Such a heating generated relatively big blisters of few centimeters size which were easily detected with THz imaging. Karpowicz et al. applied THz imaging to evaluate fire damage to a variety of carbon fiber composite samples. The THz polarization-dependent reflectivity was studied and how the polarization dependence changes versus the burn damage level was also demonstrated [START_REF] Karpowicz | Fire damage on carbon fiber materials characterized by THz waves[END_REF].

Mechanical stresses like long-term mechanical bending cycles, produce a continuous wearing out of the composite part while in-service. In consequence, the desired composites compactness and fiber matrix interaction on the composites part gets distorted, leading to detrimental effects in its performance. Stoik et al. applied THz transmissive imaging to detect the damages caused by mechanical fatigue generated by 6240 bending cycles in glass fiber composites. THz images clearly exhibited the damage areas with cracking and buckling across the central bend axis [START_REF] Stoik | Nondestructive evaluation of aircraft composites using transmissive terahertz time domain spectroscopy[END_REF].

Voids or air bubbles are usual defects that arise during the manufacturing process.

Based on the lower THz absorption of air, THz radiation is suitable for void detection in the surface and inner regions of composites. Stoik et al., demonstrated the detection of hidden circular voids of 3 mm diameter with THz imaging [START_REF] Stoik | Nondestructive evaluation of aircraft composites using reflective terahertz time domain spectroscopy[END_REF]. Moreover, composite materials are also widely used in adverse environmental conditions, i.e. offshore wind turbine blades. In such conditions, water usually diffuses into the composite part in service with time, then further degrading its performance. The relatively high permittivity of liquid water compared to other materials in the THz range enables a contrast mechanism for the detection and imaging of moisture. Jördens et al. employed THz spectroscopy to study the absorption of water into polyamide and wood plastic composite. A model for the dielectric behavior depending on the water content was developed and experimentally verified, which proved that THz waves could be considered as an ideal tool for the NDE of the water content [START_REF] Jördens | Investigation of the water absorption in polyamide and wood plastic composite by terahertz time-domain spectroscopy[END_REF].

THz imaging has shown great potentials for the NDE of fiber-reinforced composites. However, comparative studies with other NDE techniques are required in order to highlight the performance of THz imaging for this application. Despite the carbon fibers are conductive, which limits the penetration depth of THz waves, THz imaging can also be utilized to perform subsurface investigation of the carbon fiber-reinforced composites based on the THz sensitivity to the carbon fiber distortion/breakage.

Terahertz characterization of multilayered coatings on metal

The most promising industrial application is the vehicle paint quality control in the automotive industry. Conventional methods to measure the thickness of car paint layers that are commercially available, such as magnetic gauges, eddy current measurements, and ultrasonic testing, all require direct contact between the measurement sensor and the painted car surface. These measurement techniques can only cover a limited number of sampling points on selected cars and hence lack the capability to identify paint defects, monitor drying processes, and map the thickness distribution of the paint layers over a larger surface of the vehicle. New possibilities to overcome these restrictions have been shown by THz imaging, and lots of efforts have been made to improve the capability of THz imaging to characterize multilayered coating structures.

In principle, the individual thicknesses of multilayered coatings can be directly determined using time of flight measurements of ultrafast THz pulses. The absolute thickness of a layer can be determined from the optical delay between two echoes, which are reflected from its front and back surface. For optically thick samples, the reflected adjacent echoes are separated in the time domain so that the individual thickness can be calculated directly from the time delay. However, if the layer is optically thin, the echoes may partially or totally overlap in time. If the exact structure of the multilayered coating is known, efforts can be made to increase the resolution of the minimum layer thickness. In [START_REF] Yasuda | Improvement of minimum paint film thickness for THz paint meters by multiple-regression analysis[END_REF], a multiple regression analysis approach using the least-square fitting algorithm was proposed, which decreased the minimum measurable paint film thickness to 20 µm; in [START_REF] Su | Terahertz sensor for non-contact thickness and quality measurement of automobile paints of varying complexity[END_REF], Su et al. extended this numer-ical parameter fitting method by integrating the etalon effect, the absorption and dispersion of the coating materials, which further decreased the minimum measurable thickness down to 18 µm for both single and multilayered automotive paints; in [START_REF] Krimi | Highly accurate thickness measurement of multi-layered automotive paints using terahertz technology[END_REF], an advanced regression procedure with a self-calibration model is presented to measure individual automotive paint coatings in complex multilayered structures.

Due to the high robustness of the proposed self-calibration method and the genetic optimization algorithms, the approach has been applied to resolve individual layer thickness within multilayered paint samples down to 4 µm.

Due to the high sensitivity of THz waves to water contents, THz imaging can also be considered as an effective tool to monitor the drying progress in paint film.

In [START_REF] Yasui | Terahertz paintmeter for noncontact monitoring of thickness and drying progress in paint film[END_REF], adequate parameters for the drying progress can be extracted from the THz pulse-echo signal, and effectively applied to monitor the wet-to-dry transformation.

The THz imaging can be a powerful tool for quality control of the paint film on the in-process monitoring of car body painting.

Surface roughness measurement is frequently used in industrial applications. THz imaging can be used for noncontact measurement of the surface roughness of metal or dielectric materials in the order of 10 µm [START_REF] Fukuchi | Surface roughness measurement using terahertz waves[END_REF]. This range of surface roughness is too large for optical sensors, but too small for microwave sensors. The surface roughness is obtained from the effective reflectivity of the surface, which is the ratio of the spectral intensity of waves reflected from the specimen and from a reference metal plate, whose reflectivity can be assumed to be 1. In [START_REF] Jagannathan | Characterization of roughness parameters of metallic surfaces using terahertz reflection spectra[END_REF], the rms roughness of six different aluminum samples spanning a rms roughness of 5-20 µm was accurately determined by analyzing THz reflectance spectra. The rms roughness obtained by this method has good agreement with the rms roughness obtained using a surface profilometer. Moreover, it also showed that the THz reflectance spectra were able to discern two rough samples that differed in rms roughness by approximately 1 µm. It demonstrated that THz imaging can be used to identify and quantify material degra-dation in metallic surfaces and the feasibility of using THz technology in industrial applications.

THz imaging can be used to study the degradation of coated steel. If the corrosion/rusting areas under the opaque coating can be detected at an early stage, both the repair costs and damage to the substrate steel could be reduced. So far, several papers related with the corrosion, rusting and iron-oxides have been published

by Japanese researchers. Table I from [START_REF] Fuse | Evaluation of applicability of noncontact analysis methods to detect rust regions in coated steel plates[END_REF] summarizes several diagnostic techniques used for detection of rust and defects. Different noncontact optical analysis methods were also used to the detection of rust in coated steel plates. This paper concludes that neither XRF and FTIR can detect rust region hidden by the coating. Reflection intensity measurements from a sample coated with epoxy resin show that optical coherence tomography using near-infrared (IR) light and THz waves can pass through the coating. This region can be reproduced by THz imaging without contact. In contrast, neither X-ray fluorescence nor Fourier transform infrared spectroscopy can detect the rust, because of the strong X-ray and IR absorption peaks of the coating.

Typical THz imaging result in [START_REF] Fuse | Evaluation of applicability of noncontact analysis methods to detect rust regions in coated steel plates[END_REF] shows that THz imaging can effectively reveal the rust areas under the coating.

It is known that iron-oxide at room temperature is an insulator or metal depending on the composition and crystal structure. The complex refractive indices at the THz frequencies of hematite (α-F e 2 O 3 ), magnetite (F e 3 O 4 ), and wustite (F eO) are investigated in [START_REF] Hasegawa | Thickness measurement of ironoxide layers on steel plates using terahertz reflectometry[END_REF]. This paper shows that hematite and wustite are low-loss dielectrics or insulators without free carriers at room temperature. The typical THz amplitude transmittance spectra of the hematite samples is shown, and the amplitude transmittance spectra of wustite also show similar behavior similar to the hematite.

Based on the spectra, the complex refractive index of hematite can be extracted. The real refractive index is determined to be 5.5, and this value is almost independent of the frequency. Based on this value, the minimal thickness of hematite layer which can be measured is estimated about 10 µm. On the other hand, this paper also shows that magnetite is metallic at room temperature. It has been known that magnetite is transformed from metal to insulator at temperatures below 120 K or at pressures of up to 40 GPa. However, achievement of the temperature and pressure for the insulator phase is unrealistic during the process of manufacturing steel. Therefore, in this paper, annealing of the magnetite layer on the steel plate with laser irradiation is proposed. Annealing at temperatures above 580 to image the substructure of paintings on canvas by monitoring the time delay between reflections off different layers and also determine the thickness of the umber underdrawings [START_REF] Adam | Terahertz imaging of hidden paintlayers on canvas[END_REF]. In this study, the authors indicated that the smallest absolute thickness measurable with their THz-TDS system, which is about 50 µm by assuming a refractive index of 1.9 for raw umber. Furthermore, they also estimated that it was still possible to measure thicknesses of about 12 µm by means of the changes in the [START_REF] Fukunaga | Terahertz spectroscopy applied to the analysis of artists' materials[END_REF]. This study could be considered as the first application of THz reflective imaging on an actual tempera panel painting which belongs to one of the most important museums in the world.

Gilded panel paintings are a particular kind of art developed in the early medieval age. With THz reflective imaging, gold leaves beneath the paint layers were clearly observed and the layered structure of this panel painting was also revealed based on the reflections in the time domain. During the early Renaissance, painters started to prefer canvases to panels as supports for their paintings because they were much affordable and could be folded for more convenient transportation. Compared with wood panels, canvases are more transparent to THz waves, which enable THz imaging to identify hidden features, such as the artist's signature [START_REF] Seco-Martorell | Goya's artwork imaging with Terahertz waves[END_REF], under-drawings and modifications due the reuse of an earlier painting or canvas [START_REF] Koch-Dandolo | Reflection terahertz time-domain imaging for analysis of an 18th century neoclassical easel painting[END_REF][31] [START_REF] Fukunaga | THz pulsed time-domain imaging of an oil canvas painting: a case study of a painting by Pablo Picasso[END_REF]. Portable and compact up-to-date THz-TDS systems also provide the opportunities for in-situ analysis of wall paintings. THz imaging has been utilized to detect wall paintings whitewashed with lead white, which is opaque to infrared imaging and X-ray radiography. THz imaging can map pigments on wall paintings as well as evaluate the extent of cracks into the plaster [37][39]. The state-of-the-art studies of THz imaging for the NDE of actual art paintings are summarized in Table 1.1.

Besides the area of art paintings, THz imaging has also been applied for characterizing of features in stone sculpture and in architecture [START_REF] Krügener | Terahertz meets sculptural and architectural art: Evaluation and conservation of stone objects with T-ray technology[END_REF], wooden objects [START_REF] Koch | THz-imaging: a new method for density mapping of wood[END_REF] and tree rings for dendrochonological purposes [START_REF] Jackson | Terahertz pulse imaging for tree-ring analysis: a preliminary study for dendrochronology applications[END_REF], clay artifacts [START_REF] Labaune | Investigation of Terra Cotta artefacts with terahertz[END_REF], written papyrus [START_REF] Labaune | Papyrus imaging with terahertz time domain spectroscopy[END_REF], ancient mummies [START_REF] Öhrström | Terahertz imaging modalities of ancient Egyptian mummified objects and of a naturally mummified rat[END_REF] and archaeological bones [START_REF] Jackson | Terahertz pulse imaging in archaeology[END_REF]. These very promising first results reported need to be extended to a more extensive set of case studies, in order to better refine the experimental methodologies.

Scope of the thesis

The objective of this thesis is to explore the potentials and enhance the capabilities of THz imaging for the NDE and material characterization in the field of industry and culture heritage conservation science.

Chapter 1 introduces the background of this thesis. The basic knowledge of THz radiation is briefly introduced. The motivation and research problems of this project are described. The state-of-the-art of the development and applications of THz imaging in the fields of industry and cultural heritage conservation science are also summarized in this chapter.

Chapter 2 presents the main equipment, THz time-domain spectroscopy (TDS) system, utilized in this project. The main imaging contrasts, based on which the THz results are presented, are briefly described.

Chapter 3 is devoted to the NDE of fiber-reinforced composites with THz imaging. Fiber-reinforced composites are widely used in the industry to take the place of conventional materials, such as metal. In this chapter, three case studies are presents.

The first case study shows the capability of THz imaging in providing 3D quantitative information for characterization of delaminations in glass-fiber reinforced composites.

The spatial resolution, including the lateral resolution and the axial resolution of

CHAPTER 2 EQUIPMENT AND IMAGING CONTRASTS

This chapter gives a brief introduction of the main experimental equipment, as well as the basic imaging contrasts utilized in this thesis.

Terahertz time-domain spectroscopy system

A THz time-domain spectroscopy (TDS) and imaging system (Teraview TPS Spectra 3000) is employed in this thesis, shown schematically in Fig. 2.1. The GaAs photoconductive antenna is excited by an ultrafast (femtosecond) laser to produce roughly single-cycle THz pulses with bandwidth extending from 60 GHz to 3 THz. The ultrafast laser used here is an Er-doped fiber laser that emits 780 nm pulses with sub-100 femtosecond pulse duration at a repetition rate of 100 MHz and has an average output power in excess of 65 mW. Coherent detection of the THz radiation is performed in a similar photoconductive antenna circuit. By gating the photoconductive gap with a femtosecond pulse synchronized to the THz emission, a current proportional to the THz electric field is measured. By varying the optical path length, the reflected THz time-domain pulse can be sampled, resulting in both temporal amplitude and phase information. By taking the Fourier transform, spectroscopic information can be obtained. The power of the THz radiation used for measurement is below 1 µW, so no thermal strain is induced in the samples.

Both reflection and transmission imaging can be performed in this system. The sample is raster-scanned by a set of motorized stages moving in X and Y directions.

The amplitude and phase of the THz pulse transmitted or reflected through the sample is measured. Refractive-index discontinuities as a function of depth produce multiple reflections off the various associated interfaces in the samples. These reflec- In conjunction with the fiber probes on the existing TDS system, a remote stage is established to mount the probes and allow angular measurement and raster-scanning of a sample. The sample will be positioned on a horizontal motorized XY stage with the emitter and detector probes on manually repositioned mounts to allow angle coverage from reflection to transmission of the sample. This module, shown in Fig. 2.2, has the following features: (1) Measurement in both specular and diffuse reflection;

(2) Angular movement of the detector and emitter from near normal incidence accommodate reflections from rough surfaces. Manual adjustment of the incidence angle of the emitter will also be provided to change the incidence angle in reflection mode.

(3) 30 cm by 30 cm scanning area. ( 4) XY point spacing down to 50 µ m step size in the raster-scanning. ( 5) Purged lines to both emitter and detector probes.

Imaging contrast mechanisms

Generally speaking, the applications of THz imaging in the fields of industry and cultural heritage conservation science will most likely involve the the analysis of solid samples that are highly transparent to THz radiation and where contrast is achieved by a change in refractive index, absorption coefficient or scattering.

The main contrast mechanism that is typically exploited in THz imaging is the difference in refractive index between two materials. In time-domain THz imaging, the electric field E of the electromagnetic waves, rather than its intensity, is measured and the power reflectivity R is

R = E interf ace 2 E 0 2 = (n 1 -n 2 ) 2 (n 1 + n 2 ) 2 , (2.1)
where E interf ace denotes the electric field of THz waves of the reflection from the sample interface and E 0 is the reflection from a reference, and n 1 and n 2 are the refractive indices of the two media that the THz pulse is propagating through. As the THz pulse is propagating through the sample, a reflection will occur whenever there is a change in refractive index. This change in refractive index corresponds to the change of the dielectric constants.

Besides the change of the refractive index, in some crystalline materials, the intermolecular vibrational modes fall into the THz range, and therefore, a significant change in the absorption coefficient at specific frequencies can be observed, which correspond to the distinct frequencies of the vibrational modes. Polar liquids can lead to strong attenuation of the THz radiation, which provides the potential of THz imaging for moisture detection. Moreover, the conductivity σ of the sample is also related to the dielectric constants, which means, with increasing conductivity, both the absorption coefficient and refractive index also increase.

Besides the imaging contrast based on the amplitude of THz pulse, the phase information of THz pulse, such as the time of flight and time of arrival, can also be utilized as a contrast.

CHAPTER 3 TERAHERTZ NONDESTRUCTIVE EVALUATION OF FIBER-REINFORCED COMPOSITES

In this chapter, THz reflective imaging is systematically carried out for nondestructive evaluation of various damages or defects in fiber-reinforced composites. For glass fiber-reinforced composites, THz waves can be utilized to detect both surface and underlying defects, including mechanical/heat damage, voids, delaminations, intrusions, and moisture contamination. For carbon fiber-reinforced composites, due to the conductivity of carbon fibers, THz imaging can be used to detect surface or near-surface damages, such as impact-induced matrix cracking and fiber distortions/breakage. This chapter presents three case studies. The first study is to characterize forced delaminations in glass fiber-reinforced composite laminates. This is a perfect example to show the capabilities of THz imaging for the NDE of fiber-reinforced composites.

In this case, we find high-resolution, low-artifact THz C-scan and B-scan images locating and sizing the delamination in three dimensions, which enable us to determine the thicknesses of the delamination and of the layers constituting the laminate. Comparative analysis between THz imaging and ultrasonic C-scans with regard to spatial resolution is carried out demonstrating that THz imaging provides higher spatial resolution for imaging, and can be regarded as an alternative or complementary modality to ultrasonic C-scans for this class of glass fiber-reinforced composites.

The second study is to characterize a woven glass fiber-reinforced composite laminate with a small region of forced delamination, which is below the THz axial resolution, resulting in one featured echo with small amplitude in the reflected signal.

However, low-amplitude components of the temporal signal due to the ambient water vapor produce features of comparable amplitude with features associated with the delamination, which suppress the contrast of THz C-and B-scans. Wavelet shrinkage de-noising is performed to remove water-vapor features, leading to enhanced THz Cand B-scans to locate the delamination in three dimensions with high contrast.

The third study is novel, as the THz polarization is employed to enhance the imaging capability of THz radiation. In this study, THz reflective imaging is applied to characterize a hybrid fiber-reinforced composite laminate subject to low-velocity impact. The hybrid fiber-reinforced composite laminate comprises unidirectional glass/epoxy and carbon/epoxy plies with a cross-ply stack pattern. Both impactinduced intra-and inter-laminar damages are successfully detected, and the damage evolution throughout the thickness is also evaluated. The interaction between the THz polarization and carbon fiber orientation is investigated in detail. Inter-laminar damage at the interface and the intra-laminar damage close to the same interface can be differentiated via polarization-resolved imaging. is the difference between the maximum and minimum values of the reflected THz pulse in a selected time slice between 13 ps and 20 ps. This was chosen to maximize contrast between regions containing delamination and regions without delamination. The THz pulse encountering a region containing a delamination leads to higher difference between the maximum and minimum reflected amplitude in the relevant time window due to the higher difference of the refractive index, which corresponds to the lighter two circular regions in Fig. 3.1(a). 50% rule or 6 dB drop method can be utilized to estimate the size of the delamination. This method locates the pixel with the highest value in the C-scan image and assigns this a value of 100%, then colors all pixels red that have a value of at least 50% of the maximum. and analysis show that THz imaging can provide low-artifact images (time domain C-scan and B-scan) and thus three-dimensional information as well as quantitative information on delamination in glass fiber-reinforced composite laminates.

Comparison with ultrasonic C-scans

A customer-designed ultrasonic scanner fabricated by Inspection Technology Europe BV is used for ultrasonic C-scan experiment. The transducer chosen for this investigation is a focused-immersion transducer with a manufacturer-provided central frequency of 5 MHz considering both the attenuation and resolution. Ultrasonic Cscans were performed on the samples with water coupling under both transmission (pitch-catch) mode and reflection (pulse-echo) mode. Comparison between THz imaging and ultrasonic C-scans can be performed with respect to the spatial resolution of images obtained from both cases. Spatial resolution contains two parts: lateral and axial resolution. Lateral resolution is the minimum distance that can be differentiated between two point scatters across the scan plane.

The lateral resolution is high when the focal spot size of the beam is small. For THz imaging, the focal spot size of THz beam is frequency dependent, the higher the frequency, the smaller the spot size. For the THz TDS system used in this thesis, the lateral resolution at the surface is about 0.3 mm at 1 THz. For ultrasonic Cscans, lateral resolution is 1.95 mm for the focused transducers at 5 MHz. Therefore, THz imaging can achieve higher lateral resolution than ultrasonic C-scans, which can be corroborated by the comparison between the results of sizing the delamination by both techniques. THz imaging can provide more accurate results of sizing the delamination than ultrasonic C-scans, especially for the small delamination.

Axial resolution is the minimum distance that can be differentiated between two points in depth. Axial resolution is equal to half the spatial pulse length; it is high when the spatial pulse length is short. For THz imaging, the axial resolution is determined by

d = c∆T w 2n , (3.1)
where ∆ T w is the temporal width of the THz pulse. The axial resolution in the THz TDS system is estimated to be about 45 µm with ∆ T w = 0.65 ps. For ultrasonics, thee spatial pulse length is equal to the product of the number of cycles in a pulse and the wavelength. The acoustic wavelength at a frequency of 5 MHz is about 500 µm in the glass fiber composite samples. The different axial resolution can also be identified from THz and ultrasonic waveforms. As mentioned before, the various timedomain features associated with the THz pulse encountering each lamina interface as well as the delamination. By contrast, the timescale of ultrasonic pulses are quite different. The relatively long time of duration of ultrasonic pulse leads to overlapping echoes in the small thickness-to-wavelength ratio samples and makes it difficult if not impossible to clearly identify features on the length scale of the laminae thickness and delamination. The fundamental point is that the ultrasonic axial resolution is not sufficient to obtain detailed B-scan images in this case. Employing considerably higher acoustic frequencies with shorter wavelength, but the attenuation in this frequency range is prohibitively high in the samples, which in turn led to no discernible second echo signal in reflection.

Therefore, THz imaging can provide higher spatial resolution than the ultrasonic C-scan technique for imaging the glass fiber-reinforced composite samples. Ultrasonic axial resolution is not sufficient to provide the quantitative information in depth to identify the number of laminae nor the thickness of the delamination.

Enhanced terahertz imaging of small delaminations in woven glass fiber-reinforced composites

In this study, THz imaging is applied to characterize small forced delamination in woven glass fiber-reinforced composites. The forced delamination is created by inserting Upilex film with thickness only about 25 µm. THz echoes reflected from the front and rear interfaces of the forced delamination cannot be readily separated in the raw signal, leading to only one distinguishable echo in time associated with the delamination. The amplitude of this featured echo is small, and is comparable to the fluctuations resulting from the ambient water vapor. Wavelet shrinkage de-noising is performed here to suppress these fluctuations and to obtain enhanced THz C-and B-scan images for characterizing the delamination in three dimensions. In the case of a sample with forced delamination, the temporal reflected THz pulses can be estimated, based on considerations shown schematically in Fig. 3.9.

Delamination E 1st (t) E gd (t) E dg (t) Sample E 3rd (t) E in (t)
For normal incidence on an interface between two media with refractive indices n i and n j , the amplitude reflection coefficient r ij and transmission coefficient t ij are

r ij = n i -n j n i + n j (3.2) t ij = 2n i n i + n j (3.3)
In our study, the subscripts (i,j) for different media are: a-air, g-glass fiber-reinforced composites and d-forced delamination.

The reflected THz time-of-flight waveform is composed of the echoes off interfaces between various media. In principle, the temporal separation between echoes is proportional to the distance between interfaces (the thickness of the forced-delamination).

And the minimum distance discernable from the signal, viz. the axial resolution d min , is

d min = c∆T 2n d (3.4)
where c is the in-vacuo speed of light, ∆T is the temporal width of the THz pulse, and n d is the group refractive index of the medium within the forced delamination. When the forced-delamination thickness is larger than the axial resolution, two distinct THz echoes from the consecutive interfaces are well separated in time; otherwise, the two THz echoes cannot be easily distinguished. Therefore, in our case with thin delaminations below the axial resolution, the reflected THz waveform E T Hz (t) is expected to be composed of three clear features,

E T Hz (t) = E 1st (t) + E 2nd (t) + E 3rd (t) (3.5)
The first and third terms, E 1st (t) and E 3rd (t), correspond to Fresnel reflections at the bottom and top surfaces of the sample. And the second term echo, E 2nd (t), is the superposition of the two echoes returned from the interfaces associated with the delamination. In our case, we neglect the negligible dispersion in the relevent spectral range, so this featured echo E 2nd (t) can be expressed based on the incident THz pulse

E in (t), E 2nd (t) = E gd (t) + E dg (t) = t ag r gd t ga E in (t - 2n g d c )e -2αd
+ t ag t gd r dg t dg t ga E in (t -

2n g d c - 2n d d c )e -2α d e -2αd (3.6)
with d the distance between the delamination and the surface of the sample, d the thickness of the delamination, α and α the absorption coefficients in the glass fiberreinforced composites and in the forced delamination. Because the signs of the reflection coefficients (r gd and r dg ) at the interfaces between the glass fiber-reinforced composite and the delamination are opposite, the two echoes E gd (t) and E dg (t) will tend to cancel with each other, leading to the small amplitude of the associated signal E 2nd (t). Moreover, the signal amplitude decreases with decreasing delamination thickness [START_REF] Schwerdtfeger | Beating the wavelength limit: three-dimensional imaging of buried subwavelength fractures in sculpture and construction materials by terahertz time-domain reflection spectroscopy[END_REF].

Influence of water vapor

In the THz frequency region, many polar gases possess pronounced rotational transitions, leading to numerous spectral resonances. Water vapor, as the third most abundant gas in the terrestrial atmosphere, is known to have many rotational resonances in the THz frequency region [START_REF] Mittleman | Sensing with terahertz radiation[END_REF]. Therefore, THz imaging of a sample in the open air results in a combination of the sample's spectral features and ambient water-vapor resonances. In the time domain, this results in ringing-THz fluctuations-after the dominant THz pulse [START_REF] Withayachumnankul | Numerical removal of water vapour effects from terahertz time-domain spectroscopy measurements[END_REF]. Provided the features of interest are sufficiently large, these fluctuations can be ignored. However, when the real features from the sample are of comparable amplitude, they may be obscured by features associated with water vapor, leading to THz images with low contrast. Clearly, ridding the system of ambient water vapor, such as by purging with dry nitrogen, can suppress these fluc-tuations; however, it is not always possible to enclose the entire THz beam path in many practical laboratory and field applications. Several numerical approaches have been studied to remove the water-vapor fluctuations, including adaptive deconvolution [START_REF] Withayachumnankul | Numerical removal of water vapour effects from terahertz time-domain spectroscopy measurements[END_REF], absorbance spectrum fitting [START_REF] Wang | Terahertz absorbance spectrum fitting method for quantitative detection of concealed contraband[END_REF], artificial neural networks [START_REF] Ryu | Atmospheric degradation correction of terahertz beams using multiscale signal restoration[END_REF], and wavelet de-noising [START_REF] Chen | Improving extraction of impulse response functions using stationary wavelet shrinkage in terahertz reflection imaging[END_REF].

Wavelet de-noising

Wavelet shrinkage is a signal de-noising technique based on the idea of thresholding the wavelet coefficients. Mittleman [START_REF] Mittleman | Gas sensing using terahertz time-domain spectroscopy[END_REF] first suggested the use of wavelet transforms for THz signal processing because the THz pulse is similar in form to common wavelet basis functions. Compared with the general discrete wavelet transform (DW T ), the stationary wavelet transform (SW T ) is never sub-sampled and instead up-sampled at each level of decomposition. SW T decomposes a 1D signal x(n) into the approximation coefficients vector cA k,l and detail coefficients cD k,l by convolving with a low-pass filter Ψ and a high-pass filter Φ along the temporal axis at each level [START_REF] Chen | Frequency-wavelet domain deconvolution for terahertz reflection imaging and spectroscopy[END_REF],

shown in Fig. 3.10. Wavelet coefficients with small absolute values can be considered as noise, and wavelet coefficients with large absolute values are regarded as the main featured information of the signal. Removing the small absolute value coefficients by thresholding and then reconstructing the signal is expected to produce a signal in which the contribution of noise has been reduced.

In general, wavelet shrinkage de-noising with soft thresholding is performed via the following procedure [START_REF] Chen | Frequency-wavelet domain deconvolution for terahertz reflection imaging and spectroscopy[END_REF][START_REF] Ferguson | De-noising techniques for terahertz responses of biological samples[END_REF]:

1. Determine the wavelet coefficients by taking the SW T ,

[cA, cD] = SW T (x(n)) (3.7) ( )
x n ( ) where σ is the noise level and N is the number of sampling points.

1 n F ( ) 1 n Y ( ) 2 n F ( ) 2 n Y ( ) 3 n F ( ) 3 n Y 1,l cD 1,l cA 2,l
3. Threshold the wavelet coefficients cD k,l with solft-thresholding Before performing wavelet shrinkage de-noising to the reflected THz pulses, we implemented a numerical parameter fitting method based on multiple regression analysis [START_REF] Yasuda | Improvement of minimum paint film thickness for THz paint meters by multiple-regression analysis[END_REF] to make sure that the echoes between the featured echoes corresponding to the bottom and top surfaces in the reflected THz signals are mainly due to the water vapor. A typical measured THz waveform without delamination, shown in Fig. 3.22, is considered as the objective function. Echoes reflected from the bottom and top surfaces of the samples can be clearly identified in this reflected waveform. (Multiple reflections are small and correspond to features in the reflected THz pulse that occur at later times.) A model function is defined to simulate the objective function as

c Dk.l =                cD k,l -T cD k,l ≥T, cD k,l + T cD k,l ≤ -T, 0 |cD k,l | < T. (3.9
E model = aE in (t -∆t) (3.11)
where a and ∆t are the regression parameters. We consider the original THz reference signal shown in Fig. 3.12 as the incident THz pulse E in (t) and ignore the echo from the top surface in this simulation. In short, we use an amplitude-decreased and timeshifted reference signal to match the first echo in the measured signal. Initial values should be carefully set so that the regression analysis can converge to a characteristic global minimum. Then multiple regression analysis with least-squares minimization is performed to estimate the regression parameters, and the results are shown in Fig. 3.13. Signals between the Fresnel reflections from the bottom and top surfaces of the sample match quite well with the model function with regression parameters.

The coefficient of determination R 2 is also used here to provide information about the goodness of fit. In the time slice between 24 and 34 ps, the coefficient of determination equals to 0.9827, which demonstrates that these features mainly originate from the water vapor. It should be noted that featured echoes corresponding to the reflections from the internal interfaces between the fabric layers cannot be identified in the typical THz reflected waveform. Due to the complex internal structure of the woven fiber-reinforced composites with big size of fiber bundles, the echoes reflected from the internal inter-faces are very small (comparable or even smaller than the features introduced by the water vapor), and these small echoes are immersed in the reflected waveform. This is also the reason why a small mismatch can be found in Fig. 3.13 in the time slice between 27 and 29 ps, which corresponds to the location of the first internal interface between fabric layers. values of the reflected THz signal within a selected time slice between 27 and 29 ps, which corresponds to the location of the delamination. THz C-scans with wavelet denoising show higher contrast with regard to the delamination area and better enable us to differentiate the edge of the delamination from the background. This delamination is of roughly circular shape corresponding to the 12 mm diameter Upilex film disk.

Due to the big size of the fiber bundles, the thin Upilex film inside is deformed and the woven fiber fabric produces the vague grid pattern in the C-scan. We can further apply the 50% rule [START_REF] Smith | Diffraction and shadowing errors in -6 dB defect sizing of delaminations in composites[END_REF] to obtain binary THz C-scans, which only consists two areas, denoting the suspected delamination area as '1', and others as '0'. This method locates the pixel with the highest value in the C-scan and assigns to it a value of 100%, then colors all pixels red that have a value of at least 50% of the maximum.

Binary THz C-scans are frequently used for sizing delaminations [5]. The binary THz C-scans based on the data without and with wavelet de-noising are shown in Fig. 3.16.

Despite the presence of small artifacts, the binary THz C-scan based on the processed data provides a significantly better image to locate and size the delamination.

THz B-scans without and with wavelet de-noising are shown in Fig. 3.17. Fluctuations in the original THz waveforms lead to the multiple horizontal lines in the THz B-scan in Fig. 3.17(a). These horizontal lines may be misinterpreted as internal layers in the samples. By applying wavelet de-noising, the horizontal artifacts associated with water vapor are suppressed, shown in Fig. 3.17(b). Again, we apply the 50% rule to obtain binary THz B-scan images. In this case, we assign the highest value within the time slice between 27 and 29 ps as 100%, then color all pixels red that have a value of at least 50% the maximum. The binary THz B-scans based on the data without and with wavelet de-noising are shown in Fig. 3.18, which also shows that the binary THz B-scan based on processed data provides better image to locate the delamination in depth. Finally, it is important to note that this approach is of great interest in cases where undesired features associated with atmospheric water vapor of other material resonances may obscure features to be detected. In view of the difficulty and expense in purging with dry nitrogen to physically remove the water-vapor features in many cases, the demonstrated results will be of great interest for a range of laboratory-and field-based THz NDE applications.

Polarization-resolved terahertz imaging of impact damages in hybrid fiber-reinforced composites

In this study, THz imaging is firstly applied to a hybrid fiber-reinforced composite laminate, comprised of unidirectional glass/epoxy and carbon/epoxy laminae with a cross-ply stack pattern. The evolution of intra-laminar and inter-laminar damage throughout the thickness of the laminate (1.65 mm) subject to low-velocity impact is evaluated. Inter-laminar damage at the interface and the intra-laminar damage close to the same interface can be differentiated by taking advantage of the sensitivity of the carbon-fiber orientation to the THz polarization. THz C-and B-scan images are obtained to exhibit the propagation of the damage from the top to the bottom surface in three dimensions.

Terahertz polarization versus carbon fiber orientation

The conductivity of carbon fibers severely limits the penetration ability of THz waves into the material. For unidirectional carbon fiber-reinforced composites, the conductivity is anisotropic and depends on the THz polarization and fiber orientation, which can be expressed as [START_REF] Kostopoulos | Prediction and experimental validation of the electrical conductivity of dry carbon fiber unidirectional layers[END_REF] σ

(θ) = σ l cos 2 θ + σ t sin 2 θ (3.12)
where θ is the angle between the THz polarization and fiber orientation and σ l and σ t are the longitudinal and transverse conductivities, respectively. Along the fiber direction, the electric current flows through the carbon fibers, so the longitudinal conductivity depends on the conductivity of carbon fibers σ f and on the fiber volume fraction ν f ,

σ l ≈ σ f ν f (3.13)
For the transverse conductivity, because the resin material is nonconductive, the flow of electric current only occurs due to random contact between adjacent carbon fibers, which depends on the manufacturing process and the quality of the composites [START_REF] Kostopoulos | Prediction and experimental validation of the electrical conductivity of dry carbon fiber unidirectional layers[END_REF].

Therefore, the longitudinal conductivity is much higher than the transverse conductivity. Based on the literature [START_REF] Kwang-Hee | Terahertz wave approach and application on FRP composites[END_REF], longitudinal conductivity rangs from 1× 10 4 S/m to 6× 10 4 S/m, and the transverse conductivity varies from 2 S/m to 600 S/m.

By analyzing the electric conductivity, one finds that (1) when the THz polarization is parallel to the orientation of carbon fibers, the electric conductivity is maximum and the THz reflectivity also reaches the maximum; (2) when the THz polarization is perpendicular to the carbon fibers, conductivity and reflectivity achieve their minimum. The ideal amplitude reflection coefficient R ideal can be expressed as

R ideal ≈ 1 - 2ω 0 σ (3.14)
where ω is the THz frequency, and 0 is the permittivity of free space. The reflection coefficient approximately equals to 1 when THz polarization is parallel to the carbonfiber orientation.

Monitoring the reflection coefficient across the surface of carbon fiber composites with THz imaging can be utilized as a method to characterize the impact damage on the surface. When carbon fiber-reinforced composites suffer from impact damage, carbon fiber distortion and fracture will occur in the damage area, which will lead to spatial variation of the reflection coefficient as well as the polarization anisotropy.

The reflection coefficient in regions with and without impact damage can be more easily distinguished with polarization parallel to the carbon-fiber orientation [START_REF] Yang | Non-contact detection of impact damage in CFRP composites using millimeter-wave reflection and considering carbon fiber direction[END_REF].

Polarization-resolved terahertz imaging

The tested sample is a cross-ply hybrid fiber-reinforced composite laminate, shown in Fig. 3.19. The dimension of the laminate is 120 mm (length: x direction) × 120 mm (width: y direction) × 1.65 mm (thickness: z direction). The laminate is formed from prepregs of unidirectional E-glass fibers with epoxy resin, and prepregs of unidirectional carbon fibers with epoxy resin. For the prepreg of glass/epoxy, the fiber volume fraction is 60 vol.% and the resin content is 33 wt.%, the orientation of the glass fibers is along the direction of length (x direction); for the prepreg of carbon/epoxy, the fiber volume fraction is 60 vol.% and the resin content is about 42 wt.%, the orientation of the carbon fibers is along the direction of width (y direction). The stacking sequence of the laminate is [0 of the sample, and the second echo corresponds reflection at the interface between the glass/epoxy and carbon/epoxy layers with multiple reflections inside the sample following. For normal incidence on an interface between two media with refractive indices n i and n j , the amplitude reflection coefficients r ij and transmission coefficients

•G 2 /90 •C 3 ] s ,
E 1st (t) E 2nd (t) E 3rd (t) E in (t) E 1st (t) E ga (t) E ac (t)
t ij are r ij = n i -n j n i + n j (3.15) t ij = 2n i n i + n j (3.16)
In our study, we consider the THz reference signal shown in Fig. ??(a) as the incident THz pulse E T Hz (t), and we also neglect the negligibe dispersion in the relevant spectral range. The reflected THz waveform is

E T Hz (t) = E 1st (t) + E 2nd (t) + E 3rd (t) (3.17)
with

E 1st (t) = r ag E in (t) (3.18) E 2nd (t) = t ag r gc t ga E in (t - 2n g d c )e -2αd (3.19)
E 3rd (t) = t ag r gc r ga r gc t ga E in (t - 4n g d c )e -4αd (3. 20
)
where c is the in vacuo speed of light, d is the thickness of the glass/epoxy layer, n g is the refractive index of glass/epoxy layer, α is the absorption coefficient in the glass/epoxy layer, t ag and t ga are transmission coefficients from air to glass/epoxy and glass/epoxy to air, r gc are r ga reflection coefficients from glass/epoxy to carbon/epoxy and glass/epoxy to air, and we only consider one multiple reflection. In our case, with the polarization parallel to the glass fibers, the refractive index is n g, =2.124; with polarization perpendicular to the glass fibers, the refractive index is n g,⊥ =1.950. This polarization anisotropy underlies previously observed THz birefringence in unidirectional glass fiber-reinforced composites [START_REF] Palka | Precise determination of thicknesses of multilayer polyethylene composite materials by terahertz time-domain spectroscopy[END_REF].

First, we discuss the THz images when the THz pulse is incident on the top surface of the sample. Polarization-resolved C-scans obtained from the top surface the damaged region shows lower contrast due to the existence of matrix cracking and fiber distortion/fracture, which is consistent with the previous results. However, for perpendicular polarization, the damaged region shows higher contrast. This higher contrast indicates the existence of an air gap originating in the separation of the carbon/epoxy and glass/epoxy plies, i.e., delamination.

The THz waveforms in region of delamination can also be estimated, as shown in Fig. 3.22(b). Usually, the air gap associated with the delamination is thin; thus, the second echo E 2nd (t) received is the superposition of the two echoes reflected from the glass/epoxy-air interface E ga (t) and from the air-carbon/epoxy interface E ac (t), which can be expressed as We implemented a numerical parameter fitting method based on multiple regression analysis [START_REF] Yasuda | Improvement of minimum paint film thickness for THz paint meters by multiple-regression analysis[END_REF] to estimate the delamination thickness and the reflection coefficient at the interface between the air gap and carbon/epoxy layer. The measured second echo is considered as the objective function, and a model function E model is defined based on the equation (3.22) to stimulate the objective function as

E 2nd (t) = E ga (t) + E ac (t) = t ag r ga t ga E in (t - 2n g d c )e -2αd + t 2 ag r ac t 2 ga E in (t - 2n g d c - 2d gap c )e -2αd (
E model = a 1 E in (t -t 1 ) + a 2 E in (t -t 1 -∆t) (3.23)
where a 1 , a 2 , t 1 and ∆t are the regression parameters. Initial values should be carefully set so that the regression analysis can converge to a characteristic global minimum.

Then multiple regression analysis with least-squares minimization is performed to es- B-scans obtained incident from the bottom surface are shown in Fig. 3.30. Typical damage features can be observed in depth, including surface bending, inter-and intralaminar damages. Therefore, the total set of C-and B-scans exhibits the evolution of the impact-induced damage from the top to the bottom surface in three dimensions, which fits quite well with theoretical simulation results in Ref. [START_REF] Tan | Predicting low velocity impact damage and Compression-After-Impact (CAI) behaviour of composite laminates[END_REF][START_REF] Zhang | Simulating low-velocity impact induced delamination in composites by a quasi-static load model with surface-based cohesive contact[END_REF]. The size of the This method, which is verified numerically and experimentally, is able to provide a 'quasi-ideal' impulse response function, and therefore, significantly enhances the depth-resolution for resolving optically thin layers in the THz regime. The depthresolution achieved with this method is about 27 µm. In the time domain, the THz reflected signal (electric field) r(t) is the convolution of the incident THz pulse i(t) with the impulse-response function h(t), which corresponds to the structure and properties of the sample at a given two-dimensional position,

r(t) = i(t)⊗h(t). (4.1)

Deconvolution retrieves the impulse response function h(t) by applying the inverse

Fourier transform based on the convolution theorem,

h(t) = F F T -1 F F T r(t) F F T i(t) , (4.2)
where F F T denotes the Fourier transform and F F T -1 the inverse Fourier transform.

Terahertz frequency-wavelet domain deconvolution (FWDD)

Frequently, successful deconvolution cannot be expected by directly applying Eq.

(4.2), since division by small numbers will give rise to large spikes in the reconstructed impulse-response function, especially in the high frequency region, leading to severe ringing in the time domain. Therefore, deconvolution process is usually further augmented by frequency-domain filtering to suppress the high-frequency noise, which can be expressed as,

h (t) = F F T -1 F F T f (t) × F F T r(t) F F T i(t) , (4.3)
with f (t) the filter function in the time domain. In order to obtain a successful reconstruction, the temporal duration of f (t) should be short enough to resolve the time intervals between featured echoes, and f (t) should not contain extra signal cycles before or after the main peak, which will obscure the real featured echoes in the reconstructed signal; however, the selection of f (t) is also a compromise between time resolution and frequency-domain filtering [START_REF] Fletcher | Pulsed terahertz signal reconstruction[END_REF]. If the duration of f (t) is too short, its frequency spectrum will include large spikes at high frequencies, which will degrade the reconstructed signal in the time domain.

A double Gaussian filter or Wiener filtering can be selected to serve as the frequency-domain filtering [START_REF] Chen | Frequency-wavelet domain deconvolution for terahertz reflection imaging and spectroscopy[END_REF], and a tapered cosine apodisation function has also been found to work well [START_REF] Galvão | Optimization of apodization functions in terahertz transient spectrometry[END_REF]. Considering the complexity and effectiveness, a Hanning window function is chosen as the filter function f (t) in this study, and its frequency spectrum F (ω) can be expressed as

F (ω) =        e iωt 0 cos 2 ( ω 4fc ) | ω | ≤2πf c , 0 | ω | >2πf c , (4.4)
where t o corresponds to the arrival time of the main peak in the time domain and f c is the cutoff frequency. This frequency-domain filtering is easy to manipulate just by changing the cutoff frequency f c . An example of f (t) and its frequency spectrum Quite often, deconvolution only with frequency-domain filtering cannot guarantee a satisfactory signal-to-noise ratio when a relatively high value of f c is selected.

Stationary wavelet shrinkage is applied to further attenuate the residual noise. This technique decomposes a 1D signal into the approximation coefficients vector and detail coefficients by convolving with a low-pass filter and a high-pass filter along the temporal axis at each level. Wavelet coefficients with small absolute values can be considered as noise, and wavelet coefficients with large absolute values are regarded as the main featured information of the signal [54][66]. Removing the small absolutevalue coefficients by thresholding and then reconstructing the signal is expected to produce a signal in which the contribution of noise has been reduced. Sometimes, the signal after frequency-wavelet deconvolution contains slow fluctuations corresponding to the low frequency noise due to the THz source being deficient in the low THz frequency region. This kind of low-frequency noise can be canceled by subtracting the baseline of the deconvolved signal. Based on the observation of the coating surface features, four typical positions, which exemplify different failure modes, are selected in Fig. 4.2(a). The received THz signal at position [START_REF] Wang | Terahertz absorbance spectrum fitting method for quantitative detection of concealed contraband[END_REF][START_REF] Dong | Flexural strength of bidirectional hybrid epoxy composites reinforced by E glass and T700S carbon fibres[END_REF], where no evidently visible damages exist, is shown in Fig. 4.3(a1). Overlapping THz echoes at this pixel are observed due to the optically thin coating. In order to resolve the overlapping echoes and reveal the structure, THz frequency-wavelet domain deconvolution introduced previously is applied to the raw reflected THz signal at each pixel. In order to make the time resolution as high as possible, the cutoff frequency f c chosen for all the waveforms is 4 THz. In the wavelet denoising procedure, the symlet (sym4) wavelets are selected with a maximum level of 7 for the wavelet decomposition, as no significant improvement can be achieved for higher levels to justify the extra computational expense. After deconvolution, a 3D deconvolved data set is obtained. Peak detection is performed on the deconvolved signals for identifying the existence of an echo. A threshold value for peak detection is set for all pixels, above which we consider a feature as a valid peak. The first two positive peaks correspond to the echoes from the air/coating interface and the coating/substrate interface respectively. The third peak, corresponding to the second round echo, is negative due to the phase shift at the coating/air interface. We consider pixels with this kind of deconvolved signal as normal i.e., undamaged, since no features related with damages can be identified.

The optical thickness of the coating can be acquired by calculating the optical delay between the first and the second peaks, which is ∼0.5 ps. The refractive index of polyester in the THz range is ∼1.7 [START_REF] Yun-Sik | Terahertz Dielectric Properties of Polymers[END_REF], therefore, the physical thickness of the coating is ∼44 µm, which is close to the value we estimated with the optical micrograph in Optical Delay (ps) Amplitude (a.u.) The physical origin of the contrast between blisters and the surrounding undamaged regions are now discussed. Compared with the undamaged area, the relatively large optical delay between the first two positive peaks at the locations of the blisters is due to the separation between the coating and the substrate and the existence of the air gap. However, the air gap under the blisters is too small to be resolved even in the deconvolved signals. In this case, the negative peak corresponding to the echo from the coating/air interface and the positive peak corresponding to the echo from the air/substrate are not well-separated and largely cancel with each other. The result of this overlap and cancellation is only one observable positive peak with a later arrival time and smaller amplitude compared with the peak corresponding to the echo from the coating/substrate interface in undamaged areas [68][47]. Another reason, which is also responsible for the smaller amplitude of the second positive peak, is the presence of oxidization of the metal substrate in blisters locations induced by the accelerated aging. The oxidized substrate under the blisters provides a smaller reflection compared with the normal steel under the undamaged coating, where the coating and the substrate adhere tightly and an almost total reflection can be expected. Based on the analysis above, the characteristics of various failure modes, including corrosion, delamination, and blistering, have been successfully revealed using timedomain THz reflective imaging. The thickness distribution of this damaged coating can be estimated, as shown in Fig. 4.8. To do this, we assume that the refractive indices of the corrosion (metal oxide) is similar to that of the coating [START_REF] Fuse | Evaluation of applicability of noncontact analysis methods to detect rust regions in coated steel plates[END_REF]. We note in the image the metal oxide is grown in the scratch area, on both sides of which are raised areas of delaminations. Beyond this is a relatively flat area with isolated raised areas due to blisters. With this 3D image, the thicknesses associated with various failure modes, as well as the surface topology of the coating system are clearly reconstructed, exhibiting the capability of THz imaging for the quantitative NDE of polymer coatings on metals.

In this study, THz reflective imaging was demonstrated to characterize different failure modes in a polyester-coated zinc-coated steel plate. THz frequency-wavelet domain deconvolution was adapted to resolve the optically thin coating. Based on the deconvolved signals, the characteristics of various failure modes, including corrosion, delamination, and blistering, have been successfully identified. The THz deconvolved signals also enable us to evaluate the condition of adhesion, especially for the delamination and blisters, which are related with adhesion failure. The thickness distribution across the entire damaged coating, as well as its surface topology is also obtained.

These interpretations were supported by optical microscopy. Based on these results, we conclude that THz imaging, which can provide a noninvasive, noncontact, and nonionizing modality for characterizing coatings quantitatively in three dimensions, can be utilized as an effective tool for investigating the damage mechanisms and monitoring the corrosion process in polymer-coated metals.

Terahertz Sparse Deconvolution

The

Terahertz sparse deconvolution with data-resolution

In the time domain, the THz reflected signal (electric field) y(t) is the convolution of the incident THz pulse h(t) with the impulse-response function f (t), which corresponds to the structure and properties of the sample at a given point of interest,

y(t) = h(t)⊗f (t) = +∞ -∞ h(τ )f (t -τ )dτ. (4.5)
For reflective THz imaging, the incident THz pulse h(t) can be obtained by first recording the THz signal reflected from a metal plate (THz reference signal), and then multiplying the reference signal by a factor of -1 for phase correction. In practice, we should consider the discrete form of Eq.4.5 with the sampling period T s ,

y n = M -1 m=0 h m f n-m + e n , (4.6)
where y n = y(nT s ), h m = h(mT s ) and e n accounts for the noise originating from the measurement system and materials with n and m as the indices of data points, and M as the length of the data points. Let column vectors y, h, f and e collect the samples of y n , h n , f n and e n , respectively. Then Eq. (4.6) can be expressed as

y = Hf + e, (4.7)
where H is the convolution matrix whose rows are delayed versions of the reversed vector of h T or, equivalently, whose columns are delayed versions of h.

The basic idea of sparse deconvolution is to achieve the impulse response function by exploiting the sparse constraint. It aims at approximating the received THz signal y with Hf where f is a sparse sequence; that is, f has only few non-zero components. In this case, the sparse vector f can be computed by solving the l 0 regularized optimization problem, which is defined as

min f 1 2 Hf -y 2 2 + λ f 0 , (4.8)
where f 0 is the l 0 -norm of f , which is defined to be the number of nonzero entries in f , and λ is the regularization parameter, which controls the tradeoff between the sparsity of f and the residue norm.

Solving the non-convex l 0 regularized optimization problem in Eq. 4.8 is known to be nonpolynomial (NP) hard and the global optimum cannot be guaranteed. It has already been shown that this non-convex optimization problem can be approximated with a convex optimization problem by replacing the l 0 penalty with the l 1 penalty as min

f 1 2 Hf -y 2 2 + λ f 1 , (4.9)
where f 1 is the l 1 -norm of f , which is defined as the sum of the absolute values of its components. Since the l 1 norm is convex, a global optimum can be guaranteed.

The iterative shrinkage algorithm, which has been developed recently and is able to address the above optimization problem effectively, is utilized in this paper. Generally speaking, in the iterative shrinkage algorithm, each iteration involves matrix-vector multiplication involving H and H T followed by a shrinkage or soft-thresholding step.

Specifically, the general iterative procedure is given by:

f i+1 = S λτ f i -τ H T (Hf i -y) , (4.10)
where τ is an appropriate step size, which should obey

τ < 2 H T H 2 , (4.11)
in order to guarantee convergence, and the shrinkage or soft-thresholding operator S λτ is defined as

S λτ (f [n]) =            f [n] + λτ f [n] -λτ 0 | f [n] |< λτ f [n] -λτ f [n] λτ . ( 4

.12)

A thorough theoretical analysis in [START_REF] Daubechies | An iterative thresholding algorithm for linear inverse problems with a sparsity constraint[END_REF] proves the convergence of this iterative shrink-age algorithm guaranteeing that the solution is the global minimizer for convex f .

Obviously, the time-resolution of the obtained impulse response function f by sparse deconvoltuion depends upon the time-resolution of the reference signal h, which is itself determined by the discretization precision, corresponding to the data sampling period T s .

Terahertz sparse deconvolution with super-resolution

Compared with conventional deconvolution, one of the advantages of sparse deconvolution is that it can achieve super-resolution. The time-resolution of sparse deconvolution can be increased by an up-sampling approach. Although the reference signal h(t) is measured at the sampling rate of the data T s , the original discrete h, which contains M data points, can be up-sampled by factor K to form h sr with time interpolation [START_REF] Carcreff | High-resolution deconvolution applied to non destructive testing[END_REF]. In this case, we can consider that the reference signal h(t) is discretized at T s /K; therefore, the time-resolution of f (t) can also be increased by K times accordingly. The discrete convolution model can be written as the sum of K discrete convolutions as

y n = 1 K K-1 k=0 M -1 m=0 h k m f k n-m + e n , (4.13)
where h k are K sub-wavelets with sampling period T s , such that h k m = h(kT s /K + mT s ), and f k are the corresponding sparse sub-sequences with M points. In this model, the discrete convolutions based on the original data and (K -1) times more data points from the time interpolation are summed up and averaged to approximate the received signal y. In this model, we do not perform time interpolation on the received signal y, which is still discretized at T s with the original data-resolution, in order to prevent introducing additional information into y; instead, we apply a sampling and holding approach to deal with y. Then the matrix form of Eq. 4.13 is:

y = 1 K K-1 k=0 H k f k + e, (4.14)
with H k the sub-matrices obtained by taking every Kth row of H sr , which is the convolution matrix based on h sr . By taking this matrix form into the iterative shrinkage algorithm, the general iterative step should be updated to

f k i+1 = S λτ f k i -τ H kT 1 K K-1 k=0 H k f k i -y , (4.15)
where the step size τ should satisfy

τ < min k 2 H kT H k 2 (4.16)
in order to guarantee convergence. By interleaving the obtained K sub-vectors f k , the final impulse response function with super-resolution f sr , which contains KM data points, can be achieved.

Terahertz sparse deconvolution considering pulse spreading

The performance of sparse deconvolution demonstrated above is mainly limited to the deconvolution of reflective THz signals with time-invariant echoes, which assumes that the THz echoes are time-shifted, amplitude-scaled replicas of the THz reference signal.

In practice, however, the reflected THz signals can be both sparse and time-varying due to the frequency-dependent attenuation and dispersion during the propagation of THz waves in materials. For most materials, this frequency-dependent loss has a low-pass filtering effect on the propagating THz waves and results in temporal pulse spreading [73][74]. For thick multi-layered samples in particular, a shape broadening of echoes, which grows as propagation distance increases, can be clearly observed in the received THz signals. This temporal pulse spreading will definitely degrade the accuracy of the sparse deconvolution based on time-invariant echoes.

A simple but effective discrete-time pulse spreading model is designed in the following to solve the problem mentioned above. We assume that the temporal spreading of the propagating THz pulse, caused by traveling a distance slice ∆z in an attenuative and/or dispersive medium, can be modeled using a linear time-invariant (LTI) system [START_REF] Olofsson | Minimum entropy deconvolution of pulse-echo signals acquired from attenuative layered media[END_REF], whose impulse response function is ρ ∆z (t). At depth ∆z, the THz pulse h ∆z (t) can be expressed as: where h 0 (t) is the THz pulse at ∆z = 0, corresponding to the original THz reference signal. Accordingly, a model of the pulse spreading associated with traveling a multiple of this distance, z n = N ∆z, is obtained by serially connecting the above mentioned LTI system N times. The impulse response of the serially connected LTI system is modeled as an N -time self-convolution of the impulse response function of the LTI system associated with a material slice of thickness ∆z. Therefore, at depth N ∆z, the THz pulse h zn (t) will be governed by

h ∆z (t) = ρ ∆z (t)⊗h 0 (t), (4.17
h zn (t) = ρ ∆z (t)⊗ρ ∆z (t)⊗• • •⊗ρ ∆z (t)⊗h 0 (t) = ρ (N ) ∆z (t)⊗h 0 (t), (4.18)
where ρ for describing the temporal pulse spreading with a simple form:

ρ ∆z [n] =                  a n = 0 1 -2a n = 1 a n = 2 0 otherwise , (4.19)
where 0 < a 1 and a determines the severity of the pulse spreading. Note that

a = 0 yields ρ ∆z [n] = δ[n -1]
, indicating a simple time delay without temporal pulse spreading. In practice, the parameter a can be found by fitting a specific separate echo at a given depth. Let column vector ρ collects the values of ρ ∆z [n], then matrix A for describing the pulse spreading throughout the propagating distance can be expressed as

A = ρ (1) ρ (2) ρ (3) • • • ρ (n) , (4.20)
where ρ (n) represents n times self-convolution with the kernel vector ρ. Therefore, in order to include the temporal pulse spreading effect, the convolution matrix form in Eq. 4.7 should be modified to

y = HAf + e, (4.21)
where the matrix HA models the THz pulse with temporal spreading. Each column of HA now represents a THz pulse with temporal spreading for a traveling distance equals to n∆z, where n is also identical to the column number. Compared with the THz pulses in the adjacent columns, both amplitude decrease and shape broadening with respect to the traveling distance can be observed.

Accordingly, in order to deconvolve this kind of time-varying THz signal with the iterative shrinkage algorithm, the general iterative step should be further modified to

f i+1 = S λτ f i -τ (HA) T (HAf i -y) , (4.22)
where the step size τ should obey

τ < 2 (HA) T HA 2 , (4.23)
in order to guarantee convergence. It is important to note that, Eq. 4.21 indicates that the temporal pulse spreading is considered from the beginning of the received THz signal. However, it is not a common case. For example, in a typical THz reflective imaging experiment, the first columns of H correspond to the THz pulses propagating in air, which is low loss and involves no pulse spreading. In this case, A should be divided into two parts by identifying a typical column number n 0 which corresponds to the air/sample interface. To the left of column n 0 , A is composed of an identity matrix I, which is the same as setting a = 0; and then after column n 0 , by setting 0 < a 1, the temporal pulse spreading starts to be involved after the air/sample interface. In the case in which the THz pulse encounters several propagating media, one can build a block diagonal matrix A composed of the respective media matrices by setting different values of a.

Numerical and experimental verification

Numerical simulations are first performed to verify the iterative shrinkage algorithm for sparse deconvolution with both data-resolution (DR) and super-resolution (SR).

An actual THz reference signal generated by the experimental system described in Section IV, which contains 4096 data points with a sampling period T s = 0.0116 ps, is recorded and used in the simulations. An ideal and simple impulse response function

f 0 [n],
with the same sampling period and number of data points, is assumed The minimal time interval which DR sparse deconvolution is able to resolve can be identified by varying the distance between the two peaks in f 0 [n]. We determine that the impulse response function with minimal resolvable time interval f 0 [n] is as following: For the deconvolved signal with FWDD, it is obvious that this pulse spreading can also lower the performance of FWDD, as the peaks indicating the locations of interfaces become wider as the propagating distance increases. For the deconvolved signal with DR sparse deconvolution without pulse spreading, the peaks corresponding to the echoes bouncing back from the first and second papercards are quite sharp; however, additional small peaks appear at the last three interfaces, which will definitely obscure the exact locations of the interfaces. The origin of these additional peaks is the temporal pulse spreading. Without including this pulse spreading effect in the algorithm, the input signal is considered as invariant, which is the narrow THz reference signal, and thus a wider THz echo in the received signal will be treated as a combination of several narrow pulses.

f 0 [n] =            1 n = 1460 1 n = 1550 0 otherwise . ( 4
f 0 [n] =            1 n = 1460 1 n = 1485 0 otherwise , ( 4
Optical Delay (ps) Normalized Amplitude (a.u.) this algorithm is the formation of the matrix A. By peak detection, the location of the first air/papercard interface can be identified. This location is considered as the typical column number n 0 . Above n 0 , A is filled with an identity matrix, and after n 0 , A is formed based on Eq. 4.20. The key parameter a is determined by multiple trials. In our case, the criterion for the fitness of a is based on the last received THz pulse corresponding to the interface between the back of the last papercard and the air, which should satisfy the following conditions: (1) the deconvolved signal achieved f DR (t) should provide one sharp and clear pulse corresponding to the location of the last interface; (2) f DR (t) is convolved with the input signal including pulse spreading h a (t), and there should be a good fit between the convolution result h a (t) ⊗ f DR (t) and the received THz signal y 0 (t), especially for the last THz pulse. The optimal value we determine in the algorithm is a = 0.035, and the corresponding deconvolved signal is shown in Fig. 4.16(c). We observe that, by considering the pulse spreading effect, the spurious peaks, which appear in Fig. 4 In this study, sparse deconvolution based on an iterative shrinkage algorithm has been demonstrated for THz characterization of multi-layered structures. Compared with conventional deconvolution, such as FWDD, in which high-and low-frequency noises are inevitably introduced, sparse deconvolution is a pure time-domain technique, which can provide a more clear representation of the impulse response func-tion with sharp pulses. Since the time-resolution of conventional deconvolution is limited by the discretization precision, a super-resolution version of sparse deconvolution is further developed by an up-sampling approach based on time interpolation, which increases the capability of sparse deconvolution for precise estimation of the arrival times of THz echoes. In addition, the temporal pulse spreading due to the frequency-dependent loss during the THz propagation is also considered in the sparse deconvolution. A simple but effective time-domain model for describing the temporal pulse spreading effect is designed and introduced into the iterative shrinkage algorithm. This model requires little prior knowledge of the properties and structure of the materials, and can greatly improve the performance of sparse deconvolution in processing time-varying THz pulses. The algorithms for sparse deconvolution are all verified with numerical simulations and experimental measurements, which demonstrate that sparse deconvolution is an ideal and effective tool for THz nondestructive characterization of multi-layered structures.

Terahertz Deconvolution Based on Autoregressive Spectral Extrapolation

Theoretically, the reflected THz signal r(t) is the convolution of the incident THz pulse i(t) with the impulse response function h(t), which should ideally consist of M time-shifted ideal impulses, with the number M corresponding to the material structure; therefore, the Fourier transform of this ideal impulse response function is broadband and should be a sum of M complex sinusoids throughout the entire frequency spectrum. Conventional deconvolution based on direct inverse filtering aims at retrieving the impulse response function by applying the inverse Fourier transform of the transfer function H inv (f ), which is the ratio of the reflected to the incident THz spectra,

h inv (t) = F F T -1 H inv (f ) = F F T -1 F F T r(t) F F T i(t) , (4.26)
where F F T denotes the Fourier transform and F F T -1 the inverse Fourier transform, and h inv (t) is the impulse response function obtained by directly inverse filtering.

Frequently, successful deconvolution cannot be expected by directly applying Eq.4.26.

Because the THz pulse is band-limited, with a fast-decreasing spectrum outside of the passband (in our case extending from 60 GHz to 3 THz), the division of small values due to the deficiency of THz sources in the low-and high-frequency regions, will give rise to large abnormal values as high spikes and ruin the signal-to-noise ratio there; therefore, H inv (f ), as obtained from Eq.4.26 is only valid within the bandwidth with high SNR. Furthermore, these high spikes in the low-and high-frequency regions, where the SNR is low, will introduce severe low-and high-frequency noises in the impulse response function.

THz frequency-wavelet domain deconvolution (FWDD) [START_REF] Chen | Frequency-wavelet domain deconvolution for terahertz reflection imaging and spectroscopy[END_REF] is specifically designed to enhance the deconvolution process by first employing frequency-domain filtering and then further improving the SNR by wavelet de-noising. Baseline subtraction is also required to cancel the slow fluctuations corresponding to the low-frequency noise. Wiener filtering [START_REF] Chen | Frequency-wavelet domain deconvolution for terahertz reflection imaging and spectroscopy[END_REF], double Gaussian filter [START_REF] Chen | Total variation deconvolution for terahertz pulsed imaging[END_REF] and van Hann filter [START_REF] Galvão | Optimization of apodization functions in terahertz transient spectrometry[END_REF][START_REF] Dong | Polarization-resolved terahertz imaging of intra-and inter-laminar damages in hybrid fiber-reinforced composite laminate subject to low-velocity impact[END_REF] can serve as the frequency-domain filtering. However, the frequency-domain filtering will narrow the bandwidth of the impulse response function by eliminating the high spikes in the low-and high-frequency regions, and only the frequency components within the bandwidth with high SNR are kept. Therefore, pulses in the impulse response function recovered by FWDD are much wider than the ideal impulses, and consequently, limit the depth-resolution.

In this section, a novel method for enhancing the depth-resolution of THz decon-volution based on autoregressive spectral extrapolation is developed. This method consists in modeling part of the deconvolved spectrum with high SNR with an autoregressive (AR) process, and to extrapolate the remaining part of the spectrum based on this AR model. What underlies AR spectral extrapolation is the maximum entropy method which consists in determining a spectral estimation that maximizes uncertainty with respect to the unknown information, thus eliminating unreasonable constraints, but that is consistent with the known information [START_REF] Ulrych | Time series modelling and maximum entropy[END_REF]. Direct use of the maximum entropy method is marred, however, by the lack of criterion for determining the order of the model used. It has been shown [START_REF] Ulrych | Maximum entropy spectral analysis and autoregressive decomposition[END_REF] that the maximum entropy method is equivalent to the least squares fitting of an AR process for which mathematical criterion (e.g. Akaike's criterion [START_REF] Akaike | A new look at the statistical model identification[END_REF]) exist to determine the length of an AR model. The existence of such a criterion is crucial to avoid over-or under-fitting. In the following, we focus on deconvolution exploiting AR spectral extrapolation, and call this method 'AR deconvolution' for convenience. Unlike FWDD, which discards the frequency components in the low-SNR regions in order to cancel the high spikes,

AR deconvolution aims at finding the missing frequency components in the low-SNR regions using an AR model based on the data in the high-SNR regions. With AR deconvolution, the spectrum of the impulse response function throughout the entire frequency band can be estimated, then the inverse Fourier transform leads to a 'quasi-ideal' impulse response function can be achieved, and therefore, the resolution in depth can be enhanced.

Autoregressive model for terahertz spectrum

In an AR process, each data point of H i of a digitized signal is a weighted sum of its previous p points, plus a noise term N i ,

H i = p k=1 a k H i-k + N i , i > p, (4.27)
where p is the order of the AR process, a k are the AR coefficients, and H i is known within the frequency band (i L i i H ) with high SNR. This AR model can be used as a prediction filter to find an estimate Ĥi of the unknown values of H i for i > i H using the forward prediction equation,

Ĥi = p k=1 a k H i-k , i > i H . (4.28)
Similarly, we can use a backward prediction filter to find the values of

H i for i < i L , Ĥi = p k=1 b k H i+k , i < i L , (4.29)
where b k are the coefficients for the backward prediction filter. The optimum values for the coefficients a k and b k are determined by minimizing the squared error between the model and the N BW (N BW = i H -i L + 1) available data points with high SNR.

As mentioned above, the frequency spectrum of the ideal impulse response function, of a multi-layered system, should be the superposition of a limited number of complex sinusoids, each corresponding to a layer interface. It has already been shown that [START_REF] Ulrych | Time series modelling and maximum entropy[END_REF], for a noiseless case, this kind of signal, which consists of a sum of complex sinusoids, can be modeled as an AR process with an order equal to twice the number of sinusoids; for a noisy case, this kind of signal should be modeled as an AR process with an order much higher than the number of sinusoids. Akaike's Information Criterion (AIC) [START_REF] Akaike | A new look at the statistical model identification[END_REF], which is based on the principle of entropy maximization in information theory, provides a measure for the selection of the model order p, and the AIC of order p can be calculated using the residual sums of squares from regression

S 2 p , (AIC) p = N ln(S 2 p ) + 2p, (4.30)
where N is the number of data points, and ln is the natural logarithm. AIC deals with the trade-off between the goodness of fit of the model and the complexity of the model, and according to Akaike's theory, the most accurate model has the smallest AIC value. For the THz frequency spectrum, both the forward and backward prediction filters are needed to find the missing data in the low-and high-frequency regions, and the AR coefficients are obtained by minimizing the sum of the forward and backward prediction errors 2 , (3) data within [i P H , i H ] are used for the fitting to determine the a k . The Burg method is utilized to minimize this error term [START_REF] Shakibi | Resolution enhancement of ultrasonic defect signals for crack sizing[END_REF]. By adding the Levinson-Durbin constraint [START_REF] Ortigueira | Global versus local minimization in least-squares ar spectral estimation[END_REF], the Burg method enables the AR coefficients to be determined by a fast recursive algorithm, and guarantees a stable prediction filter, which is important to estimate the spectrum throughout the entire frequency band.

2 = i P L i=i L |H i - p k=1 b k H i+k | 2 + i H i=i P H |H i - p k=1 a k H i-k | 2 , (4.31

Numerical and experimental verification

Teraview TPS Spectra 3000 is employed in this study to perform THz reflective imaging at almost normal incidence in order to obtain the data used in the following

f 0 h 0 f 0 + h 0 + e n (1525, 0.7) (1500, 1)
(1650, -0.5) The assumed frequency spectrum H 0 , which ideally consists of a sum of three complex sinusoids and is not bandlimited throughout the frequency band (from 0 THz to 85.99 THz), is obtained by the Fourier transform of h 0 . In Fig. 4.18(a), H 0 is compared with H inv , which is the direct division of the spectra of r 0 and f 0 .

By comparison, the low-SNR frequency bands of H inv which contain high spikes corresponding to low-and high-frequency noises, and the high-SNR frequency band the detailed data allocation for this AR process. After fitting this AR model to the N BW available data, we apply this AR model as a prediction filter to extrapolate the data to find the missing data in the regions below 0.18 THz and above 1.5 THz (up to 85.99 THz). Note that this is well outside the bandwidth of our imaging system; the ability to recover data out to this high frequency depends on our having sufficient knowledge of the frequency spectrum to construct an accurate AR model.

H 0 H inv H AR h 0 h inv h AR
The estimated frequency spectrum H AR based on this AR model is shown in red in Fig. 4. [START_REF] Yasui | Terahertz paintmeter for noncontact monitoring of thickness and drying progress in paint film[END_REF]. By comparison between H AR and H 0 , we can find out that the estimated frequency components in H AR match the assumed data in H 0 quite well, which verifies the effectiveness of this AR model to recover the missing data. By simply performing the inverse Fourier transform of H AR , the deconvolved signal h AR , can be achieved.

In Fig. 4.19, h AR is compared with h F W DD , which is the deconvolved signal obtained by FWDD, and the assumed impulse response function h 0 . For FWDD, van Hann filter is selected to serve as the frequency-domain filter to suppress the highfrequency noise, and the maximum value of the cutoff frequency we can set is f c = 3.5

THz in order to obtain a satisfactory SNR of the deconvolved signal. More details about the FWDD algorithm we use can be found in [START_REF] Dong | Terahertz Quantitative Nondestructive Evaluation of Failure Modes in Polymer-Coated Steel[END_REF]. With FWDD, only the first and the third interface can be identified, and the first assumed layer with time interval in the high-SNR regions. In this way, the entire spectrum of the impulse response function is estimated, and a 'quasi-ideal' impulse response function with enhanced depth-resolution is achieved. We find out AR deconvolution is very suitable to deal with THz TDS signals, as the THz spectrum provides a large bandwidth with high SNR to establish a high-order AR model. It is also important to note that the deconvolution result is sensitive to the selection of parameters and SNR when a high-order AR model is utilized. This method is verified numerically and experimentally with a one-layered polymer coating with thickness of 22.5 µm, which cannot be resolved by FWDD. The thickness is successfully resolved by AR deconvolution, demonstrating that AR deconvolution enables us to beat the limitation of the THz wavelength and enhance the depth-resolution for resolving the optically thin layers in the THz regime.

CHAPTER 5 TERAHERTZ REFLECTOMETRY OF ART PAINTINGS

Studies at the intersection between the quantitative sciences and the humanities have been especially fruitful in the last few decades with the availability of new spectroscopic tools that permit noninvasive studies of art. In particular, physical characterization of art using emerging tools may provide information that conclusively confirms or refutes judgments based on connoisseurship. Determination of the physical characteristics, not only across a painting but also in depth, is one of the most important procedures to gain insight into its structure. The layer-by-layer structure, or stratigraphy, reveals the sequential application of the preparatory and ground layers on the support (e.g., canvas or wood panel), pictorial layers, and varnish, as well as of possible subsequent revisions or restorations. A detailed knowledge of the stratigraphy provides a basis for evaluation of its authenticity and attribution, insight into historical or artist-specific techniques for art-historical studies, as well as the recognition of any decay and of consequent conservation and/or restoration requirements. For paintings, stratigraphic analysis can reveal the way in which the paint layers are applied, and consequently, tell us how the artist worked. The conventional approach to obtain information on the stratigraphy is based on the characterization of the cross-sections of micro-samples taken from the objects with standard micro-analytical tools, such as visible-light and electron microscopy, energy-dispersive X-ray spectroscopy, Raman and infrared spectroscopy [START_REF] Reischig | High-resolution non-invasive 3D imaging of paint microstructure by synchrotron-based X-ray laminography[END_REF]. This approach is invasive, resulting in the destruction of the integrity of the painting. Various noninvasive and noncontact modalities which can provide in-situ quantitative information in depth, such as confocal X-ray fluorescence (with elemental distribution contrast) [START_REF] Reiche | Depth profiling reveals multiple paint layers of Louvre Renaissance paintings using non-invasive compact confocal micro-X-ray fluorescence[END_REF], femtosecond pump-probe microscopy (with molecular and structural constrast) [START_REF] Villafana | Femtosecond pump-probe microscopy generates virtual crosssections in historic artwork[END_REF], nuclear magnetic resonance (with 1 H abundance constrast) [START_REF] Presciutti | Noninvasive nuclear magnetic resonance profiling of painting layers[END_REF], and optical coherence tomography (with structural constrast) [START_REF] Cheung | Ultra-high resolution Fourier domain optical coherence tomography for old master paintings[END_REF], are under active research. Although with micrometer-level high resolution, these methods have limited penetration in depth, or imply probing a small region of interest, as opposed to an extended area, and therefore, cannot generate a 3D global mapping of the layer structure of a painting [START_REF] Alfeld | Mobile depth profiling and sub-surface imaging techniques for historical paintings-A review[END_REF]. obtained with the deconvolved signals.

For Type IV, shown in Fig. 5. 6(d), no extra featured peak can be identified.

Therefore, we conclude that there is only one paint layer on the paperboard or the thickness of the underlying layer is too thin to be detected in the THz regime.

Correlation between terahertz and X-ray images

The deconvolved data can also provide quantitative information on the painting, such as the layer thicknesses. The optical thickness of the paint layers can be acquired by calculating the optical delay between the air/paint and paint/paperboard interfaces.

The distribution of the paint-layer thickness can reveal in which areas the artist applied more layers of paint. (In our case, the thick paint application with a palette knife resulted in considerable impasto that enables us to check the reasonableness Whereas Fig. 5.10 provides a measure of the optical thickness of the painting and . Therefore, we can conclude that THz imaging can be utilized to obtain information that is similar to that contained in conventional X-ray imaging. Based on our study, we conclude that THz frequency-wavelet deconvolution can be an effective tool for the stratigraphic and subsurface investigation of art paintings, and may also be applied for the characterization of other stratified systems.

In addition, one striking point comes out of this study is the high similarity between the THz and X-ray images. THz imaging of the thickness distribution of the paint exhibits a high degree of correlation with the X-ray transmission image.

Moreover, THz imaging also reveals the spots in the paperboard which cannot be identified in the X-ray image. Compared with X-ray imaging, THz imaging can be carried out in reflection, so that there is no need access to both sides of a painting; THz imaging can provide information in depth and THz data contains spectral information which can potentially be used to identify different pigments. Therefore, our results open up the way for the use of non-ionizing THz imaging as a potential substitute for ionizing X-ray analysis in nondestructive evaluation of art paintings.

Terahertz characterization of the stratigraphic details of a 17th century Italian oil painting on canvas

One lacuna in the past success of THz stratigraphic characterization of paintings, however, is the field of pre-19th century easel paintings, where the paint layer thicknesses are usually smaller than 50 µm. This characteristic paint-layer thickness is optically thin in the THz regime, since it is much less than the time over which the THz pulse propagates within its duration, corresponding to the depth resolution of a typical THz-TDS system. Nonetheless, there may be spectral information present at the relevant short wavelengths that is obscured in the raw signal. In the context of THz reflectometry, when dealing with optically thin paint layers, the THz echoes resulting from the various interfaces between layers will partially or even totally overlap in time and thus these echoes will merge rather than be distinct. In addition, the depth resolution achieved by FWDD is not high enough to characterize the stratigraphy of typical easel paintings before the 19th century. Consequently, to our knowledge, the detailed stratigraphy of pre-19th century easel paintings has not been clearly revealed by THz reflectometry, as the paint layers in easel paintings are usually very thin in the THz regime, especially for the 16th and 17th century easel paintings. to ∼100 µm. Nonetheless, as we shall see, due to the nature of the painting's layer structure, we can reconstruct the stratigraphy on a scale significantly below ∼100 µm. This is also the reason the time-domain THz C-scan in Fig. 5.14(a) contains both the surface and subsurface features.

As mentioned above, the detailed stratigraphy associated with the painting itself is not evident in the raw data. Nonetheless, the sought for information is indeed contained in the data; it is a matter how to extract it in order to reconstruct the stratig- FWDD (detailed procedure in [START_REF] Dong | Terahertz Quantitative Nondestructive Evaluation of Failure Modes in Polymer-Coated Steel[END_REF]) is also employed to process the raw data for comparison. It is important to note that, in the deconvolution process, we consider the THz reference signal as the input and the reflected THz signal as the output; therefore, the actual impulse response function associated with reflection coefficients should be obtained by multiplying the deconvolved signal by a factor of -1 for phase correction.

Sparsity-based deconvolved signals

The (b3). The physical origin of this peak is the existence of an additional layer in these areas; however, this layer is too thin to be resolved even in the deconvolved signal, which generates only one observable positive peak. We summarize the presence of this additional peak, and thus delaminated regions, in the binary THz C-scan in Fig. Quantitative information in depth, such as the physical thicknesses of each resolved layer, can be estimated based on the knowledge of corresponding refractive indices.

Starting with the canvas, a ground layer (gesso) is applied to seal the canvas and to create a smooth surface on which to paint. The imprimatura, literally the 'first paint layer', follows the ground that would seal the oil-absorbent gesso layer. Without the imprimatura, paint directly applied would soak into the surface and be difficult to control. The underpainting layer provides a proper foundation of the scene/subject matter, which was typically painted in a dark and muted monochrome tone, usually consisting of umber, as brown underpainting has often be used in oil painting right from the 15th to 17th centuries. Such an approach was widespread to the point that the entire tonality and compositions of paintings frequently accounted for this dark underpainting. THz B-and C-scans clearly reveal the features of the ground, imprimatura, and the underpainting layers, which are fairly uniform across the painting.

Assuming the refractive index of gesso is about 1.52 [START_REF] Gomez-Sepulveda | History of Mexican Easel Paintings from an Altarpiece Revealed by Non-invasive Terahertz Time-Domain Imaging[END_REF], the average physical thickness of the ground layer is about 132.7 µm. For other oil-based paint layers, although the difference of refractive index between each layer is sufficient to produce the THz reflections, we assume a mean refractive index of 1.85 [START_REF] Gomez-Sepulveda | History of Mexican Easel Paintings from an Altarpiece Revealed by Non-invasive Terahertz Time-Domain Imaging[END_REF] in order to estimate the physical thickness of each paint layer. Based on this assumed refractive index, we estimate the average thickness of imprimatura is about 34 µm and the average thickness of the underpainting layer is about 38 µm. The presence of these three preparatory layers is confirmed by optical microscopy near the painting's edge, shown in Fig. 5.21, where these layers are exposed.

The pictorial layers are applied subsequent to the underpainting. They contain the visually evident composition of the finished painting, and consist of additional applications of background color, the figure of the Vièrge, facial features, hands, the gossamer veil, and other find details. In the THz C-scans, textural features due to the craquelure first become pronounced in the pictorial layer. This is also where we is estimated as 96 µm. This thick region is identified as a retouchings of the varnish.

Visual inspection shows that these regions exhibit an anomalous texture.

In Based on this work, THz reflectometry shows great potentials in providing in-situ 3D quantitative information for a broad range of art-historical studies, as well as for conservation, restoration, and authentication.

Perspective

The potential applications of THz imaging in the fields of industry and cultural heritage conservation are being explored continuously.

For the NDE of fiber-reinforced composites, THz imaging of carbon fiber-reinforced composites should be emphasized, since carbon fiber-reinforced composites are most widely used in the aerospace and automotive industries to take the place of metal.

Although the conductivity of carbon fibers limits the penetration depth of THz waves, depth can be extracted; however, it is not possible to extract spectral information at a depth of sufficient fidelity. Therefore, more efforts should be made to resolve the spectral signature of the material at depth. Finally, there is significant need for further development of more powerful signal processing routines to allow for real-time analysis of terahertz time-domain signals.

Publications

Peer-reviewed publications:

Chapter 4 : 4 . 3 . 2 Chapter 5 :Chapter 6 : 13 xOF FIGURES 1 . 1

 4432561311 Terahertz Deconvolution For Stratigraphic Characterization of Multilayered Structures . . . . . . . . . . . . . . . 4.1 Terahertz Deconvolution Based on Inverse Filtering . . . . . . . . . . 4.1.1 Terahertz frequency-wavelet domain deconvolution (FWDD) . 4.1.2 Characterization of failure modes in polymer-coated steel with FWDD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Terahertz Sparse Deconvolution . . . . . . . . . . . . . . . . . . . . . 4.2.1 Terahertz sparse deconvolution with data-resolution . . . . . . 4.2.2 Terahertz sparse deconvolution with super-resolution . . . . . 4.2.3 Terahertz sparse deconvolution considering pulse spreading . . 4.2.4 Numerical and experimental verification . . . . . . . . . . . . 4.3 Terahertz Deconvolution Based on Autoregressive Spectral Extrapolation 4.3.1 Autoregressive model for terahertz spectrum . . . . . . . . . . viii Numerical and experimental verification . . . . . . . . . . . . Terahertz Reflectometry of Art Paintings . . . . . . . . . 5.1 Terahertz stratigraphic and subsurface investigation of a 20th century Italian oil painting on paperboard . . . . . . . . . . . . . . . . . . . . 5.1.1 Terahertz raw images of After Fishing . . . . . . . . . . . . . 5.1.2 Reconstruction of stratigraphy based on FWDD . . . . . . . . 5.1.3 Correlation between terahertz and X-ray images . . . . . . . . 5.2 Terahertz characterization of the stratigraphic details of a 17th century Italian oil painting on canvas . . . . . . . . . . . . . . . . . . . . . . .5.2.1 Terahertz raw images of Vièrge en priere . . . . . . . . . . . 5.2.2 Sparsity-based deconvolved signals . . . . . . . . . . . . . . . 5.2.3 3D global mapping of detailed stratigraphy . . . . . . . . . . . Conclusion and Perspective . . . . . . . . . . . . . . . . . . 6.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix LIST OF TABLES 1.1 State-of-the-art THz imaging of actual art paintings . . . . . . . . . . LIST The terahertz portion of the electromagnetic spectrum. . . . . . . . . 2.1 Schematic diagram of THz time-domain spectroscopy (TDS) system in both reflection and transmission configurations. . . . . . . . . . . . . 2.2 Image of the advanced variable angle reflection/transmission scan module. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 (a) THz C-scan image of sample 1 with the contrast mechanism associated with the peak-to-valley difference of the reflected THz pulse within the time window 13-20 ps. (b) THz C-scan image after applying 50% rule for sizing the delaminations. . . . . . . . . . . . . . . . . . . 3.2 THz reflection B-scan images through the cross-sections labeled in Fig.3.1(a) with the contrast mechanism associated with the amplitude of the reflected THz pulse. . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Three-dimensional THz imaging for sample 1. . . . . . . . . . . . . . 3.4 Time traces of detected reflected THz signals (a) in the absence of delamination and (b) with delamination. . . . . . . . . . . . . . . . . 3.5 THz reflection images of representative samples. THz C-scans (a1) to (a6), B-scans (b1) to (b6), and time-domain waveforms (c1) to (c6) corresponding to samples 23, 10, 60, 46, 51, and 57, respectively. . . . 3.6 Ultrasonic imaging C-scan results for samples 51 and 57. (a) 51 in transmission, (b) 57 in transmission, (c) 51 in reflection, and (d) 57 in reflection modes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Ultrasonic time-domain waveforms without and with delamination for sample 51 in (a) transmission mode and (b) reflection mode. . . . . . xi 3.8 Ultrasonic B-scan images for (a) sample 51 with x position at 18 mm, (b) sample 57 with x position at 18 mm. . . . . . . . . . . . . . . . . 3.9 Estimated reflected THz pulses in region with delamination indicating the observed features associated with reflections off various interfaces.3.10 Schematic level-3 decomposition of a signal with stationary wavelet transform (SW T ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.11 The photograph of the woven E-glass fiber reinforced nylon matrix laminate. Inset: Amplification of the photograph to show the size of the fiber bundles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.12 THz reference signal with and without wavelet de-noising in (a) time domain and (b) frequency domain. The inset in (a) shows the fluctuations in the signal due to ambient water vapor. . . . . . . . . . . . . 3.13 Multiple parameter regress analysis of measured THz waveform without delamination to identify the water-vapor effect. . . . . . . . . . . 3.14 Typical reflected THz waveforms with and without wavelet de-noising in the region (a) without and (b) with delamination. . . . . . . . . . 3.15 THz C-scan images based on THz waveforms (a) without and (b) with wavelet de-noising. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.16 Binary THz C-scan images based on THz waveforms (a) without and (b) with wavelet de-noising. . . . . . . . . . . . . . . . . . . . . . . . 3.17 THz B-scan images based on THz waveforms (a) without and (b) with wavelet de-noising. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.18 Binary THz B-scan images based on THz waveforms (a) without and (b) with wavelet de-noising. . . . . . . . . . . . . . . . . . . . . . . . 3.19 Schematic diagram of hybrid fiber-reinforced composite sample indicating dimensions and internal structure. . . . . . . . . . . . . . . . . 3.20 Schematic diagram (edge view) of sample subjected to low-velocity impact damage with highlighted damage types. . . . . . . . . . . . . 3.21 Ultrasonic C-scan results (a) in transmission and (b) in reflection modes. The ultrasonic waves are incident from the top surface of the sample. The contrast is based on the absolute value of the amplitude of the second-round signal in transmission and the second echo in reflection. xii 3.22 Estimated reflected THz waveforms (a) in undamaged region and (b) in region with delamination. . . . . . . . . . . . . . . . . . . . . . . . 3.23 Typical time-of-flight waveforms from an undamaged region with THz polarization parallel and perpendicular to the carbon fibers. . . . . . 3.24 THz C-scans of the top surface with polarization perpendicular (a) and parallel (b) to the carbon-fiber orientation. . . . . . . . . . . . . . . . 3.25 THz C-scans of the Interface I with polarization perpendicular (a) and parallel (b) to the carbon-fiber orientation. . . . . . . . . . . . . . . . 3.26 THz B-scans incident on the top surface (along section y=0) and with polarization perpendicular (a) and parallel (b) to the carbon-fiber orientation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.27 THz C-scans of the bottom surface with polarization perpendicular (a) and parallel (b) to the carbon fiber orientation. . . . . . . . . . . . . 3.28 THz C-scans of the Interface II with polarization perpendicular (a) and parallel (b) to the carbon fiber orientation. . . . . . . . . . . . . . . . 3.29 Temporal waveforms for perpendicular THz polarization at selected positions on sample (see Fig. 11). . . . . . . . . . . . . . . . . . . . . 3.30 B-scans incident from the bottom surface (along section y=0) and with polarization perpendicular (a) and parallel (b) to the carbon-fiber orientation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Hanning window function with typical values, t 0 =10 ps and f c =4 THz, in the time domain and its Fourier transform (power spectrum) in the inset. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 (a) Visible photograph of the surface of the coating sample with four typical positions highlighted to represent different failure modes. This photograph is also labeled with the pixel number which is the same as the THz C-scans in the following section. (b) Optical micrograph from the edge of the coated sample to roughly estimate the thicknesses of the coating and the steel plate. . . . . . . . . . . . . . . . . . . . . . 4.3 The THz raw signals [(a1) and (b1)] and deconvolved signals [(a2) and (b2)] at positions (50, 60) and (70, 190), with the corresponding representations of round-trip echoes in [(a3) and (b3)]. . . . . . . . . xiii 4.4 (a) THz C-scan based on the raw signals with the contrast mechanism as the maximum amplitude of the received signal, which is similar to the optical photograph in Fig. 4.2(a); (b) THz C-scan based on the deconvolved signals associated with the delamination, which indicates the regions with delamination and the physical thicknesses of the delamination across the coating plane. . . . . . . . . . . . . . . . . . . . 4.5 THz raw signals [(a1) and (b1)] with the corresponding deconvolved signals [(a2) and (b2)] at positions (79, 115) and (21, 121) where a blister is visually present. . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 (a) THz C-scan based on the optical delay between the first positive peak and second positive peak in the deconvolved signals, which represents the optical distance between the coating surface and the top surface of the metal substrate; (b) THz C-scan based on the amplitude of the second positive peak in the deconvolved signals, which indicates the adhesive condition. . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Optical micrographs associated with different failure modes after peeling off the coating. (a) Oxidized substrate in the blistering area; (b) Oxidized substrate in the delamination area; (c) Comparison between the steel substrate at the undamaged area and the blistering area by peeling off the coating at one blister area and the adjacent undamaged area; (d) Grown metal oxide due to corrosion along the scratch. . . . 4.8 THz image of the thickness distribution, as well as the surface topology, of the coating. Different failure modes, including corrosion, delamination, and blistering, have been characterized quantitatively in three dimensions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9 Comparisons between the assumed (a)f 0 [n] and (b)f 0 [n], and the deconvolution results obtained by sparse deconvolution and FWDD respectively. Insets are the zoom-in images of corresponding boxed peaks. 4.10 Comparisons between the assumed (a)f 0 [n] and (b)f 0 [n], and the deconvolution results obtained by DR sparse deconvolution, SR sparse deconvolution with K = 4 and FWDD respectively. Insets are the zoom-in images of corresponding boxed peaks. . . . . . . . . . . . . . 4.11 Two samples with multi-layered structures. (a) Sample A: a threelayered structure, which is composed of one copy paper, air gap and one polymer coating; (b) Sample B: a seven-layered structure, which is composed of four different papercards with air gaps between them. xiv 4.12 The THz reference signal used in the deconvolution of experimental THz received signals, which contains 512 data points with sampling period T s = 0.093 ps. The inset is the frequency spectrum of this reference signal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.13 The raw and deconvolved signals for Sample A. (a) The received THz signal from Sample A; (b) The deconvolved signal with FWDD; (c) The deconvolved signal with DR sparse deconvolution. . . . . . . . . 4.14 Comparison between the DR deconvolved sigal and the SR deconvolved signals with the up-sampling factor K = 2, 4 and 8. Two insets are the zoom-in version of the first and fourth peaks. . . . . . . . . . . . . . . 4.15 Comparsion between the received signal and the re-convolution signal. (a) Comparison between the re-convolution without considering the pulse spreading effect and the actual received signal, and the inset is the zoom-in of the last echo; (b) Comparison between the re-convolution considering the pulse spreading effect and the actual received signal, and the inset is the zoom-in of the last echo; (c) The deviation between h(t)⊗f DR (t) and y 0 (t) in (a); (d) The deviation between h a (t)⊗f DR (t) and y 0 (t) in (b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.16 Comparison between the deconvolved signals with and without considering the temporal pulse spreading effect. (a) The deconvolved signal with FWDD; (b) The deconvolved signal with sparse deconvolution without considering the pulse spreading; (c) The deconvolved signal with sparse deconvolution including pulse spreading; (d) The zoom-in of the last three interefaces in (b); (e) The zoom-in of the last three interefaces in (c). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.17 The assumed impulse response function h 0 and the simulated received THz signal r 0 . The inset shows the actual THz reference signal f 0 . . . 4.18 (a) Comparison between the assumed frequency spectrum H 0 , the deconvolved frequency spectrum by direct inverse filtering H inv and the estimated frequency spectrum by AR spectral extrapolation H AR . (b) The zoom-in of (a), which shows the detailed data allocation for the AR model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.19 Comparison of the simulated deconvolution results obtained by AR deconvolution with p = 44, FWDD and the assumed impulse response function. The inset shows the AIC values with different model orders. xv 4.20 (a) The deconvolution results based on FWDD and AR deconvolution. The inset shows the received THz signal reflected from the poly-coated steel sample; (b) The estimated frequency spectrum (up to 5 THz) based on autogressive spectral extrapolation. . . . . . . . . . . . . . . 5.1 Images of After Fishing by Ausonio Tanda. (a) Visible photograph of After Fishing, and (b) X-ray transmission image of After Fishing. . . 5.2 THz C-scan image of After Fishing in time domain based on the raw data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 THz C-scan images of After Fishing in frequency domain at (a) 0.4, (b) 0.6, (c) 0.8, and (d) 1.2 THz. . . . . . . . . . . . . . . . . . . . . 5.4 THz imaging results of the 'Spots' location in the paperboard. . . . 5.5 THz frequency-wavelet deconvolution result for a typical THz waveform.111 5.6 Four different types of the deconvolved signals with (a) (d) corresponding to Type I Type IV. . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 THz C-scans based on the Type I signal and the main peak indicating the surface feature of the painting. . . . . . . . . . . . . . . . . . . . 5.8 THz C-scans based on the Type II and III signals indicating the existence of the underlying layer. . . . . . . . . . . . . . . . . . . . . . . 5.9 Comparison between THz B-scans based on the raw data and binary THz B-scans based on the deconvolved data. . . . . . . . . . . . . . 5.10 THz image of the thickness distribution of the applied paint layers calculated with an estimated refractive index of the pigments. . . . . 5.11 Raking light image of After Fishing. . . . . . . . . . . . . . . . . . . . 5.12 The fused image of the X-ray image in red and THz thickness distribution image in green to distinguish the areas of similar intensity in yellow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.13 Images of Vierge en priere. (a) Visible photograph of the Viège en priere; Image of Viège en priere obtained by UV Fluorescence; (c) Image of Viège en priere obtained by IR Reflectography. . . . . . . . xvi 5.14 THz C-scans based on the raw THz data based on the following contrast mechanisms in the reflected signal: (a) peak-to-valley amplitude; (b) integrated spectrum between 0.5 and 1.0 THz; (c) peak-to-valley amplitude between 13 to 18 ps. The horizontal line at Y = 150, marked cross section, is in reference to the B-scan in Fig. 5.15. Contrast is normalized to one. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.15 The THz B-scan based on the raw signal with the cross-section Y = 150. Two typical THz reflected signals are plotted corresponding to the THz B-scan, with the waveform at pixel (89, 150) in green and the waveform at pixel (124, 150) in blue. . . . . . . . . . . . . . . . . . . . . . . . . 5.16 Typical THz reflected signals and the sparsity-based deconvolved signals. Figures 5.16(a1), (a2), and (a3) show examples of raw reflected signals (black) and signals reconstructed from the sparse-deconvolution and reference signals. Figures 5.16

1 INTRODUCTION 1 . 1

 111 (b1), (b2), and (b3) show the corresponding sparsity-based deconvolution signals (red); Fig. 5.16(b1) also shows the deconvolved signal achieved by FWDD for comparison. . . 5.17 Comparison for optical delay corresponding to the layers above the canvas between (a) the THz B-scan based on the raw data (Fig. 5.15 on an expanded vertical scale) and (b) the binary THz B-scan based on the deconvolved data with the cross-section Y = 150 in which a valid peak is assigned value '1' and the other positions '0' regardless of the sign or height of the peak. . . . . . . . . . . . . . . . . . . . . . . . . 5.18 Varnish features revealed by THz sparsity-based deconvolved signals. (a) THz C-scan based on the amplitude of the first peak of the deconvolved signal. The contrast is normalized to one; (b) THz C-scan based on the optical delay between the first and second peaks of the deconvolved signal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.19 Features of the pictorial layer revealed by THz sparsity-based deconvolved signals. (a) THz C-scan based on the amplitude of second peak of the deconvolved signals; (b) THz C-scan based on the amplitude of the peak corresponding to the interface between the pictorial and underpainting layers; (c) binary THz C-scan indicating the positions with the existence of delaminations; (d) THz C-scan based on the optical delay between the peaks corresponding to the thickness of the pictorial layer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii 5.20 Features of the underpainting, imprimatura, and ground layers revealed by THz C-scans based on sparsity-based deconvolved signals. THz Cscans based on the amplitude of (a1) the fourth, (b1) fifth, and (c1) sixth peak. Contrast is normalized to one. THz C-scans based on the optical delay between (a2) the third and fourth peaks, (b2) fourth and fifth peaks, and (c2) fifth and sixth peaks. . . . . . . . . . . . . . . . 5.21 The optical microscopic image of a small region at the edge of Vièrge en priere. The applications of ground, imprimatura, and the underpainting can be clearly identified. . . . . . . . . . . . . . . . . . . . . 5.22 White-light raking images of Vièrge en priere with the light source (a) placed on the left side and (b) the top side of the painting. . . . . . . xviii SUMMARY Thanks to the emergence of powerful light sources and highly sensitive detectors operating in the terahertz (THz) spectral region, the traditional THz 'gap' (0.1-10 THz) is shrinking fast nowadays. Recent advances in photonics and electronics enables the development of compact but sophisticated THz time domain spectroscopy (TDS) systems. Today, pulsed THz imaging based on THz-TDS systems is investigated in many areas spanning from nondestructive evaluation (NDE), industrial quality control, medical imaging and security sectors to studies of fundamental physics and cultural heritage. Despite the on-going broad applications of THz imaging, a THz 'killer application'-a novel, innovative use in which THz imaging exhibits higher performance than any other existing techniques, is still waiting to be clearly identified. Therefore, the research problems of this thesis aim at the exploration of the potential of THz imaging in the fields of industry and cultural heritage conservation science, and developing theories and techniques to enhance the capabilities of THz imaging to solve practical problems in NDE and material characterization. The thesis is outlined as follows. Chapter 1 introduces the background of this thesis. The basic knowledge of THz radiation is briefly introduced. The motivation and research problems of this project are described. The state-of-the-art of the development and applications of THz imaging in the fields of industry and cultural heritage conservation science are also summarized in this chapter. Chapter 2 presents the main equipment, the THz time-domain spectroscopy (TDS) system, utilized in this project. The main imaging contrasts, based on which the THz results are presented, are briefly described. Chapter 3 is devoted to NDE of fiber-reinforced composites with THz imaging. Fiber-reinforced composites are widely used in the industry to take the place of conxix ventional materials, such as metal. In this chapter, three case studies are presented. The first case study shows the capability of THz imaging in providing 3D quantitative information for characterization of delaminations in glass-fiber reinforced composites. The spatial resolution, including the lateral resolution and the axial resolution of THz reflective imaging are discussed in detail in this study. The second case study takes the water-vapor influence into account, especially for THz characterization of small delaminations in woven glass-fiber reinforced composites. Wavelet de-noising is applied to cancel the noise due to the atmospheric water-vapor. In view of the difficulty and expense in purging with dry nitrogen to physically remove the watervapor features in many cases, the demonstrated results will be of great interest for a range of laboratory-and field-based THz NDE applications. In the third case study, polarization-resolved THz imaging is firstly applied to a hybrid fiber-reinforced composite laminate. The evolution of intra-and inter-laminar damage throughout the thickness of the laminate subject to low-velocity impact is evaluated. Inter-laminar damage at the interface and the intra-laminar damage close to the same interface can be differentiated by taking advantage of the sensitivity of the carbon-fiber orientation to the THz polarization. Comparative studies with ultrasonic imaging are also involved in the chapter. Chapter 4 focuses on developing advanced THz deconvolution techniques to enhance the capability of THz imaging for the stratigraphic characterization of multilayered structures with optically thin layers in the THz regime. Generally, layers with thicknesses smaller than 100 µm cannot be easily resolved based on the raw THz signals, due to the depth-resolution achieved by a typical THz TDS system, which further limits the applications of THz imaging. Deconvolution, if applied properly, can yield sub-wavelength or sub-pulse-width depth resolution. In the context of THz imaging, three kinds of deconvolution methods are developed in this chapter. Conventional deconvolution, such as THz frequency-wavelet domain deconvolution (FWDD), xx which is based on inverse filtering is introduced. The depth-resolution achieved by FWDD is about 55 µm in air. FWDD is used in a case study for the characterization of various failure modes in polymer-coated. However, depth-resolution achieved by FWDD cannot satisfy all of the expected applications of THz imaging. In order to further enhance the depth-resolution, sparse representation is exploited, and sparse deconvolution based on an iterative shrinkage algorithm is developed. With an upsampling approach, sparse deconvolution with super-resolution is also developed to overcome the time resolution limited by the sampling period in the measurement. A simple but effective time-domain model for describing the temporal pulse spreading due the frequency-dependent loss is also designed and introduced into the algorithm, which significantly improves the performance of sparse deconvolution in processing time-varying pulses during the propagation of THz waves in materials. The depthresolution achieved by sparse deconvolution is about 45 µm in air. Finally, a novel method for enhancing the depth-resolution of THz deconvolution based on autoregressive spectral extrapolation is developed. An autoregressive process is modeled based on the THz frequency components with high SNR, and the missing frequency components in the low SNR regions are extrapolated based on the autoregressive model.In this way, the entire THz frequency spectrum of the impulse response function is recovered. This method is able to provide a 'quasi-ideal' impulse response function, and therefore, significantly enhances the depth-resolution for resolving optically thin layers in the THz regime. The depth-resolution achieved by this method is about 27 µm in air. The deconvolution techniques developed and presented in this chapter greatly enhance the capability for resolving optically thin layers in the THz regime, and provide a powerful tool to the entire THz community for a broad applications. Chapter 5 is addressed to THz reflectometry of art paintings. Based on the advanced deconvolution techniques developed in Chapter 4, THz reflectometry with enhanced depth-resolution presents great potentials for the characterization of stratigxxi raphy in historical paintings in this chapter. Compared with other techniques, THz reflectometry can provide 3D global mapping of the stratigraphy quantitatively. Two historical paintings are well-studied via THz reflectometry. The first one is a 19th century Italian oil painting on paperboard, After Fishing. Based on the deconvolved signals achieved by FWDD, the stratigraphy and subsurface features are clearly revealed. The second historical painting is a 17th century Italian easel painting, Vièrge en priere. Sparsity-based THz reflectometry is applied to characterize the detailed stratigraphy. Based on the sparsity-based deconvolved signals, the detailed layer structure, including the canvas, the ground, imprimatura, underpainting, pictorial, and varnish layers are clearly identified. In addition, a hitherto unidentified restoration of the varnish has been found. It is important to note that, it is the first time to our knowledge THz reflectometry has resolved multiple layers in an easel painting. Based on this work, the full promise of THz reflectometry to provide a global and detailed account of an easel painting's stratigraphy has been unlocked by exploitation of the sparse deconvolution. Chapter 6 is divided into three sections. Firstly, the conclusion part summarizes the results and contributions of this thesis. Secondly, further possible research directions are proposed in the perspective part. The peer-reviewed publications and conferences attended during the time of Ph.D. study are listed at the end of this chapter. xxii CHAPTER Motivation and research problems The terahertz (THz) portion of the electromagnetic spectrum, shown in Fig.1.1, extends from approximately 100 GHz to 10 THz, and lies between the microwave and infrared; the wavelength in this range is 3 mm to 30 µm. THz waves can penetrate numerous nonmetallic materials that may be opaque in the range of visible and infrared light. Moreover, as nonionizing radiation, THz waves present minimal known health risks. The potential and suitability of the THz technology for practical applications such as nondestructive evaluation (NDE) and material characterization, have been blocked for many years due to the technical difficulty of producing efficient sources and detectors. However, with the development of ultrafast components in both photonics and electronics, the situation has evolved drastically in the last two decades. This evolution, has enabled the introduction of the THz technology to the NDE and material characterization fields, quickly highlighting the advantages of the THz technology for the evaluation and characterization of advanced materials and complex structures.

Figure 1 . 1 :

 11 Figure 1.1: The terahertz portion of the electromagnetic spectrum.

  Rutz et al. showed that the fiber content can be characterized by measuring the real refractive index of the composites and the fiber orientation can also be estimated by analyzing THz absorption at different polarizations[1]. Jördens et al. studied the birefringent properties of plastics filled with differing amounts of short glass fibers and measured the preferential fiber orientations as well as the fraction of oriented fibers based on the THz birefringence of the composites[6].Due to the remarkable properties of THz waves, THz imaging has already been explored for detecting various damages and defects during the manufacturing and maintenance of composites. For composites reinforced by non-conductive fibers, such as glass fibers and ultra-high molecular weight polyethylene (UHMWPE) fibers, THz imaging can be utilized to detect both surface and underlying defects. For carbon fiber composites, due to the conductivity of carbon fibers, THz imaging has largely been applied to detect surface or near-surface damages.Impact damage is a typical damage that composites can suffer both in service and during maintenance due to many different types of collisions. The resulting impactinduced damage in composite laminates involves a combination of intra-laminar damage and inter-laminar damage, which leads to the separation of adjacent plies (delamination). Delaminations are considered as the most important failure mechanism, because they can severely degrade the strength and the integrity of the structure, and may propagate undetected during service leading to a significant decrease in stability and durability. Ospald et al. systematically studied a wide range of samples based on glass fiber composites with various types of defects with THz waves[7]. Impact damages (mainly fiber distortions and matrix cracking) caused by four different impact energies between 15 and 40 J were clearly observed and delaminations (also artificial inserts) in different depths of a sandwich structure were also successfully detected.Im et al. studied the penetration ability of THz waves in unidirectional carbon fiber composites and found that it depends on the relation between the THz polarization and carbon fiber orientation[8]. Bezborodov et al. characterized the impact damage on the surface of carbon fiber composites by monitoring the reflection coefficients across the surface and showed that the reflection coefficient in regions with and without impact damage can be more easily distinguished with polarization parallel to the carbon fiber orientation[9]. Palka et al. applied THz reflective imaging to evaluate an UHMWPE composite sample punctured by a projectile and 3D distribution of the delaminations inside the sample were clearly revealed[START_REF] Palka | 3D non-destructive imaging of punctures in polyethylene composite armor by THz time domain spectroscopy[END_REF].

Figure 2 . 1 :

 21 Figure 2.1: Schematic diagram of THz time-domain spectroscopy (TDS) system in both reflection and transmission configurations.

Figure 2 . 2 :

 22 Figure 2.2: Image of the advanced variable angle reflection/transmission scan module.

3. 1 Figure 3 . 1 :

 131 Figure 3.1: (a) THz C-scan image of sample 1 with the contrast mechanism associated with the peak-to-valley difference of the reflected THz pulse within the time window 13-20 ps. (b) THz C-scan image after applying 50% rule for sizing the delaminations.

Fig. 3 .Figure 3 . 2 :Figure 3 . 3 :

 33233 Figure 3.2: THz reflection B-scan images through the cross-sections labeled in Fig.3.1(a) with the contrast mechanism associated with the amplitude of the reflected THz pulse.

Figure 3 . 4 :

 34 Figure 3.4: Time traces of detected reflected THz signals (a) in the absence of delamination and (b) with delamination.

Fig. 3 .Figure 3 . 5 :Figure 3 . 6 :Figure 3 . 7 :Figure 3 . 8 :

 335363738 Fig.3.6 shows the ultrasonic C-scan images for samples 51 and 57 in both transmission and reflection, showing that ultrasonic C-scans can also successfully identify the delamination areas. The contrast for ultrasonic C-scans are based on the absolute value of the amplitude of the ultrasonic transmitted or reflected wave. The typical transmitted and reflected waveforms obtained with and without the delamination are compared in Fig.3.7. Note the much larger timescale of the ultrasound pulses versus the THz pulses. In the transmission mode shown in Fig.3.7(a), the transmitted waveform in the windowed time slice is chosen to provide sharper contrast for the ultrasonic C-scans, because this transmitted waveform travels the samples three times. In the reflection mode shown in Fig.3.7(b), the second echo which is also in the windowed time slice, is chosen to obtain the ultrasonic C-scan images, since the first echo is mainly the specular reflection from the sample surface.With the ultrasonic waveforms in reflection mode, the ultrasonic B-scan images can also be obtained, shown in Fig.3.8. In the B-scan images, we can still clearly

Figure 3 . 9 :

 39 Figure 3.9: Estimated reflected THz pulses in region with delamination indicating the observed features associated with reflections off various interfaces.

Figure 3 . 10 :

 310 Figure 3.10: Schematic level-3 decomposition of a signal with stationary wavelet transform (SW T ).

) 4 .

 4 Perform the inverse stationary wavelet transform (ISW T ) to recover the time domain signal x(n), x(n) = ISW T ([cA, c D]) (3.10) 3.2.3 Enhanced terahertz imaging with wavelet de-noising A woven (2 × 2 twill weave) E-glass fiber reinforced nylon matrix laminate, shown in Fig. 3.11, is employed in this study. The dimensions of the laminate are 50 mm (length) × 50 mm (width) × 1.54 mm (thickness), and the surfaces are flat. The laminate is composed of three glass fabric layers stacked as (0 • /90 • ) with fiber volume fraction about 50 vol.%. The size of the fiber bundles is 1200 tex (weight in grams per kilometer of yarns), shown in the inset of Fig. 3.11. Delamination was introduced between the first and second fabric layer by incorporating a Upilex (ultra-high heatresistant polymide) release film disk with thickness about 25 µm and diameter about 12 mm prior to consolidation of the laminate.

Figure 3 . 11 :

 311 Figure 3.11: The photograph of the woven E-glass fiber reinforced nylon matrix laminate. Inset: Amplification of the photograph to show the size of the fiber bundles.

Figure 3 . 12 :

 312 Figure 3.12: THz reference signal with and without wavelet de-noising in (a) time domain and (b) frequency domain. The inset in (a) shows the fluctuations in the signal due to ambient water vapor.

Figure 3 . 13 :

 313 Figure 3.13: Multiple parameter regress analysis of measured THz waveform without delamination to identify the water-vapor effect.

Figure 3 . 14 :Figure 3 . 15 :Figure 3 . 16 :

 314315316 Figure 3.14: Typical reflected THz waveforms with and without wavelet de-noising in the region (a) without and (b) with delamination.

Figure 3 .

 3 Figure 3.17: THz B-scan images based on THz waveforms (a) without and (b) with wavelet de-noising.

Figure 3 . 18 :

 318 Figure 3.18: Binary THz B-scan images based on THz waveforms (a) without and (b) with wavelet de-noising.

Figure 3 . 19 :Figure 3 . 20 :Figure 3 . 21 :

 319320321 Figure 3.19: Schematic diagram of hybrid fiber-reinforced composite sample indicating dimensions and internal structure.

Figure 3 . 22 :

 322 Figure 3.22: Estimated reflected THz waveforms (a) in undamaged region and (b) in region with delamination.

Figure 3 . 23 :

 323 Figure 3.23: Typical time-of-flight waveforms from an undamaged region with THz polarization parallel and perpendicular to the carbon fibers.

Figure 3 . 24 :Figure 3 . 25 :Figure 3 . 26 :

 324325326 Figure 3.24: THz C-scans of the top surface with polarization perpendicular (a) and parallel (b) to the carbon-fiber orientation.

Figure 3 . 27 :Figure 3 . 28 :

 327328 Figure 3.27: THz C-scans of the bottom surface with polarization perpendicular (a) and parallel (b) to the carbon fiber orientation.

Figure 3 . 29 :

 329 Figure 3.29: Temporal waveforms for perpendicular THz polarization at selected positions on sample (see Fig. 11).

Figure 3 . 4 TERAHERTZ

 34 Figure 3.30: B-scans incident from the bottom surface (along section y=0) and with polarization perpendicular (a) and parallel (b) to the carbon-fiber orientation.

4. 1

 1 Terahertz Deconvolution Based on Inverse Filtering For samples with layered structure, time-domain THz imaging can provide information in depth by analyzing the reflected THz signals with an incident approximately single-cycle THz pulse. Due to dielectric variations with depth, reflected temporal THz echoes associated with the Fresnel coefficients between various layers are recorded as a function of transverse position in amplitude and time delay. The echo amplitudes provide information on the various refractive indices, while the time delays between THz echoes provide the optical thickness of successive layers. With a knowledge of the refractive indices of the corresponding materials, the physical layer thickness can be extracted. When the layer thickness is optically thin, THz echoes will partially or totally overlap; therefore, the amplitude and time delay cannot be directly extracted from the time-domain waveform. In this case, THz deconvolution can be utilized to resolve the overlapping echoes and reconstruct the intrinsic impulse-response function, and hence the physical structure of the sample studied.

Figure 4 . 1 :

 41 Figure 4.1: Hanning window function with typical values, t 0 =10 ps and f c =4 THz, in the time domain and its Fourier transform (power spectrum) in the inset.

F

  (ω) for typical parameters (t 0 = 10 ps and f c = 4 THz) is shown in Fig. 4.1.

4. 1 . 2

 12 Characterization of failure modes in polymer-coated steel with FWDD In this subsection, a case study about the implementation of FWDD for real industrial application is presented. THz reflective imaging is applied to characterize the failure modes in a polymer coating on a steel plate, and FWDD is implemented to resolve the temporally overlapping echoes in order to quantitatively characterized of different failure modes, viz. corrosion, delamination, and blistering. THz images in three dimensions clearly exhibit the coating thickness distribution across the entire damaged coating, highlighting the THz features associated with different failure modes, thus demonstrating that THz imaging can be considered as an effective modality for characterizing damage mechanisms in polymer coatings on metals. In this study, THz imaging is performed in reflection at almost normal incidence. Before imaging the sample, a THz reference signal, shown in Fig.4.12, was recorded by setting a bare metal plate (i.e., an excellent THz reflector) at the sample position. The tested sample, shown in Fig. 4.2(a), is a one-layered polymer (polyester) coating on a zinc-coated steel substrate. Prior to the application of the polymer coating, a thin (∼5 µm) proprietary primer coating was applied. Both the primer and polymer were roll coated onto the substrate. The coating was initially scratched through the coating in the center, and after a multi-month accelerated-corrosion exposure, various types of failure, including corrosion, delamination, and blistering, are visually evident in the coating. Optical microscopy was employed to estimate the thickness of the coating, shown in Fig. 4.2(b). (The primer coating was not evident in the optical micrographs.) The thickness of the coating is about 0.05 mm based on observation from the edge of the sample. We then proceeded with the THz imaging. This sample was raster-scanned by a set of motorized stages moving in x and y directions in 0.2 mm steps over a 33 mm × 55 mm region of the sample plane, corresponding to 165 × 275 pixels. Each recorded reflected temporal THz waveform contains 4096 data points, and the signal is averaged over 10 shots per pixel. After completing the scanning, a 3D volume raw data set was acquired.

Figure 4 . 2 :

 42 Figure 4.2: (a) Visible photograph of the surface of the coating sample with four typical positions highlighted to represent different failure modes. This photograph is also labeled with the pixel number which is the same as the THz C-scans in the following section. (b) Optical micrograph from the edge of the coated sample to roughly estimate the thicknesses of the coating and the steel plate.

Figure 4 . 3 :

 43 Figure 4.3: The THz raw signals [(a1) and (b1)] and deconvolved signals [(a2) and (b2)] at positions (50, 60) and (70, 190), with the corresponding representations of round-trip echoes in [(a3) and (b3)].

Fig. 4

 4 Fig.4.2(b).

Figure 4 . 4 :

 44 Figure 4.4: (a) THz C-scan based on the raw signals with the contrast mechanism as the maximum amplitude of the received signal, which is similar to the optical photograph in Fig. 4.2(a); (b) THz C-scan based on the deconvolved signals associated with the delamination, which indicates the regions with delamination and the physical thicknesses of the delamination across the coating plane.

Figure 4 . 5 :Figure 4 . 6 :

 4546 Figure 4.5: THz raw signals [(a1) and (b1)] with the corresponding deconvolved signals [(a2) and (b2)] at positions (79, 115) and (21, 121) where a blister is visually present.

Figure 4 . 7 :

 47 Figure 4.7: Optical micrographs associated with different failure modes after peeling off the coating. (a) Oxidized substrate in the blistering area; (b) Oxidized substrate in the delamination area; (c) Comparison between the steel substrate at the undamaged area and the blistering area by peeling off the coating at one blister area and the adjacent undamaged area; (d) Grown metal oxide due to corrosion along the scratch.

Figure 4 . 8 :

 48 Figure 4.8: THz image of the thickness distribution, as well as the surface topology, of the coating. Different failure modes, including corrosion, delamination, and blistering, have been characterized quantitatively in three dimensions.

  reflected signals from multi-layered structures are a class of very special signals comprised of a limited number of echoes, and the corresponding impulse response functions have a sparse representation, which means only a limited number of data points have non-zero values. This feature enables us to exploit the sparse constraint and retrieve the impulse response function by sparse deconvolution. Compared with conventional deconvolution, sparse deconvolution is a pure time-domain method, therefore, there is no introduction of the high-and low-frequency noises mentioned above, and it can be expected to achieve a more clear representation of the material structures with super-resolution. For one-dimensional NDE problems, super-resolution techniques seek to recover a high-resolution signal sequence from one or a set of low-resolution acquisition systems. However, sparse deconvolution has not been studied in detail nor has it been extensively used to process THz NDE signals, and to our knowledge, only one attempt has been reported so far. In[START_REF] Parrott | Terahertz pulsed imaging in vivo: measurements and processing methods[END_REF], sparse deconvolution is briefly introduced and used to process THz signals reflected from a human palm.In this section, the principle of sparse deconvolution and a computationally efficient iterative shrinkage algorithm for sparse deconvolution are demonstrated to process THz signals reflected from multi-layered structures. In order to increase the time-resolution, which is limited by the data sampling period, a super-resolution model for sparse deconvolution is developed by an up-sampling approach. Based on the assumption that the THz echoes are the time-shifted, amplitude-scaled replicas of the THz reference signal, both FWDD and sparse deconvolution are mainly limited to deconvolve reflected THz signals with time-invariant pulses. However, in practice, the temporal pulse spreading caused by frequency-dependent attenuation and dispersion during the propagation of THz waves in materials does occur and, therefore, lowers the performance of deconvolution. In order to solve this problem, a simple but effective time-domain model for describing the pulse spreading is designed and introduced into the algorithm for sparse deconvolution to enhance its ability to deconvolve time-varying echoes. Both numerical simulations and experimental measurements are performed to prove the effectiveness of sparse deconvolution, and the results are also compared with those obtained by FWDD.

  )

  represents N times self-convolution with the kernel function ρ ∆z (t).In the discrete model, the distance slice ∆z = cT s corresponds to the data sampling period T s , with c the propagating speed of THz waves in the material. Because ∆z is small, the temporal spreading of the THz pulse between the adjacent distance slices must be relatively small, which implies that the discrete form of the impulse response function ρ ∆z [n] should be close to a Dirac function delayed one sampling period. This means ρ ∆z [n] ≈ δ[n -1]. In this paper, we model the impulse reponse function ρ ∆z[n]

. 24 )Figure 4 . 9 :

 2449 Figure 4.9: Comparisons between the assumed (a)f 0 [n] and (b)f 0 [n], and the deconvolution results obtained by sparse deconvolution and FWDD respectively. Insets are the zoom-in images of corresponding boxed peaks.

. 25 )

 25 in which the time interval between two peaks equals 25T s . The deconvolution results are shown in Fig.4.9(b). We observe that two peaks can be identified based on the results from DR sparse deconvolution, while only one peak can be located based on the results from FWDD. It is important to note that, in practice, for both FWDD and sparse deconvolution, the minimal thickness resolution, corresponding to the depthresolution, is dependent on the coherence length, the sampling frequency, and the SNR. Next, SR sparse deconvolution is utilized to recover f 0 [n] and f 0 [n] based on signals with lower time-resolution. Both f 0 [n] and f 0 [n] are convolved with the reference signal h[n], and after convolution, the signals are under-sampled by factor 8 to simulate the received signals y 0 [m] and y 0 [m], which contain only 512 data points corresponding to a data sampling period T s = 8T s . In this case, we consider the timeresolution of y 0 [m] and y 0 [m] as the data-resolution. Both DR sparse deconvolution and SR sparse deconvolution are performed based on y 0 [m], y 0 [m] and h [m], which is accordingly the under-sampled version of h[n] by a factor 8. In this simulation, the maximum under-sampling factor is limited by the Nyquist sampling frequency to ensure all the information in the received signal is sampled. The deconvolution results are shown in Fig. 4.10 for both cases. The DR deconvolution results, f DR [m] and f F W DD [m], which are obtained from DR sparse deconvolution and FWDD respectively, cannot recover the assumed impulse response functions with enough accuracy, as the locations of the peaks in f DR [m] and f F W DD [m] do not exactly match the peaks in f 0 [n]. In order to increase the accuracy and timeresolution, SR sparse deconvolution based on Eq. 4.15 is employed, and the superresolution achieved is determined by the up-sampling factor K. In our simulations, with K = 4, the SR deconvolution results f SR [k] successfully recover the assumed impulse response function f 0 [n], as the locations of the peaks in f SR [k] exactly coincide with the assumed peaks in f 0 [n]. It is noted that the accuracy of f SR [k] is also affected by the start point in the under-sampling period; in this simulation, with only 4 times up-sampling, we can achieve an accurate recovery. The results thus show that SR sparse deconvolution can overcome the limitation of data-resolution and recover the impulse response function with super-resolution.

Figure 4 . 10 :

 410 Figure 4.10: Comparisons between the assumed (a)f 0 [n] and (b)f 0 [n], and the deconvolution results obtained by DR sparse deconvolution, SR sparse deconvolution with K = 4 and FWDD respectively. Insets are the zoom-in images of corresponding boxed peaks.

Figure 4 . 11 :

 411 Figure 4.11: Two samples with multi-layered structures. (a) Sample A: a threelayered structure, which is composed of one copy paper, air gap and one polymer coating; (b) Sample B: a seven-layered structure, which is composed of four different papercards with air gaps between them.

Figure 4 . 12 :

 412 Figure 4.12: The THz reference signal used in the deconvolution of experimental THz received signals, which contains 512 data points with sampling period T s = 0.093 ps. The inset is the frequency spectrum of this reference signal.

Figure 4 . 13 :Figure 4 . 14 :

 413414 Figure 4.13: The raw and deconvolved signals for Sample A. (a) The received THz signal from Sample A; (b) The deconvolved signal with FWDD; (c) The deconvolved signal with DR sparse deconvolution.

Figure 4 . 15 :

 415 Figure 4.15: Comparsion between the received signal and the re-convolution signal. (a) Comparison between the re-convolution without considering the pulse spreading effect and the actual received signal, and the inset is the zoom-in of the last echo; (b) Comparison between the re-convolution considering the pulse spreading effect and the actual received signal, and the inset is the zoom-in of the last echo; (c) The deviation between h(t) ⊗ f DR (t) and y 0 (t) in (a); (d) The deviation between h a (t) ⊗ f DR (t) and y 0 (t) in (b).

Figure 4 . 16 :

 416 Figure 4.16: Comparison between the deconvolved signals with and without considering the temporal pulse spreading effect. (a) The deconvolved signal with FWDD; (b) The deconvolved signal with sparse deconvolution without considering the pulse spreading; (c) The deconvolved signal with sparse deconvolution including pulse spreading; (d) The zoom-in of the last three interefaces in (b); (e) The zoom-in of the last three interefaces in (c).

  .16(b), are eliminated, and all the interfaces are clearly located by sharp pulses. Furthermore, the accuracy of the deconvolved signals are confirmed by the re-convolution and the comparison with the actual received THz signal y 0 (t), shown in Fig. 4.15. By carefully checking the deviation between the re-convolution and the actual received signal, shown in Fig. 4.15(c) and (d), the deconvolved signal with pulse spreading is more accurate, since smaller ripples in the fluctuations of the deviation can be observed in the region of late-coming echoes. The zoom-in insets of the last THz echoes in Fig. 4.15(a) and (b) further prove the validity of this algorithm.

  ) with respect to the individual a k and b k . In this process, the N BW available data are divided into three parts: (1) data within [i P L , i P H ] are the p data which are kept the same and used as the base of the AR model; (2) data within [i L , i P L ] are used for the fitting to find b k ;

  numerical and experimental study. The THz reference signal f 0 [n], with n as the index of data points, is recorded by setting a metal plate at the sample position, shown in the inset of Fig.4.17. The data sampling period in the measurement is set to T s = 0.0116 ps. Each recorded reflected temporal THz waveform contains 4096 data points, and the signal is averaged over 10 shots. With this setting, the entire frequency spectrum obtained by Fourier transform is from 0 THz to 85.99 THz, which also contains 4096 data points.

Figure 4 . 17 :

 417 Figure 4.17: The assumed impulse response function h 0 and the simulated received THz signal r 0 . The inset shows the actual THz reference signal f 0 .

Figure 4 .Fig. 4 . 19 .

 4419 Figure 4.18: (a) Comparison between the assumed frequency spectrum H 0 , the deconvolved frequency spectrum by direct inverse filtering H inv and the estimated frequency spectrum by AR spectral extrapolation H AR . (b) The zoom-in of (a), which shows the detailed data allocation for the AR model.

  25T s (corresponding to an air gap with thickness of 43.5 µm) cannot be resolved. In contrary, h AR is able to resolve all the interfaces, and exhibits a 'quasi-ideal' impulse response function compared with the assumed impulse response function h 0 . Based on this simulation, we can conclude that the minimum thickness resolved by THz reflective imaging, is enhanced by AR deconvolution, since the frequency components throughout the entire spectrum are estimated and recovered. It is important to note that the minimum thickness (time interval) resolution , corresponding to the depthresolution, is mainly dependent on the bandwidth of the THz source, the sampling frequency and the SNR. In our simulations, the minimum time interval which can be resolved by AR deconvolution with the SNR = 10 dB is 15T s (corresponding to an air gap with thickness of 26.1 µm); on the other hand, the minimum time interval resolved by FWDD with the same SNR is 30T s . Therefore, the depth-resolution can be increased by a factor of two using AR deconvolution.

Figure 4 . 19 :

 419 Figure 4.19: Comparison of the simulated deconvolution results obtained by AR deconvolution with p = 44, FWDD and the assumed impulse response function. The inset shows the AIC values with different model orders.

Figure 4 .

 4 Figure 4.20: (a) The deconvolution results based on FWDD and AR deconvolution. The inset shows the received THz signal reflected from the poly-coated steel sample; (b) The estimated frequency spectrum (up to 5 THz) based on autogressive spectral extrapolation.

Figure 5 . 1 : 2 PixelFigure 5 . 3 :Figure 5 . 4 :Figure 5 . 5 :

 512535455 Figure 5.1: Images of After Fishing by Ausonio Tanda. (a) Visible photograph of After Fishing, and (b) X-ray transmission image of After Fishing.

Figure 5 . 6 :Figure 5 . 7 :Figure 5 . 8 :Figure 5 . 9 :

 56575859 Figure 5.6: Four different types of the deconvolved signals with (a) (d) corresponding to Type I Type IV.

Fig. 5 .Figure 5 . 10 :

 5510 Fig. 5.11 to support our assumption. The raking light image provides information on the surface topography, which also highlights the physical thickness distribution of the applied paint layers on the paperboard. Areas with thicker applied paint layers, as determined in Fig. 5.11, also show high contrast in Fig. 5.10. Relatively small differences in the refractive indices of various paint layers at their mutual interfaces produce the Fresnel coefficients resulting in the echoes we observe. Still, as we argue, we are able to obtain a reasonable estimate of layer thicknesses and overall thickness of the paint application.

Figure 5 . 11 :

 511 Figure 5.11: Raking light image of After Fishing.

Figure 5 . 12 :

 512 Figure 5.12: The fused image of the X-ray image in red and THz thickness distribution image in green to distinguish the areas of similar intensity in yellow.

Figure 5 . 13 :Figure 5 . 14 :Fig. 5 .Figure 5 . 15 :

 5135145515 Figure 5.13: Images of Vierge en priere. (a) Visible photograph of the Viège en priere; Image of Viège en priere obtained by UV Fluorescence; (c) Image of Viège en priere obtained by IR Reflectography.

  raphy. Sparsity-based time-domain deconvolution based on the shrinkage algorithm is utilized to process the 3D volume raw data. After deconvolution, a sparsity-based impulse-response function is achieved, which entirely depends on the stratigraphy and provides a new imaging parameter with enhanced depth resolution. In the left-hand frames are shown three examples of raw THz signals received at various pixels (black curves) in Fig. 5.16(a1), (a2), and (a3). The corresponding sparsity-based impulse response functions are shown in the right-hand frames, Fig. 5.16(b1), (b2), and (b3).

Figure 5 . 16 :

 516 Figure 5.16: Typical THz reflected signals and the sparsity-based deconvolved signals. Figures 5.16(a1), (a2), and (a3) show examples of raw reflected signals (black) and signals reconstructed from the sparse-deconvolution and reference signals. Figures 5.16(b1), (b2), and (b3) show the corresponding sparsity-based deconvolution signals (red); Fig. 5.16(b1) also shows the deconvolved signal achieved by FWDD for comparison.

  Figure 5.16: Typical THz reflected signals and the sparsity-based deconvolved signals. Figures 5.16(a1), (a2), and (a3) show examples of raw reflected signals (black) and signals reconstructed from the sparse-deconvolution and reference signals. Figures 5.16(b1), (b2), and (b3) show the corresponding sparsity-based deconvolution signals (red); Fig. 5.16(b1) also shows the deconvolved signal achieved by FWDD for comparison.

Figure 5 . 17 :

 517 Figure 5.17: Comparison for optical delay corresponding to the layers above the canvas between (a) the THz B-scan based on the raw data (Fig. 5.15 on an expanded vertical scale) and (b) the binary THz B-scan based on the deconvolved data with the cross-section Y = 150 in which a valid peak is assigned value '1' and the other positions '0' regardless of the sign or height of the peak.

Figure 5 . 18 :

 518 Figure 5.18: Varnish features revealed by THz sparsity-based deconvolved signals. (a) THz C-scan based on the amplitude of the first peak of the deconvolved signal. The contrast is normalized to one; (b) THz C-scan based on the optical delay between the first and second peaks of the deconvolved signal.

5. 19 (

 19 c). In this image, white regions indicate the presence of the additional peak, black its absence. The white regions are highly correlated with the craquelure; this is not entirely surprising, since stresses associated with craquelure formation and the channels open to the infiltration of moisture and other contaminants may also lead to delamination localized there. Continuing to consider subsequent peaks in the sparsity-based deconvolved signal, features associated with the underpainting, imprimatura, and ground layers are investigated based on the THz C-scans in Fig. 5.20. Figures 5.20

  (a1), (a2), and (a3) provide C-scans based on the amplitudes of the fourth, fifth, and sixth peaks, respectively. Lines associated with the craquelure are still evident due to the shadow effects associated with reduced signals reaching subsequent layers depending on THz reflectivity and scattering from layers above. We note that the fourth and fifth peaks cannot be resolved in the deconvolved signals in all the pixels across the painting, corresponding to the black regions in the THz C-scans. Most of the black regions appear in the woman's profile. This might be due to the multiple paint applications in the profile, which makes the underneath layers thinner than the depth resolution achieved. In Figs.5.20(b1), (b2), and (b3) are shown C-scans computed from the optical delay between the third and fourth, the fourth and fifth, and the fifth and sixth peaks, respectively, if resolved. We note that, for Fig.5.20(c), the optical delay is calculated based on the peak corresponding to the ground/canvas interface and the preceding peak which can be resolved. Based on the results, we conclude that the Master of the Vièrge en prier applied layers of fairly uniform thickness.Sparsity-based THz reflectometry provides detailed three-dimensional information, enabling us to reconstruct the detailed stratigraphy of the Vièrge en priere.

Figure 5 . 19 :

 519 Figure 5.19: Features of the pictorial layer revealed by THz sparsity-based deconvolved signals. (a) THz C-scan based on the amplitude of second peak of the deconvolved signals; (b) THz C-scan based on the amplitude of the peak corresponding to the interface between the pictorial and underpainting layers; (c) binary THz C-scan indicating the positions with the existence of delaminations; (d) THz C-scan based on the optical delay between the peaks corresponding to the thickness of the pictorial layer.

Figure 5 . 20 :

 520 Figure 5.20: Features of the underpainting, imprimatura, and ground layers revealed by THz C-scans based on sparsity-based deconvolved signals. THz C-scans based on the amplitude of (a1) the fourth, (b1) fifth, and (c1) sixth peak. Contrast is normalized to one. THz C-scans based on the optical delay between (a2) the third and fourth peaks, (b2) fourth and fifth peaks, and (c2) fifth and sixth peaks.

Figure 5 . 21 :

 521 Figure 5.21: The optical microscopic image of a small region at the edge of Vièrge en priere. The applications of ground, imprimatura, and the underpainting can be clearly identified.

  summary, sparsity-based THz reflectometry proposed in this study clearly revealed the detailed stratigraphy of a 17th century easel painting with layer-thicknesses smaller than 50 µm, including the varnish, pictorial, underpainting, imprimatura, and the ground layers. Retouching of varnish, as well as age-induced craquelure in the pictorial layer are also successfully characterized. The results achieved by sparsity-based THz reflectometry are supported by various techniques. It is important to emphasize that in many cases a single imaging modality provides only very limited information about an art object, and that in practice what can be learned about an artwork by combining several techniques frequently yields more than the sum of what might be learned by applying each technique in a vacuum. Sparsity-based THz reflectometry promises to provide an effective in-situ 3D quantitative imaging modality for a broad range of cultural heritage objects, and an invaluable contribution to art-historical studies, as well as for conservation, restoration, and authentication.The full promise of THz imaging, of which the performance is enhanced by the theories and techniques developed in this thesis, to provide a noninvasive, noncontact, and nonionizing modality for the NDE and material characterization in the field of industry and cultural heritage are well-exhibited.For the NDE of fiber-reinforced composites, THz imaging shows the capability for providing 3D quantitative characterization of delaminations in glass-fiber reinforced composite laminates. High-resolution, low-artifact THz C-and B-scan images locating and sizing the delaminations are obtained, which enable us to determine the thicknesses of the delamination and the layers constituting the laminate. The influence of atmospheric water-vapor is taken into account when characterizing small delaminations in woven glass-fiber reinforced composites. Wavelet shrinkage de-noising is applied to remove water-vapor features, leading to enhanced THz C-and B-scans to locate the delaminations in three-dimensions with high contrast. In view of the difficulty and expense in purging with dry nitrogen to physically remove the water-vapor features in many cases, the demonstrated results will be of great interest for a range of laboratory-and field-based THz NDE applications. In addition, polarization-resolved THz imaging is firstly applied to a hybrid fiber-reinforced composite laminate. The evolution of intra-and inter-laminar damage throughout the thickness of the laminate subject to low-velocity impact is evaluated. Inter-laminar damage at the interface and the intra-laminar damage close to the same interface can be differentiated by taking advantage of the sensitivity of the carbon-fiber orientation to the THz polarization.These studies demonstrate that THz imaging can be regarded as an alternative or complementary modality for the NDE of fiber-reinforced composites. This thesis makes a great contribution to the THz community by developing advanced THz deconvolution techniques to enhance the capability of THz imaging for the stratigraphic characterization of multi-layered structures with optically thin layers in the THz regime. Previously, layers with thicknesses smaller than 100 µm could not be easily resolved and characterized based on the raw THz signals, due to the depth-resolution achieved by a typical THz TDS system, which further limited the applications of THz imaging. Deconvolution, if applied properly, can yield sub-wavelength or sub-pulse-width depth resolution. In the context of THz imaging, three kinds of deconvolution methods are developed in this chapter. Conventional deconvolution, such as THz frequency-wavelet domain deconvolution (FWDD), which is based on inverse filtering is introduced. The depth-resolution achieved by FWDD is about 55 µm in air. A case study for the characterization of various failure modes in polymer-coated with FWDD is also presented. However, depth-resolution achieved by FWDD cannot satisfy the real applications of THz imaging. In order to further enhance the depth-resolution, sparse representation is exploited, and sparse deconvolution based on an iterative shrinkage algorithm has been developed. With an up-sampling approach, sparse deconvolution with super-resolution is also developed to overcome the time resolution limited by the sampling period in the measurement.A simple but effective time-domain model for describing the temporal pulse spreading due the frequency-dependent loss is also designed and introduced into the algorithm, which significantly improves the performance of sparse deconvolution in processing time-varying pulses during the propagation of THz waves in materials. The depthresolution achieved by sparse deconvolution is about 45 µm in air. Finally, a novel method for enhancing the depth-resolution of THz deconvolution based on autoregressive spectral extrapolation is developed. An autoregressive process is modeled based on the THz frequency components with high SNR, and the missing frequency components in the low SNR regions are extrapolated based on the autoregressive model.In this way, the entire THz frequency spectrum of the impulse response function is recovered. This method is able to provide a 'quasi-ideal' impulse response function, and therefore, significantly enhances the depth-resolution for resolving optically thin layers in the THz regime. The depth-resolution achieved by this method is about 27 µm in air. The deconvolution techniques developed and presented in this thesis greatly enhance the capability for resolving optically thin layers in the THz regime, and provide a powerful tool to the entire THz community for a broad applications.Finally, based on the advanced deconvolution techniques developed in this thesis, the capability of THz reflectometry for the characterization of stratigraphy in historical paintings is significantly improved. Compared with other existing techniques, THz reflectometry with enhanced depth-resolution can provide 3D global mapping of the stratigraphy quantitatively. The most promising result in this thesis is that the detailed stratigraphy of a 17th century Italian easel painting, Vièrge en priere, is successfully revealed via sparsity-based THz reflectometry. The detailed layer structure, including the canvas, the ground, imprimatura, underpainting, pictorial, and varnish layers are clearly identified. In addition, a hitherto unidentified restoration of the varnish has been found. It is important to note that, it is the first time to our knowledge THz reflectometry has resolved multiple layers in an easel paintings.

  the enhanced THz imaging by deconvolution is expected to perform sub-surface investigations of carbon fiber-reinforced composites. The damage mechanisms induced by tension or bending in woven fiber-reinforced composites will be well-studied via THz imaging. The main damage mechanisms introduced by tension are complex and can be classified into micro-structural damage within the fiber strands and the macroscopic damages within the composites. Considering the wavelength of THz waves, the macroscopic damage mechanisms including transverse cracks in weft, cracks in purematrix regions, delamination between warp and weft and between adjacent layers, warp tensile failure and finally fracture, are expected to be revealed by THz imaging. By mapping the material parameters under THz different frequencies, THz images in transmission or reflection are expected to show the cumulative damage patterns across the sample plane. Damage indicators for the existence of various damages and defects due to tension or bending are also expected to develop, which is benefical for identifying and detecting damages at the initial stage of the fatigue life of woven-fabric composites. For the THz imaging in cultural heritage conservation science, there are still several open challenges to be faced. The knowledge of the THz fingerprints of materials used in artworks is rare. Contributions should be made to the development of a global reliable and standardized database in order to improve the quantitative analysis and material identification. THz studies of archaeological and historical objects will be carrying on. Correlation between THz images and images from other NDE techniques will be made to reveal the various features of artworks. The deconvolution methods developed in this thesis have already reached the limitation of the depth-resolution considering the THz wavelength. In order to further increase the depth resolution, producing THz pulses with shorter time-duration or establishing THz interferometry imaging techniques might be necessary. Current deconvolution procedure is based on the THz reference signal and the reflected THz signals. Blind deconvolution in the THz context is still needed in order to get rid of the influence of the reference signal and deconvolve the time-varying echoes during the propagation of THz waves. With deconvolution, the structural information at

TABLE OF CONTENTS

 OF Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Imaging contrast mechanisms . . . . . . . . . . . . . . . . . . . . . . Chapter 3: Terahertz Nondestructive Evaluation of Fiber-reinforced Composites . . . . . . . . . . . . . . . . . . . . . . . . . . . Terahertz 3D quantitative imaging . . . . . . . . . . . . . . . 3.1.2 Comparison with ultrasonic C-scans . . . . . . . . . . . . . . . 3.2 Enhanced terahertz imaging of small delaminations in woven glass fiber-reinforced composites . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Influence of water vapor . . . . . . . . . . . . . . . . . . . . . 3.2.2 Wavelet de-noising . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Enhanced terahertz imaging with wavelet de-noising . . . . . . 3.3 Polarization-resolved terahertz imaging of impact damages in hybrid fiber-reinforced composites . . . . . . . . . . . . . . . . . . . . . . . .

x List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi Chapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Motivation and research problems . . . . . . . . . . . . . . . . . . . . 1 1.2 State-of-the-art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2.1 Terahertz imaging of fiber-reinforced composites . . . . . . . . 4 1.2.2 Terahertz characterization of multilayered coatings on metal . 8 1.2.3 Terahertz imaging in cultural heritage conservation science . . 1.3 Scope of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2: Equipment and Imaging Contrasts . . . . . . . . . . . . . 2.1 Terahertz time-domain spectroscopy system . . . . . . . . . . . . . . 2.2 3.1 Terahertz imaging of forced delaminations in glass fiber-reinforced composites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii 3.1.1 3.3.1 Terahertz polarization versus carbon fiber orientation . . . . . 3.3.2 Polarization-resolved terahertz imaging . . . . . . . . . . . . .

Table 1 .

 1

		1: State-of-the-art THz imaging of actual art paintings
	Ref. Types Painting Name	Year
	[26] Panel Badia Polyptych by Giotto di Bondone	2010
	[29] Panel San Giovenale Triptych by Masaccio	2015
	[30] Panel the Virgin with the Child and a Saint in Sicily Italy	2015
	[31] Panel A Garden in front of a Country Seat by David Teniers	2015
	[32] Panel The Lamentation over the Dead Christ by Fra Angelico	2016
	[33] Easel Sacrifice to Vesta by Francisco de Goya	2013
	[34] Easel The Dying Messalina and Her Mother by Nicolai Abildgaard 2015
	[35] Easel Homme au chapeau by Pablo Picasso	2015
	[36] Easel Santo Entierro de Nuestro Senor altarpiece, Mexico	2017
	[37] Wall	Wall painting of Dazhao Monastery, China	2010
	[38] Wall	Wall painting of Catalhoyuk, Turkey	2013
	[39] Wall	Wall painting of Nebbelunde Church, Denmark	2015
	time separation of 0.15 ps. However, in order to achieve these values, more elaborate
	signal processing methods are required.
	Considering the application of THz imaging on actual art paintings, one of the
	most important application of THz imaging in the art conservation science is the case
	study performed at the end of 2008 on the panel painting Polittico di Badia by Giotto
	di Bondone, on display at the Uffizi Gallery in Florence, Italy

Ultrafast Laser SAMPLE SAMPLE Reflection Mode Transmission Mode Optical Delay Line Probe Pulse Pump Pulse THz Transmitter Photoconductive Antenna THz Receiver LT-GaAs Lock-in Amplifier Current Pre-Amplifier THz Pulse

ACKNOWLEDGEMENTS

First and foremost I would like to express my deep sense of gratitude and appreciation to my Ph.D. advisor Prof. David. S. Citrin. It has been a honor to be

THz is still acceptable; consequently, the cutoff frequency f c can be selected slightly higher than 1.2 THz to obtain higher time resolution in the deconvolution procedure.

By contrast with the X-ray transmission image in Fig. 5.1(b), one striking feature observed are several small and blurred spots randomly distributed in the frequencydomain THz C-scans; these spots are not observed in the X-ray transmission image.

By carefully checking the THz waveforms at relevant pixels, small echoes can be found in the time interval 12 to 28 ps, corresponding to the paperboard support. The THz B-scan together with the THz C-scan clearly locate the four highlighted spots in three dimensions, shown in Fig. 5.4. These spots are a result of defects in the paperboard, which is speculated may be due to oil and/or biological growth. Because organic materials containing mainly light atoms present low radiological contrasts, they are difficult to detect by standard X-ray systems [START_REF] Bessou | Advantage of terahertz radiation versus X-ray to detect hidden organic materials in sealed vessels[END_REF]. THz B-scans based on the raw data cannot provide the stratigraphic detail, since separation of the paint layers above the paperboard cannot be resolved. Artifacts (weak horizontal lines) due to ambient water-vapor and the system noise also lower the quality of the THz B-scans.

In this study, in order to tease out the detailed stratigraphic information that is actually contained in the reflected THz signal, the sparse representation is introduced achieved by deconvolution, we gain 3D quantitative insight into the detailed stratigraphy above the canvas throughout the painting, including the varnish, pictorial layer, underpainting, imprimatura, and the ground layer. In addition, we identify delaminations in the pictorial layer associated with age-induced craquelure, and locate a hitherto unidentified extensive restoration of the painting. We emphasize, this is the first time to our knowledge THz reflectometry has resolved multiple layers in an easel painting with such thin layers. Although we focus on easel paintings, the proposed modality can be applied to a wide range of culture heritage objects and provides invaluable information for art-historical studies, as well as for conservation, restoration, and authentication.

The painting in this study was chosen as a typical painting for the period that observe the presence of delamination, closely associated with the craquelure. The surface morphology due to the craquelure can also be seen in white-light raking images shown in Fig. 5 It was typical in the 17th century finally to apply varnish (a natural resin) to saturate the paint colors and protect the surface. The craquelure continues to be pronounced in the varnish, as is evidenced in the corresponding THz C-scans. The application of the varnish is not uniform across the painting. Assuming the refractive index of the varnish is about 1.6 [START_REF] Yun-Sik | Terahertz Dielectric Properties of Polymers[END_REF], the physical thickness of the varnish at pixel (89, 150) is about 35 µm. It is important to note that the thicker region near the Vièrge's head is clearly identified, which exhibits the same shape as the dark areas in the UV fluorescence image in Fig. 5.13(c). The physical thickness at pixel (185, 158)