Edson Cardozo De Oliveira

Beatriz Castillo López De Larrinzar

Chushuang Xiang

N D Lanzillotti-Kimura

Antonio García-Martín

N D Lanzillotti- Kimura

Chiral

Chiral acoustoplasmonics

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Phonons 2023

Paris, France The possibility of creating and manipulating nanostructured materials encouraged the exploration of new strategies to control electromagnetic properties. Among the most intriguing nanostructures are those that respond differently to helical polarization, i.e., exhibit chirality [START_REF] Rodríguez-Álvarez | Tunable circular dichroism through absorption in coupled optical modes of twisted triskelia nanostructures[END_REF][START_REF] Auguié | Fingers Crossed: Optical Activity of a Chiral Dimer of Plasmonic Nanorods[END_REF]. Here, we present a simple structure based on crossed elongated nanobars with the long axis parallel to the polarization plane forming a given angle between them and vertically aligned, as shown in Fig. 1(a). We theoretically demonstrate that, in such structure, light-handedness defines the dominating cross-section absorption or scattering, with a 200% difference from its counterpart (scattering or absorption). The proposed chiral system opens the way to enhanced coherent phonon excitation and detection. In general, the generation of acoustic phonons is optimized by maximizing the absorption, while the detection is enhanced by engineering the scattering properties [START_REF] Lanzillotti-Kimura | Polarization-controlled coherent phonon generation in acoustoplasmonic metasurfaces[END_REF][START_REF] O'brien | Ultrafast acousto-plasmonic control and sensing in complex nanostructures[END_REF]. In chiral structures, this could be achieved by a simple pump-probe experiment using circularly polarized light at the same wavelength. The use of structures sustaining chiral optical modes appears as a natural, yet unexplored strategy to address these conditions. These results constitute one of the first steps towards harvesting chirality effects in the design and optimization of efficient and versatile acoustoplasmonic transducers.

Chiral acoustoplasmonics

Figures

  Figures

Figure 1 :

 1 Figure 1: (a) Sketch of the geometrical layout of the chiral structure. Two gold nanobars separated by a distance D = 80 nm, with the long axis parallel to the polarization plane forming a given angle between them and vertically aligned. The bottom bar is embedded in SiO 2 while the top bar sits on the surface of SiO 2 . (b) Absorption and scattering crosssections for right (RCP) and left (LCP) circularly polarized incident beam.

  Figure 1: (a) Sketch of the geometrical layout of the chiral structure. Two gold nanobars separated by a distance D = 80 nm, with the long axis parallel to the polarization plane forming a given angle between them and vertically aligned. The bottom bar is embedded in SiO 2 while the top bar sits on the surface of SiO 2 . (b) Absorption and scattering crosssections for right (RCP) and left (LCP) circularly polarized incident beam.

Edson Rafael Cardozo de Oliveira a

	B. Castillo López de Larrinzar	b , C. Xiang	a , N. D. Lanzillotti-Kimura	a , A. García-Martín	b
	a. Université Paris-Saclay, CNRS, Centre de Nanosciences et de Nanotechnologies, 10 Boulevard
	Thomas Gobert, 91120 Palaiseau, France
	b. Instituto de Micro y Nanotecnología IMN-CNM, CSIC, CEI UAM + CSIC, Isaac Newton 8, Tres
	Cantos, Madrid 28760, Spain
	edson.cardozo@c2n.upsaclay.fr


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Augustin Feussom Tcheumeleu

Laurent Millet

Damien Rius

Adam A Ali

Yves Bergeron

Pierre Grondin

Sylvie Gauthier

Olivier Blarquez

An 8500-year history of climate-fire-vegetation interactions in the eastern maritime black spruce-moss bioclimatic domain, Québec, Canada

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Shlomo Mehari

Daniel A Cohen

Daniel L Becerra

Haojun Zhang

Claude Weisbuch

James S Speck

Shuji Nakamura

Steven P Denbaars

Semipolar III-Nitride laser diodes for solid-state lighting

Keywords: laser diodes, III-Nitride, semiconductor lasers, transparent conductive oxides, optical loss, laser lighting

Several hurdles to further enhance the performance of semipolar III-Nitride laser diodes are addressed in this work. Particularly, we focused on improving their high operating voltage by thinning the p-GaN cladding layer and utilizing a transparent conductive oxide p-contact. On-wafer optical absorption measurements showed that a further reduction in voltage with thinner p-GaN was limited by increased optical loss due to increased mode overlap with the ITO/metal anode. In separate attempts to minimize bulk-related optical losses, we implemented a new design that consisted of an AlGaN electron blocking layer (EBL) placed remotely from the quantum wells (QWs) and a low p-waveguide Mg doping profile. A very low optical loss of about 2 cm -1 was extracted but the net improvement in differential efficiency was limited by lower internal injection efficiency due to carrier accumulation in the p-waveguide region. With an optimized design, that consisted of a lightly doped EBL close to the QWs and a UID p-waveguide, an improved light output power of 1.4 W at 1.5 A and a low threshold current density of 1.2 kA/cm 2 were obtained.

INTRODUCTION

III-Nitride laser didoes (LDs) are poised to become essential components in a broad range of applications such as in medical diagnostics, quantum technologies, and lighting systems. [START_REF] Najda | GaN laser diodes for quantum sensors, optical atomic clocks and systems[END_REF][START_REF] Murayama | Watt-Class Green (530 nm) and Blue (465 nm) Laser Diodes[END_REF][START_REF] Masui | Recent improvement in nitride lasers[END_REF] Specifically, in high-intensity white lighting systems, such as in automobile headlights and stage lighting, III-Nitride LDs offer much higher spatial brightness and no efficiency droop above threshold compared to LEDs. [START_REF] Pourhashemi | Pulsed high-power AlGaN-cladding-free blue laser diodes on semipolar (20-2-1) GaN substrates[END_REF] To realize most of these applications, however, the power conversion efficiency (PCE) of visible III-Nitride LDs, which is still far lower than that of other III-V LDs, [START_REF] Crump | Efficient High-Power Laser Diodes[END_REF] must be further improved. Research efforts in recent years, primarily focused on improving the material quality of violet to blue c-plane LDs, have shown great progress. [START_REF] Kawaguchi | Optical-loss suppressed InGaN laser diodes using undoped thick waveguide structure[END_REF] But at longer wavelengths the picture looks far less rosy. A promising route to improve the PCE, especially at longer wavelengths, is to grow the quantum wells (QWs) on non-c-plane directions, which is predicted to improve the radiative efficiency and gain. [START_REF] Scheibenzuber | Calculation of optical eigenmodes and gain in semipolar and nonpolar InGaN/GaN laser diodes[END_REF] Among the most prominent advantages of QWs grown on non-basal directions are: high indium incorporation, reduced hole effective mass due to anisotropic strain, and increased electron-hole wavefunction overlap due to lower polarization-related electrical fields. [START_REF] Feezell | Semipolar (20-2-1) InGaN/GaN Light-Emitting Diodes for High-Efficiency Solid-State Lighting[END_REF] To fully exploit this potential, it is crucial to thoroughly understand the underlying physical mechanisms that deteriorate the LD's operation. In this context, we present here our recent work on blue III-Nitride LDs grown on semipolar GaN substrates. We carefully examined and analyzed the main hurdles that impeded their performance, focusing on improving the continuous-wave (CW) operating voltage and reducing the high optical losses. We discuss first our effort to mitigate with the high operating voltage. The voltage was greatly reduced by thinning down the p-GaN cladding layer thickness and replacing the metallic contacts with a tin-doped indium oxide (ITO) p-contact. The limitations of this approach with regard to a tradeoff between low voltage and low optical loss are discussed. We then present results from LDs with very low optical losses (~2 cm -1 ) that were obtained by redesigning our p-side waveguide region. The effects of carrier accumulation and tradeoff with lower injection efficiency are discussed.

DEVICE FABRICATION

The epitaxial layers in this work were grown by metal organic chemical vapor deposition (MOCVD) on a free-standing semipolar ( ) 202 1 n-GaN substrate, provided by Mitsubishi Chemical Corporation. A schematic cross-section of our typical edge-emitting LDs is shown in Figure 1. For blue LDs the multiple quantum well (MQW) region was designed with two 3.5 nm In 0.18 Ga 0.82 N quantum wells and a single 7 nm GaN barrier layer. An additional 20 nm UID GaN layer was incorporated on either side of the active region. The n-side and p-side waveguides consisted of a 65 nm In 0.07 Ga 0.93 N layer compositionally graded to GaN over 10 nm at each side. Their Si and Mg doping levels were about 1x10 19 cm -3 , respectively. A 10 nm Mg-doped (4x10 19 cm -3 ) p-Al 0.28 Ga 0.72 N layer was implemented as an electron blocking layer (EBL) between the active region and the p-side waveguide. To study the impact of p-cladding design on the operating voltage, the p-GaN cladding layer's thickness was varied and was capped with a 15 nm p+ GaN contact layer (Mg was ramped to 1x10 20 cm -3 ).

The samples were then processed into an edge-emitting laser structure consisting of a tin-doped indium oxide (ITO) pcontact, as detailed in Ref. 9. LDs were tested under pulse and continuous-wave (CW) electrical injection. The pulsed electrical and luminescence characteristics were measured using a pulse width of 1 µsec with a repetition rate of 10 kHz. The light output power was measured by collecting the light emitted with a calibrated Si photodiode and integrating sphere.

p-pad

RESULTS AND DISCUSSION

Impact of p-GaN on LD's operating voltage

To reduce the operating voltage, we explored the impact of the p-side cladding design. Unlike in LEDs, the Mg doping levels in the p-GaN layer cannot be too high or else it would increase the optical loss. Instead, we thinned down the p-GaN thickness and replaced the metallic p-contact with a low absorption ITO contact. As shown in Figure 2a), when the p-GaN thickness was decreased from 650 nm to 250 nm the CW operating voltage significantly decreased from 7.2 V to 5.4 at 10 kA/cm 2 without impacting the light output characteristic (not shown here). A further reduction in the p-GaN thickness from 250 nm to 150 nm (with a different set of LDs that consisted of a lower n-side Si doping of 1x10 18 cm -3 ) an additional decrease in the operating voltage was observed, as shown in the CW light-current-voltage (L-I-V) characteristics of 8x1200 μm LDs (Figure 2b)). However, the light output power at this point started to deteriorate. Its peak intensity dropped from 1.7 W for 250 nm p-GaN to 1.5 W for 150 nm p-GaN. To better understand this tradeoff behavior, namely, degradation of the light output power with an improved operating voltage, we have studied the absorption spectra of these devices using the segmented contacts method. [START_REF] Blood | Characterization of Semiconductor Laser Gain Media by the Segmented Contact Method[END_REF] The aim was to identify differences in absorption losses due to a possible optical mode penetration into the ITO/metal contacts. No rear facet was used, and an extended length of absorbing region insured there was no reflection from the back edge of the ridge. The light output was collected using a set of objective lenses and a single mode optical fiber connected to an array spectrometer. The absorption spectra were obtained using equation ( 1), [START_REF] Blood | Characterization of Semiconductor Laser Gain Media by the Segmented Contact Method[END_REF] assuming a uniform structure, ( ) ( )

1 ln meas i meas I back L I front α α ⎛ ⎞ -Γ - = ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ , (1)
where I meas refers to the ASE intensity measured when a certain segment is injected (front or back), α Γ is the modal absorption ( Γ and α are the confinement factor and the absorption coefficient of the gain medium, respectively), and i α is the internal optical modal loss.

The absorption spectra of the LDs with 250 nm and 150 nm p-GaN measured at 2 kA/cm 2 are compared in Figure 3 b).

From the asymptotic value at long wavelengths of the absorption curves we find an increase in the optical loss from about 19 cm -1 for the LDs with 250 nm p-GaN to about 25 cm -1 for LDs with 150 nm p-GaN. In an attempt to understand the contributions of the p-contact (about 200 nm ITO) and the top metallic p-pads (Ti/Au 30/1000 nm) to the increased optical loss, we carried out optical mode profile simulations for the LDs' structures with 250 nm and 150 nm p-GaN. Comparing the two calculated mode profiles, as shown in Figure 4, we find that the overlap with the ITO contact layer increased from 0.06% to 0.14% and the overlap with the metallic p-pads increased from 5x10 - 6 % to 5x10 -4 % when the p-GaN thickness was decreased from 250 nm to 150 nm, respectively. These variations might seem low, but considering the relatively high absorption coefficient of these layers the optical loss differences can be well explained. Our standard un-annealed ITO contacts (In 2 O 3 :Sn 2 O 3 90:10%wt) have an absorption coefficient of ~1700 cm -1 (Figure 5, left red column), and the metallic p-pads absorption coefficient is in the range of ~10 6 cm -1 , we thus estimated an increased loss of about 1.4 cm -1 in the ITO layer and about 5 cm -1 in the metallic p-pads, comparing the 250 nm and 150 nm p-GaN designs. Obviously, a straightforward approach to mitigate this increased loss, if one wants to further reduce the operating voltage by further reducing the p-GaN thickness, is to increase the ITO thickness and reduce its absorption coefficient. One such approach have explored to reduce the ITO's absorption coefficient is by changing its In 2 O 3 to Sn 2 O 3 weight ratio from our baseline ratio of 90:10 %wt to lower compositions. As shown in Figure 5, the absorption coefficient changed considerably with the tin %wt. With a 99.5:0.5 %wt ratio, the absorption coefficient decreased by an order of magnitude, while the resistivity increased only slightly (from 3.3x10 -4 Ωcm to 6.4x10 -4 Ωcm). Since the transport in edge-emitting LDs is vertically this increased resistivity has no impact on the operating voltage. Further work is still required to incorporate this new contact schemes into a LD structure and test their contribution to a lower optical loss.

ITO

The role of p-waveguide design in LD's operation

In this section, we discuss the impact of the p-side waveguide design, particularly, its Mg doping profile and EBL location, on the LD's light output characteristics. Figure 6 a) shows the calculated optical mode profile along our standard LD structure together with the estimated optical losses in the different layers (shaded green). As a result of freecarrier absorption, [START_REF] Kioupakis | Free-carrier absorption in nitrides from first principles[END_REF] the highly Mg-doped regions, i.e., the p-AlGaN and the p-InGaN layers, contribute significant optical losses. To reduce these apparent high optical losses, we moved the EBL location remotely from the QWs (with slightly lower Mg doping levels) and also eliminated the Mg doping from the p-side InGaN waveguide. A comparison between the Mg doping profile in our standard LD's structure and in the new design is shown in the secondary ion mass spectroscopy (SIMS) depth profiles in Figure 6 b). The peak Mg doping level was reduced from 4x10 19 cm -3 to 8x10 18 cm -3 , and the InGaN p-waveguide was changed to a UID doping level. We processed LDs that had either a high Mg doping profile and EBL close to the QWs or a low Mg doping profile and a remote EBL. From the analysis of their absorption spectra, we find a significant reduction in the optical loss from about 20 cm -1 in the standard LD's design to about 2 cm -1 in the new design (Figure 7 a)). Despite this improvement in optical loss the net differential efficiency was fairly similar, as shown in the L-I curves of these two designs in Figure 7 b). This is attributed to a reduction in injection efficiency: electrons escaping the quantum wells accumulate in the lower bandgap InGaN waveguide, where they recombine with abundant holes traveling to the wells. The resulting carrier sink exacerbates electron leakage from the wells. The increase in threshold current density from 1.8 kA/cm 2 to 2.2 kA/cm 2 is consistent with this decrease in injection efficiency. In our latest attempt to suppress a possible carrier sink while still benefiting from lower optical losses, we also explored a LD design that consisted of a close EBL (with about half of Mg doping levels) and a UID InGaN p-WG. As shown in Figure 7 b), with this design the threshold current density remarkably decreased to 1.2 kA/cm 2 , indicating that the injection efficiency was improved. No reliable absorption spectra were taken yet for these LDs to estimate the impact on optical loss. But considering that the slope efficiency was also improved to 1.2 W/A, the optical losses are possibly lower than the standard LDs. Overall this optimized design enabled an increased light output power of 1.4 W at 1.5 A.

CONCLUSIONS

Several impediments to high power conversion efficiency in high power blue semipolar III-Nitride LDs were evaluated and addressed. The high CW operating voltage was found to decrease significantly with the thickness of the p-GaN cladding layer and further improvement was limited mainly by optical mode penetration into the metallic p-pads. A lower absorbing and thick ITO scheme might further improve the voltage. Optical losses were correlated with highly Mg doped regions and were optimized to enhance injection efficiency by suppressing carrier accumulation in the p-WG region. Further enhancement of the injection efficiency for the low optical loss LDs remains a challenge that needs to be addressed.

Figure 1 .

 1 Figure 1. Schematic cross-section of an edge-emitting laser diode grown on a semipolar GaN substrate.

Figure 2

 2 Figure 2. a) Current voltage characteristics of LD structures with p-GaN cladding layer thicknesses of 650 nm and 250 nm. b) CW L-I-V curves of 8x1200 μm LDs with 250 nm and 150 nm p-GaN.

Figure 3 a

 3 Figure 3 a) shows a schematic top-view illustration of the segmented contacts structure. Two segments of equal length (L=300 μm) were patterned. A front contact, closer to the output facet, and a back contact, away from the output facet were separated by a 3 μm gap. An 8 μm wide front facet (laterally angled by 11° to suppress feedback) was incorporated to emit amplified spontaneous emission (ASE).No rear facet was used, and an extended length of absorbing region insured there was no reflection from the back edge of the ridge. The light output was collected using a set of objective lenses and a single mode optical fiber connected to an array spectrometer. The absorption spectra were obtained using equation (1),[START_REF] Blood | Characterization of Semiconductor Laser Gain Media by the Segmented Contact Method[END_REF] assuming a uniform structure,

Figure 3

 3 Figure 3. a) Top-view illustration of the segmented contacts structure and optical setup, b) absorption curves of LDs with a 250 nm and a 150 nm p-GaN thickness measured at 2 kA/cm 2 .

ITOFigure 4 .

 4 Figure 4. Comparison of the calculated optical mode profiles in LDs structures with a 250 nm and a 150 nm p-GaN thickness. The zoom-in figures (right side) show the increased overlap of the optical mode with the ITO contact and the metallic p-pads as the p-GaN thickness is reduced to 150 nm (distance scales on all graphs).

Figure 5 .

 5 Figure 5. ITO's absorption coefficient as a function of its composition (In 2 O 3 to Sn 2 O 3 weight percentage). The left red column is our baseline solid composition of 90:10 %wt.

Figure 6

 6 Figure 6. a) Optical mode profile calculation along our baseline LD layers with their associated optical loss (shaded green). b) Mg SIMS depth profile comparison between our baseline LD design, which consisted of a close EBL and high Mg levels, and our new LD design that consisted of a remote EBL and a UID InGaN p-waveguide.

Figure 7

 7 Figure 7. a) Absorption spectra of two LDs, one with an EBL close to the QWs and a high Mg doping profile, and a second with a remote EBL and a low Mg doping profile. b) L-I curves of LDs with different p-WG designs.

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ACKNOWLEDGEMENTS

This work was supported in part by the Solid State Lighting and Energy Electronics Center (SSLEEC) at UCSB. The information, data, or work presented herein was funded in part by the Advanced Research Projects Agency-Energy (ARPA-E), U.S. Department of Energy, under Award Number DE-AR0000671. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof. S. Mehari acknowledges partial support from the Andrew and Erna Finci Viterbi Foundation.


Aurélien Renault
email: aurelien.renault@orange.com

Vincent Lemaire
email: vincent.lemaire@orange.com

Alexis Bondu
email: alexis.bondu@orange.com

Dominique Gay
email: dominique.gay@univ-reunion.fr

Automatic Feature Engineering for Time Series Classification: Evaluation and Discussion

Time Series Classification (TSC) has received much attention in the past two decades and is still a crucial and challenging problem in data science and knowledge engineering. Indeed, along with the increasing availability of time series data, many TSC algorithms have been suggested by the research community in the literature. Besides state-of-the-art methods based on similarity measures, intervals, shapelets, dictionaries, deep learning methods or hybrid ensemble methods, several tools for extracting unsupervised informative summary statistics, aka features, from time series have been designed in the recent years. Originally designed for descriptive analysis and visualization of time series with informative and interpretable features, very few of these feature engineering tools have been benchmarked for TSC problems and compared with state-of-the-art TSC algorithms in terms of predictive performance. In this article, we aim at filling this gap and propose a simple TSC process to evaluate the potential predictive performance of the feature sets obtained with existing feature engineering tools. Thus, we present an empirical study of 11 feature engineering tools branched with 9 supervised classifiers over 112 time series data sets. The analysis of the results of more than 10000 learning experiments indicate that feature-based methods perform as accurately as current state-of-the-art TSC algorithms, and thus should rightfully be considered further in the TSC literature.

I. INTRODUCTION

The goal of Time Series Classification (TSC) is to assign a class label y from a set Y = {y i } of predefined labels to an unlabeled time series X = [x 1 , x 2 , . . . , x m ] which is an ordered set of real values. The associated machine learning task is to train a classifier function f on a labeled time series data set D = {(X 1 , y 1 ), (X 2 , y 2 ), . . . , (X n , y n )} in order to map the space of possible input series to the class labels of Y . Then, for an incoming unlabeled time series X, the prediction of the assignment is given by f (X).

Formulated as such, TSC has been identified as a top-10 challenging problem in data mining research [START_REF] Yang | 10 challenging problems in data mining research[END_REF] and has received much attention in the literature. This particular attention is due to the overwhelming amount of available time series data [START_REF] Esling | Time-series data mining[END_REF]. Indeed, in many scientific fields, measurements are taken over time. The resulting collection of ordered data is represented as time series. Thus, times series data arise from many real-world applications: e.g., in the UCR/UEA archive [START_REF] Bagnall | The UEA & UCR time series classification repository[END_REF], audio signal, electrocardiogram, encephalogram, human activity, image or motion classification are among the most frequent tasks. In order to solve these tasks, hundreds of TSC algorithms have been suggested in the past two decades [START_REF] Bagnall | The great time series classification bake off: a review and experimental evaluation of recent algorithmic advances[END_REF]. Various paradigms have been exploited in the TSC literature; from simple distance-based Nearest Neighbors to deep learning architectures and complex hybrid ensemble methods. The very latest contributions [START_REF] Middlehurst | Hive-cote 2.0: a new meta ensemble for time series classification[END_REF] indicate that five methods, HIVECOTE (ensemble methods HC1 [START_REF] Lines | Time series classification with hivecote: The hierarchical vote collective of transformation-based ensembles[END_REF] and HC2 [START_REF] Middlehurst | Hive-cote 2.0: a new meta ensemble for time series classification[END_REF]), convolutional kernels based ROCKET [START_REF] Dempster | Rocket: exceptionally fast and accurate time series classification using random convolutional kernels[END_REF], tree ensemble TS-CHIEF [START_REF] Shifaz | Ts-chief: a scalable and accurate forest algorithm for time series classification[END_REF] and deep learning based Inception-Time [START_REF] Ismail Fawaz | Inceptiontime: Finding alexnet for time series classification[END_REF] achieve top predictive performance.

Besides the numerous TSC algorithms, another research area has been developed in the recent years, namely time series Feature Construction (FC). FC approaches are attractive as they regroup methods for extracting informative and interpretable summary statistics from time series. Features might be diverse to capture various properties of the time series, e.g., seasonality, trends, autocorrelation, etc, and thus can be adapted to various application domains. The pioneering work HCTSA [START_REF] Fulcher | Highly comparative timeseries analysis: The empirical structure of time series and their methods[END_REF] allows to extract more than 7000 unsupervised time series features summarizing properties of the distribution of values in a time series, correlation properties, entropy and complexity measures, how properties of a time series change over time, etc. Since then, several unsupervised feature engineering tools have been independently developed in various programming languages: e.g., C/Python-based CATCH22 [START_REF] Lubba | catch22: Canonical time-series characteristics[END_REF]; Python-based FEATURETOOLS [START_REF] Kanter | Deep feature synthesis: Towards automating data science endeavors[END_REF], TSFRESH [START_REF] Christ | Distributed and parallel time series feature extraction for industrial big data applications[END_REF], [START_REF] Christ | Time series feature extraction on basis of scalable hypothesis tests (tsfresh -A python package)[END_REF], TSFEL [START_REF] Barandas | Tsfel: Time series feature extraction library[END_REF]; and R-based TSFEATURES [START_REF] Hyndman | tsfeatures: Time series feature extraction[END_REF], FEASTS [START_REF] O'hara-Wild | feasts[END_REF]. Originally designed for descriptive analysis and visualization of time series, their use for TSC problems has been less studied: e.g., B. Fulcher et al. [START_REF] Fulcher | hctsa: A computational framework for automated time-series phenotyping using massive feature extraction[END_REF] suggest a linear classifier based on the feature set generated by HCTSA and on a much smaller set, CATCH22 [START_REF] Lubba | catch22: Canonical time-series characteristics[END_REF]. However, to our knowledge, no extensive study has been led on the effectiveness of existing feature engineering tools for TSC problems and how they compare with the leading TSC performers.

In this article, we address these shortfalls and suggest several simple learning processes to assess the effectiveness of feature construction for TSC. Our two-step learning processes consist in : (i) transforming the original time series data into feature-based tabular data using an unsupervised feature engineering tool (or a combination of tools), and (ii) learning a supervised classifier on the obtained tabular data. Figure 1 summarizes the final results of three selected processes compared with current leaders on 112 UEA/UCR data sets. Using feature construction is then comparable with the current state of the art TSC algorithms in terms of predictive performance (exhaustive details of the experiments are provided in Section III). The rest of the paper is organized as follows: in Section II, we position our feature construction approach w.r.t. related work in TSC and feature engineering. The empirical evaluation and comparison are reported in Section III. We conclude in Section IV with some perspectives for feature construction in TSC problems.

II. BACKGROUND

One of the simplest, but also very popular method for TSC problems is k-Nearest-Neighbors coupled with a similarity measure, e.g., Euclidean distance or dynamic time warping. Following these baselines, the TSC literature has grown tremendously. With the main focus on efficiency and predictive performance, several paradigms have been exploited:

• similarity-based: generally coupled with 1-Nearest Neighbor (1-NN), several similarity measures have been experimented for TSC problems. Elastic similarity measures provide the best predictive performance as they allow to tackle with local distortions and misalignments in time series [START_REF] Lines | Time series classification with ensembles of elastic distance measures[END_REF]. • interval-based: from random intervals extracted from time series, a new feature vector is build and a traditional classifier is trained. Successful representatives of this type of method are the ones using multiple intervals and ensemble classifier, see e.g. [START_REF] Baydogan | Time series representation and similarity based on local autopatterns[END_REF].

• shapelet-based: the search for sub-sequences highlighting class separation not depending on when they occur in a serie has attracted many research efforts since [START_REF] Ye | Time series shapelets: a new primitive for data mining[END_REF]. • dictionary-based: these algorithms build words from sliding windows, then a classifier is learned on the transformed data made of histograms of word counts [START_REF] Large | On time series classification with dictionary-based classifiers[END_REF] • deep learning based: following the success of deep learning for images and videos, several neural network architectures have been suggested for TSC [START_REF] Ismail Fawaz | Deep learning for time series classification: a review[END_REF].

• ensemble-based: to combine the performance of existing approaches, meta-ensemble classifiers have been designed and achieved top predictive performance. For example, the Collective of Transformation-based Ensembles (CoTE [START_REF] Bagnall | Time-series classification with COTE: the collective of transformation-based ensembles[END_REF]) and its successors HC1 [START_REF] Lines | Time series classification with hivecote: The hierarchical vote collective of transformation-based ensembles[END_REF] and HC2 [START_REF] Middlehurst | Hive-cote 2.0: a new meta ensemble for time series classification[END_REF] fall in this category and are also providing structured overviews of the TSC literature. As the TSC literature already offers well-structured overviews of existing TSC approaches, in the rest of the background part, we will concentrate on the recent top performers. The following algorithms are also the contenders in the experimental section.

A. Top TSC performers

ROCKET family One of the most promising method from the past few years is the ROCKET (Random Convolutional KErnel Transformation) model ( [START_REF] Dempster | Rocket: exceptionally fast and accurate time series classification using random convolutional kernels[END_REF]), which achieved SOTA performances with just a fraction of the time needed for competing approaches. The idea of ROCKET is to use a high number of random convolutional kernels with random lengths, dilations, padding and weights, then convolve them with the time series samples to finally extract two features per kernel, which are the max and the ppv (proportion of positive values) for each sample. A linear classifier is then trained on generated features, either a Ridge or a Logistic regression depending on the data set size we are dealing with: logistic regression will generally be preferred working with bigger data sets.

An extension of the ROCKET framework has been further proposed by the authors in MiniROCKET ( [START_REF] Dempster | Minirocket: A very fast (almost) deterministic transform for time series classification[END_REF]), which is now used as the default ROCKET variant in most analysis. The goal of MiniROCKET is to reduce, and almost eradicate randomness from the pipeline. In order to do this, authors have restricted much more the kernels parameters and only extract the ppv for each sample, making the algorithm almost deterministic. Hence, a pre-defined set of 84 kernels has been chosen (among the 512 possible ones), which is supposed to balance accuracy and computational cost. Within this framework, MiniROCKET achieved about the same level of accuracy as ROCKET but way faster (up to 75x faster on larger data sets), making this approach even more competitive when it comes to computation time and scalability.

One more ROCKET variant has been developed, this time focusing on predictive performance rather than speed, namely MultiROCKET ( [START_REF] Tan | Multirocket: Multiple pooling operators and transformations for fast and effective time series classification[END_REF]). This algorithm is significantly improving the overall performance by extracting four features per kernel: adding to ppv the mpv (mean of positive values), the mipv (mean of indices of positive values) and the lspv (longest stretch of positive values). Additionally, those pooling operators are extracted for one alternative time series representation, the first order difference. This improvement comes at the cost of training time, which is approximately 2x slower, generating by default 5x more features than MiniROCKET. InceptionTime, a Deep Learning approach The Incep-tionTime algorithm is one of the current most accurate DL architecture for TSC ( [START_REF] Ismail Fawaz | Inceptiontime: Finding alexnet for time series classification[END_REF]). InceptionTime is an ensemble of five Inception networks which are using Inception-V4 modules ( [START_REF] Szegedy | Inception-v4, inception-resnet and the impact of residual connections on learning[END_REF]) which themselves combine classic Inception blocks as introduced in [START_REF] Szegedy | Going deeper with convolutions[END_REF] and Residual Networks (ResNet) ( [START_REF] He | Deep residual learning for image recognition[END_REF]). The InceptionTime network architecture consists in two residuals blocks, each of which containing three Inception modules, it is then followed by a global averaging pooling layer and a fully-connected layer with the softmax activation function. For further information about DL for TSC, we refer to the extensive analysis of [START_REF] Ismail Fawaz | Deep learning for time series classification: a review[END_REF], which review the main DL algorithms for TSC. Hybrid/Ensemble methods Ensemble approaches aim at combining the performance of several classifiers to built its own prediction. Those types of algorithms are currently holding the state-of-the-art in term of accuracy on the usual UEA & UCR benchmark. The Time Series Combination of Heterogeneous and Integrated Embedding Forest (TS-CHIEF) ( [START_REF] Shifaz | Ts-chief: a scalable and accurate forest algorithm for time series classification[END_REF]), the second most accurate algorithm, builds random forest of decision trees whose splitting functions are time series specific and based on similarity measures, bag-of-words representations, and interval-based transformations. The Hierarchical Vote Collective of Transformation-Based Ensembles HIVE-COTE (HC [START_REF] Lines | Time series classification with hivecote: The hierarchical vote collective of transformation-based ensembles[END_REF], HC2 [START_REF] Middlehurst | Hive-cote 2.0: a new meta ensemble for time series classification[END_REF]) is the current best performer, training independently and combining several classifiers, i.e., Shapelet Transform [START_REF] Bostrom | Binary shapelet transform for multiclass time series classification[END_REF], Temporal Dictionary Ensemble aka TDE [START_REF] Middlehurst | The temporal dictionary ensemble (TDE) classifier for time series classification[END_REF], an ensemble of ROCKET estimators called Arsenal and the interval based Diverse Representation Canonical Interval Forest aka DrCIF [START_REF] Middlehurst | The canonical interval forest (CIF) classifier for time series classification[END_REF]. Hybrid ensemble classifiers are the most accurate TSC algorithms to date. However, the top performance comes at the price of high memory and CPU resources. As a result, they are generally at least an order of magnitude slower than the other contenders (see Table III).

B. Feature Extraction tools for time series

Besides previous algorithms dedicated to TSC problems, some research works proposed to automatically extract unsupervised features from time series in order to analyze and visualize the data. Most of them though, are rarely considered it comes to classification task and are less mentioned in TSC literature. In this part, we will briefly review the currently available packages/methods which are extracting some understandable features from time series data and which could actually be useful for classification. Tsfresh, which stands for Time Series FeatuRe Extraction on basis of Scalable Hypothesis tests, is a library proposing to use 63 usual times series characterization in order to extract, at most, around 800 features per sample ( [START_REF] Christ | Distributed and parallel time series feature extraction for industrial big data applications[END_REF]). It actually provides 3 pre-defined features dictionaries whose length ranging from 10 (minimal set-up, very efficient extraction) to 800 features. The package also provides some relevance filter based on some statistical tests to only keep the most informative variables among the ones you have extracted. The features computation can be parallelized, theoretically making the method more scalable. tsfel:The Time Series Feature Extraction Library is a more recent work which, similarly extracts a bunch of features from time series. Its specification is that it allows you to separate your data into user-specified window length and provides some better tools in order to analyse the temporal complexity of the extracted features. In tsfel, one can compute features according to their domain, the available domains are: temporal, spectral and statistical one or to extract them all at once. catch22 & hctsa: The CAnonical Time series CHaracteristics (catch22) is a subset of 22 variables from the 7730 available features in the hctsa toolbox ( [START_REF] Fulcher | hctsa: A computational framework for automated time-series phenotyping using massive feature extraction[END_REF]), which is originally a matlab package, able to extract up to nearly 7700 time series features. This limited set of features has been empirically selected over a large collection of datasets in order to choose the most explanatory ones and are, as well, minimally redundant. This approach take just a fraction of the computation time needed to compute and analyse the all hctsa set of features. Authors exhibit the fact that concerning classification, the average performance reduction is only around 7% ( [START_REF] Lubba | catch22: Canonical time-series characteristics[END_REF]). tsfeatures: Originally a R package ( [START_REF] Hyndman | tsfeatures: Time series feature extraction[END_REF]), the tsfeatures library is used to extract various type of features for time series data. The package implements around 40 methods in order to extract features, ranging from entropy to hurst exponent. A python version of the package is also available, we did not perform any time performances comparison between the two versions. feasts: The Feature Extraction and Statistics for Time Series (feasts) ( [START_REF] O'hara-Wild | feasts[END_REF]), looks very similar to tsfeatures. This R package provides tools for the analysis and visualization of time series. There is no python implementation of the package.

Featuretools is an open-source Python library, which performs auto-features engineering based on relational tables. The library is built upon a feature discovery algorithm called Deep Feature Synthesis ( [START_REF] Kanter | Deep feature synthesis: Towards automating data science endeavors[END_REF]), which uses some aggregation and transform functions applied over several related tables, to create features. Working with temporal data with featuretools is quite natural, it includes many date based operations as well as some of the tsfresh functions. While the features computation can be perform on multiple threads, the discovery though, cannot ; which make the method not very scalable when generating a large number of features. Additionally, one can remark that there is no way to prevent features generation from overfitting, generate very complex features may indeed results in learning some training set specificities and thus degrade classification performances. Generalised signatures: The signature transform is an infinite collection of statistics for sequential data representation and/or feature extraction derived from rough paths theory. The signature can be thought as a moment generating function, as each term in the collection has a particular (geometrical) meaning as a function of data points. Usually, one compute the N truncated signature of x = (x 1 , ..., x n ), with x i ∈ R d and linearly interpolate x into the path f = (f 1 , ..., f d ) : [0, 1] -→ R d . The signature of depth N of x is defined as follows:

Sig N (x) =         • • • 0≤t 1 ≤...≤t k ≤1 k j=1 df i j dt (t j )dt 1 ...dt k    1≤i 1 ,...,i k ≤d      1≤k≤N (1)
The depth-1 terms (k = 1), for example, simply represent the total increments (difference between end and start point) over each dimension. When it comes to TSC, the generalised signature pipeline ( [START_REF] Morrill | A generalised signature method for multivariate time series feature extraction[END_REF]), which has primarily been designed for multivariate series (d > 1), computes the signature transform over some hierarchical dyadic windows, to finally concatenate the outputs into the feature vector. This approach has shown to be competitive compared to state-of-the-art classifiers.

III. EXPERIMENTAL VALIDATION

In order to explore the potential of feature construction for TSC problems, the experiments carried out in this paper have been designed to answer the following questions:

• (Q 1 ) How do the various FC tools compare with each other in terms of predictive performance and time efficiency ? • (Q 2 ) In which extent using the new features perform better than raw original data ? • (Q 3 ) Is it beneficial to combine several feature engineering tools ? • (Q 4 ) How does feature construction based methods compare with state of the art TSC methods ? Aiming at the full reproducibility of the experiments, we first present the details of our experimental protocol then discuss the obtained results. A dedicated webpage with the source code and scripts to run the experiments is also available 1 .

A. Experimental protocol

Data sets. We have conducted our experiments on the usual time series classification benchmark. We ran our experiments on 112 data sets out of the 128 ones which are available in the UCR Time Series Classification Archive ( [START_REF] Bagnall | The UEA & UCR time series classification repository[END_REF], [START_REF] Dau | The UCR time series archive[END_REF]). Indeed, data sets with missing values or with times series of unequal length have been excluded from the current study, as well as the Fungi data set which only contains one instance per class in the training set. Feature engineering tools. As one of the main goal is to compare several feature extraction methods, it is more suitable to separate the feature generation step from the classification one. Thus, we decided to include in this study only the unsupervised methods, which can be easily used with any classifier (see Table I). We removed deep learning approaches from our study, since generative models like AutoEncoders require very high CPU resources -which leads to hardly fair comparison -and, as highlighted in [START_REF] Ismail Fawaz | Deep learning for time series classification: a review[END_REF], their predictive 1 https://github.com/aurelien-renault/Automatic-Feature-Engineering-for-TSC performance on TSC problems is mitigated. In addition, based on the TSF, CIF algorithms, as in [START_REF] Middlehurst | The freshprince: A simple transformation based pipeline time series classifier[END_REF], we have included two methods for extracting some set of descriptive statistics on random sampled intervals i.e. subseries. Nevertheless, since we want to be able to use any off-the-shelf classifier, we are extracting the whole set of features on all the sampled intervals rather than learning one Decision tree on some features subsets as is TSF, CIF.

Finally, we tested a large panel of libraries which are the following:

• ROCKET and its variants ( [START_REF] Dempster | Rocket: exceptionally fast and accurate time series classification using random convolutional kernels[END_REF], [START_REF] Dempster | Minirocket: A very fast (almost) deterministic transform for time series classification[END_REF], [START_REF] Tan | Multirocket: Multiple pooling operators and transformations for fast and effective time series classification[END_REF]),

• Intervals Transform (which simply apply a summary transformation over multiple random intervals) • Intervals Catch22 (catch22 features from random intervals)

• TsFreh ([13]), • Tsfel ([15]), • Catch22 ([11]), • TsFeatures ([16]), • FeatureTools ([12]), • Signature ([33]) • Feasts ([17]),
• HCTSA ( [START_REF] Fulcher | hctsa: A computational framework for automated time-series phenotyping using massive feature extraction[END_REF]). Experimental parameters and classifiers. In order to keep the comparison as fair as possible, we evaluated the different methods limiting the number of features to 1000. Of course, not all libraries generate this number of extracted features (see Table I), tsfresh for example, cannot generates more than 794 features without explicitly telling it what to compute. When available, the transformations algorithms have been run using the sktime package ( [START_REF] Löning | sktime: A unified interface for machine learning with time series[END_REF]). Hyperparameters values are reported in Table II. All experiments have been computed using a single thread of a 4-core 2GHz Intel Core i5-1038NG7 CPU, except for hctsa which used 4 workers, with 16Go RAM. Once all the features matrices have been computed, we applied 9 different classifiers, Random Forests (100, 500 trees), Logistic Regression (ℓ 2 , elasticnet penalties), XGBoost, SVM (linear, rbf kernel), 1 Nearest Neighbors and Rotation Forest with every other parameters being the default ones, from the sklearn library ( [START_REF] Pedregosa | Scikit-learn: Machine learning in python[END_REF]), except for the Rotation Forest classifier, which is in the sktime package [START_REF] Bagnall | Is rotation forest the best classifier for problems with continuous features?[END_REF]. Evaluation methodology. We use pre-defined train/test sets available from the UCR archive and the accuracy as the predictive performance evaluation measure. This way, the comparison with state of the art methods like HC1, HC2, TSChief and InceptionTime -which are computationally expensive -is feasible as we report the results from the original papers. To compare different approaches over several data sets, critical difference diagrams have been drawn using the posthoc Nemenyi rank based test. Although [START_REF] Benavoli | Should we really use post-hoc tests based on mean-ranks?[END_REF] suggests to use the Wilcoxon signed-rank test with the Holm correction in place of the Nemenyi one, the diagrams drawn this way have been found to be unreadable quite often as it contained some overlapping cliques, as the Wilcoxon signed-rank test is not based on mean ranks, the analysis could have been confusing. Though, we display, when needed, alongside critical Nemenyi

✓ ✗ ✓ ✗ catch22 [11] 22 ✓ ✗ ✓ ✓ Constructed featuretools [12] inf ✓ ✗ ✓ ✗ fears [35] inf ✗ ✓ ✓ ✗ Signature gen. signature inf ✓ ✗ ✗ ✓ Convolution shapelet [36] n shapelets ✗ ✗ ✗ ✓ rocket [7] inf ✓ ✗ ✓ ✓ Symbolic (SFA)
BOSS & cie ( [START_REF] Schäfer | The boss is concerned with time series classification in the presence of noise[END_REF], [START_REF] Large | On time series classification with dictionary-based classifiers[END_REF], [START_REF] Middlehurst | Scalable dictionary classifiers for time series classification[END_REF])

c l ✗ ✗ ✗ ✓ WEASEL [40] c 2l ✗ ✗ ✗ ✓ (SAX) BOP [41] c l ✓ ✗ ✗ ✓ Intervals TSF [42] 3r √ m ✗ ✓ ✗ ✓ CIF [43] 25r √ m ✗ ✓ ✗ ✓ Deep Learning AE/VAE [44] latent dim ✓ ✗ ✗ ✗ TS2Vec [45] ✓ ✗ ✗ ✗

B. Experimental results and Discussion

Performance comparison of feature engineering tools. Due to space limitation, we only report charts for three classifiers: Random Forests, Rotation Forests and Logisitc Regression. For a default Random Forest classifier, the best performing methods are the ROCKET-like methods, tsfresh, hctsa and the random intervals extracting the catch22 features.(see figure 2). When dealing with linear classifiers though, the ROCKETlike approaches become significantly better than the other tested methods as they are likely to generate independent kernels (see Figure 3). On the other side, the tsfresh/hctsa features are more likely to be correlated, which can partially explain that best classifiers are non linear ones such as Random Forest or XGBoost. It's also worth noting that data is standardized before being processed by linear classifiers, as they usually embed penalization, which make some of the extracted features useless.

We are also displaying the results obtained for the Rotation Forest classifier [START_REF] Lin | Rotation-invariant similarity in time series using bag-of-patterns representation[END_REF] (see Figure 4), which is actually the best performer for the great majority of the tested libraries, even if generally less scalable. Running times comparison. When considering computation time, we observe a clear trend: as expected, all the ROCKETfeatures based methods outperform the others in term of both time per feature and overall run time. In average the ROCKETlike methods take less than half a second for one given data set. Among the less scalable methods, we find libraries as hctsa, feasts or tsfeatures, as those are designed for analysis and 5a), feasts take more than 4 seconds to extract one feature, which make them the worst scalable libraries among the tested methods. Furthermore, one can notice that bringing featuretools from 50 to 1000 generated features slow down the algorithm, doubling the time/feature metric (see Figure 5b), which validate the previously mentioned hypothesis made In order to put some more perspective on our results we analyze the performance when using only the raw time series, i.e. each time point is considered as one feature, as well as the concatenation of the raw data with the extracted features and compare those with the previously obtained results, only using the computed features. Considering these three strategies, the first conclusion is that, extracting features, no matter which method is used, outperforms practically every time a classification only based on the raw time series. As well, Figure 6 demonstrates that, on average, across all strategies for all classifiers, adding the raw data to the feature matrix is performing significantly better. A more in-depth analysis actually shows that this improvement can be explained by the libraries which tend to generate correlated features. Indeed, in those cases, the best classifiers are the ones that are less sensitive to feature correlation and redundancy, i.e. Random Forest or XGBoost; adding the values of the time series (which could be correlated), when the length of the considered time series remains under a certain threshold, is improving performance in a significant way. However, as the time series length grows, this adding tends to add more noise and, at some point, harm performances, in Figure 7, we are displaying the performance only considering data sets for which the length of the time series are below/above the median length of the UCR repository.

Additionally, one can notice that, only working with data sets whose time series length are beyond the third quartile (around 28 datasets), the overall performance is much degraded and adding the raw data is then ranked behind the FTS strategy. The methods for which linear classifiers were the best never benefit from the raw data: even if some classifiers handle this incorporation better (adding a ℓ 1 penalty for example), they are able, at best, not to degrade the performance too much.

D. Combining libraries

In this part, we define one strategy in order to figure out if performance can be improved by combining several feature engineering tools. To keep things simple, the tested approach consists in gradually stacking the different methods, ranked by one given metric, here accuracy, obtained with a vanilla Random Forest classifier. We are not considering the whole set of ROCKET-like methods any longer, we decided to only retain the 1000 features created by MiniROCKET in a fisrt experiment, before completely deleting MiniROCKET features Fig. 7: Accuracy results for all strategiess on all classifiers considering time series length in a second one. Indeed, as ROCKET represents, a TSC approach on its own, we wanted to address whether or not our features set could still be competitive not using this great performing classifier. Additionally, to reduce computation time and redundancy, the expanded version of featuretools (with 1000 features) is also removed, as it did not provide any more significant information than its lighter version (with 50 features). It seems intuitive that stacking the different features ones onto others will probably add redundancy; in this way classifier like random/rotation forest seem to better fit this purpose ( [START_REF] Bagnall | Is rotation forest the best classifier for problems with continuous features?[END_REF]), we only reports the results for the Rotation Forest in what follows as it provided best performance so far. For this classifier, the best stacking strategy both in terms of ranks and mean accuracy (see Figure 8a) is the ones using are not performing well on their own i.e. catch22, feasts and tsfeatures, which are also the ones creating a limited number of features. When removing MiniROCKET features, some of those libraries become informative and the best strategy, without MiniROCKET, is now containing the all libraries set, only excluding tsfeatures (see Figure 8b). The top performers of both of these experiments are called Features and Features noROCKET respectively in what follows. Those strategies though, are not performing significantly better that stacking the two or three best libraries ones onto others.

E. Comparison with state of the art TSC approaches

We are now interested in comparing the obtained performance using our previously defined feature generation libraries with state-of-the-art methods, we can see that our methods are not the worst performers (See Figure 9). With the Rotation Forest algorithm (with default parameters) our main approach, noted Features, is ranked second among the six others methods, with a the fourth mean accuracy on 112 data sets from the UCR Archive with default train/test splits. Actually, under Nemenyi test, the method is the only one not significantly performing worst than the current most accurate method, namely HIVE-COTE 2.0 (HC2). Figure 10 gives additional results by comparing Features to four of the best TSC using 1vs1 scatter plots with standard ±5% to highlight notable difference of performance.

As the selected approach contains in itself some SOTA method with MiniROCKET features, we are also providing results when optimizing combining strategy without it. The Features noRocket approach is, under Holm's correction, as InceptionTime, not significantly worst than HC2: it remains competitive, only loosing 0.2% of mean accuracy comparing to its MiniROCKET version. Replacing MiniROCKET features comes also at the price of around 12 hours more in total training time from features extraction step.

Concerning scalability, Table III demonstrates that the optimal combination of features take a relatively long time to train on the default splits, indeed, it includes the worst scalable methods, i.e. the non-python ones such as hctsa and feasts. When using only the python available libraries (Features python), we are roughly dividing the total time by 2, it comes at the cost of exactly 0.1% of average accuracy, not loosing any rank compared with the SOTA classifiers. This approach though, becomes significantly worst than HC2. Moreover, it is worth-noting that these train time values do not include features extraction step on the test set, which would actually considerably increase the total time (see Table IV). That being said, even by considering complete feature extraction on both train and test set, our approaches remain substantially faster than HC2 for similar predictive performance on the default split, according to both Nemenyi test and Holm's corrected Wilcoxon signed-rank test.

IV. CONCLUSION

In this paper, we aim at exploring the potential predictive power of feature construction (FC) for time series classification (TSC). To this end, we have designed simple processes to branch existing feature engineering tools with standard classifiers. We have led extensive experiments resulting in the comparison of 11 feature engineering tools branched with Fig. 10: Scatter plot of accuracy results comparison: our approach "Features" vs state of the art TSC algorithms HiveCote2 (HC2), ROCKET, InceptionTime and TS-CHIEF 9 classifiers over 112 TSC problems -totalling more than 10000 learning experiments. The analysis of the experimental results indicates that (i) using feature engineering leads to better predictive performance for a given standard classifier, (ii) combining several unsupervised feature engineering tools with the rotation forest classifier is comparable with the best TSC performers in terms of predictive performance while demanding reasonable computing resources. The predictive and efficiency results of the suggested approaches also indicate that feature construction fot TSC is a valuable option to pursue. Indeed, it might be naturally extended to multivariate TSC problems -extracting the same feature set over each dimension or, e.g., randomly sampling some subsets of features in the case of high dimensional data sets. Also, another perspective would be to filter redundant or correlated features to speed up the learning phase.

Fig. 1 :

 1 Fig. 1: Critical difference diagram of FC variants against current state of the art on 112 UCR/UEA data sets. The average rank is plotted and each contender is annotated by its mean accuracy results. Thick lines linking group of classifiers indicate no significant difference in predictive performance. This demonstrates that suggested FC processes (in red) are among the best TSC performers.

  Fig. 2: Random Forest

  Fig. 4: Rotation Forest

Fig. 5 :

 5 Fig. 5: Computation time performances

  Fig. 6: Performance comparison: adding the raw time series or not labeled with mean accuracy

  (a) ts length < 315 (b) ts length > 315 (c) ts length > 720

Fig. 8 :

 8 Fig. 8: Critical diagrams labeled with mean accuracy Only the first and last library are displayed, one underscore stands for one in-between library like in the following : TSFRESH + + HCTSA = TSFRESH + INTERVALS C22 + HCTSA.

  Fig. 9: Predictive performance comparison with state-of-theart TSC algorithms

TABLE I :

 I Comparison table of several time series transformation methods. (max features: maximum number of generated features, unsup: unsupervised features extraction, embedded clf: use a specific classifier, i.e. can not be used with any off-theshelf classifier, repr: use of different TS representations, -sktime: present in sktime framework (0.13.0)

	Methods	Parameters	Nb features
	rocket	n kernels : 500	1000
	minirocket	n kernels : 1000	924
	multirocket	n kernels : 125 n f eatures per kernel : 6	1008
		n intervals : 100
	intervals	agg : [mean, min, max, sum, med, std	1000
		count, skew, quant(0.25), quant(0.75)]
	intervals c22	n intervals = 45	1000
		agg : [mean, min, max, sum,
	featuretools	median, std, count, skew] transf : [D, DD, CU M SU M,	50
		DCU M SU M, ACF, P S]
	featuretools 1k col combinations : [add, substr, mult]	1000
	signature	window name : dyadic window depth : 4	930
	hctsa	n f eatures : 1000	1000
	catch22		22
	feasts		33
	tsfeatures	def ault parameters	37
	tsfel		142-390
	tsfresh		789

TABLE II :

 II

Libraries' hyperparameters values diagrams, the binary matrix showing explicitly the Holm adjusted pairwise Wilcoxon comparison for each considered methods (see subpart (b) of figures). In these charts, the small black squares identify pairs of approaches that do not differ significantly in performance.

TABLE III :

 III Run time to train on a single split on the 112 UCR datasets. For all competitors algorithms, the value for each dataset is the median taken over 30 different resamples. For our approaches, written in red, the value is the one for the default train/test split.

	Algorithm	Extraction Extraction Classifier
		train (h)	test (h)	Fit (h)
	Features python	15.82	37	9.33
	Features	34.05	79.58	12.11
	Features noROCKET 38.74	90.1	8.88

TABLE IV :

 IV Run time details of the proposed approaches. Training time is the sum of first and third column.


Suman Kumar Maji
email: kumarmaji@iitp.ac.in

Hussein Yahia

Image Denoising in Fluorescence Microscopy using Feature based Gradient Reconstruction

Keywords: Image Denoising, Fluorescence Microscopy, Poisson-Gaussian Noise, Singularity exponents

Purpose: The utility of fluorescence microscopy imaging comes with the challenge of low resolution acquisitions which severely limit information extraction and quantitative analysis. Image denoising is a technique that aims to remove noise from microscopy acquisitions by taking into account prior statistics of the corrupting noise. In this paper, we propose an image denoising technique for fluorescence microscopy imaging. Approach: The proposed technique is based on the principle of multifractal feature extraction from a noisy sample followed by a reconstruction technique from these features. It is observed that by following a proper hierarchical classification procedure, meaningful features can be extracted from a noisy image. A denoised image is then estimated from this sparse feature set through proper formulation of an optimization problem.

Results: Experiments are performed on both synthetic image databases as well as on real fluorescence microscopy data. Superior denoising results, in comparison to multiple comparing techniques, validate the potential of the proposed approach.

Conclusion:

The proposed method gives superior denoising results for low resolution fluorescence microscopy image acquisitions and can be used for post processing of data by biologists.

Introduction

Over the decade, fluorescence microscopy has become a biologists tool of choice for in vivo analysis of cellular structures. However, due to various physical constraints like short exposure time, photo-toxicity, photo-bleaching, device imperfections, etc. the resulting image is a low resolution noisy acquisition. This seriously hampers further analysis like cell segmentation, [START_REF] Gharipour | Segmentation of cell nuclei in fluorescence microscopy images: An integrated framework using level set segmentation and touching-cell splitting[END_REF] recognition, [START_REF] Almasi | Joint volumetric extraction and enhancement of vasculature from low-snr 3-d fluorescence microscopy images[END_REF] reconstruction, 3 etc of the acquired samples. Noise in fluorescent microscopy acquired samples is characterized by a mixed Poisson-Gaussian (MPG) statistics. [START_REF] Boulanger | Patch-based non-local functional for denoising fluorescence microscopy image sequences[END_REF][START_REF] Montagner | An unbiased risk estimator for image denoising in the presence of mixed poisson-gaussian noise[END_REF][START_REF] Maji | Joint denoising-deconvolution approach for fluorescence microscopy[END_REF][START_REF] Maji | A variational model for poisson gaussian joint denoising deconvolution[END_REF][START_REF] Zhang | A poisson-gaussian denoising dataset with real fluorescence microscopy images[END_REF] Image denoising is a technique that specifically addresses noise removal in a sample through proper knowledge of the noise source and its distribution.

1 Multiple denoising strategies have been proposed in the literature for improving the quality of the low resolution fluorescent microscopy samples. These strategies range from Bayesian based formulations, [START_REF] Chouzenoux | A convex approach for image restoration with exact poisson-gaussian likelihood[END_REF][START_REF] Gao | Bayesian joint super-resolution, deconvolution, and denoising of images with poisson-gaussian noise[END_REF] wavelet based techniques, [START_REF] Luisier | Image denoising in mixed poisson-gaussian noise[END_REF][START_REF] Blu | The sure-let approach to image denoising[END_REF] non-local patch based techniques [START_REF] Boulanger | Patch-based non-local functional for denoising fluorescence microscopy image sequences[END_REF][START_REF] Baudes | A non-local algorithm for image denoising[END_REF] to variance stabilization based approaches, [START_REF] Makitalo | Optimal inversion of the generalized anscombe transformation for poisson-gaussian noise[END_REF][START_REF] Azzari | Variance stabilization for noisy+estimate combination in iterative poisson denoising[END_REF] to name a few. Bayesian based techniques rely on deriving an appropriate noise restoration model using Bayes principle, taking into account the statistics of the nature of the noise. In this regard, the denoising model in [START_REF] Chouzenoux | A convex approach for image restoration with exact poisson-gaussian likelihood[END_REF] is particularly remarkable as it derives a convex optimization model using Bayes formulation, for fluorescent microscopy samples, based on the Poisson-Gaussian likelihood. Wavelet based denoising schemes operate on manipulating the wavelet coefficients within the framework of forward and inverse wavelet transform. The SURE-LET [START_REF] Blu | The sure-let approach to image denoising[END_REF] approach is one such technique. It is an orthonormal wavelet thresholding based denoising scheme that takes into account the interscale dependencies between the wavelet coefficients for removing noise. This scheme has been extended to a PURE-LET [START_REF] Luisier | Image denoising in mixed poisson-gaussian noise[END_REF] scheme, where in the authors have generalized the concept of Haar wavelet transform to fit into a series of linear expansion of thresholds. Patch based non-local denoising techniques, like, [START_REF] Boulanger | Patch-based non-local functional for denoising fluorescence microscopy image sequences[END_REF] work on the principle of exploiting patch-based redundancy within a given sample. The NDSAFIR technique [START_REF] Boulanger | Patch-based non-local functional for denoising fluorescence microscopy image sequences[END_REF] is one such method, where the authors take into account the spatio-temporal image patches of observed fluorescent samples for minimizing an objective non-local energy functional. Stein's unbiased risk estimator (SURE) is a similarity criteria that is often used in image denoising for predicting optimal solutions. In PG-URE [START_REF] Montagner | An unbiased risk estimator for image denoising in the presence of mixed poisson-gaussian noise[END_REF] the authors extend the concept of SURE to address the case of Poisson-Gaussian noise removal in fluorescent samples. Another popular category of image denoising technique are the variance stabilization (VST) based techniques. [START_REF] Makitalo | Optimal inversion of the generalized anscombe transformation for poisson-gaussian noise[END_REF][START_REF] Azzari | Variance stabilization for noisy+estimate combination in iterative poisson denoising[END_REF] VST based methods make use of appropriate transformations in order to convert the signal dependent noisy component into signal independent noise. VST+BM3D [START_REF] Azzari | Variance stabilization for noisy+estimate combination in iterative poisson denoising[END_REF] and GAT+BM3D [START_REF] Makitalo | Optimal inversion of the generalized anscombe transformation for poisson-gaussian noise[END_REF] are two such popular denoising schemes for fluorescent microscopy samples that make use of the Anscombe transform as a stabilizier for the signal-dependent Poisson noise.

Deep learning based denoising techniques have also become popular in recent years. [START_REF] Belthangady | Applications, promises, and pitfalls of deep learning for fluorescence image reconstruction[END_REF][START_REF] Moen | Deep learning for cellular image analysis[END_REF] A content-aware image restoration based on deep learning is proposed in [START_REF] Weigert | Content-aware image restoration: pushing the limits of fluorescence microscopy[END_REF] for denoising fluorescence microscopy samples. A three dimensional deep neural network based residual channel attention denoising network for fluorescence microscopy is proposed in. [START_REF] Chen | Three-dimensional residual channel attention networks denoise and sharpen fluorescence microscopy image volumes[END_REF] In addition to these, self-supervised denoising networks that work on the principle of learning only from the single noisy image is also widely used for fluorescence microscopy denoising. The Self2Self 20 model is one such network that uses Bernoulli sampling as a dropout strategy for denoising images. The Neighbor2Neighbor (Nb2Nb) [START_REF] Huang | Neighbor2neighbor: Self-supervised denoising from single noisy images[END_REF] is another approach that uses a random neighbor up-sampler to generate training data for effective denoising.

The denoising technique that we propose in this paper is based from ideas encountered in the analysis of multifractal systems. [START_REF] Turiel | The multifractal structure of contrast changes in natural images: From sharp edges to textures[END_REF][START_REF] Turiel | Microcanonical multifractal formalism -a geometrical approach to multifractal systems: Part 1. singularity analysis[END_REF][START_REF] Turiel | Numerical methods for the estimation of multifractal singularity spectra on sampled data: A comparative study[END_REF][START_REF] Badri | Fast and accurate texture recognition with multilayer convolution and multifractal analysis[END_REF] These are scale-free complex systems whose characteristic quantities display a power-law behaviour and are characterized by values called singularity exponents (SE's). [START_REF] Maji | Edges, transitions and criticality[END_REF] Natural scale-free complex systems exhibit multifractal traits [START_REF] Falconer | Techniques in fractal geometry[END_REF] implying thereby that signals acquired from such systems can be characterized by multifractals, an approach to complex systems which has been validated over a large dataset. [START_REF] Turiel | The multifractal structure of contrast changes in natural images: From sharp edges to textures[END_REF] The denoising scheme that we propose is a two-step concept where in the first step we extract a sparse feature map through multifractal analysis on the acquired noisy image. In the second step we build an optimization problem to estimate a denoised image from the gradients confined to the extracted feature map.

The contribution of this paper are as follows:

• We formulate a novel denoising algorithm that uses multifractal analysis to extract meaning-ful features from a noisy image, and then reconstruct a denoised image from these features.

Under the hypothesis of a multifractal system, the integrated signal's gradient of the original image has a power-law behaviour whose exponent at each point x of the signal's domain, h(x), known as a singularity exponent (SE), can be computed using a local correlation measure.

• We recall the computation of the local correlation measure and of the set of lowest exponents, L ∞ , which is is of fundamental significance in signal processing as it allows the reconstruction of a new signal from the SEs.

• We design a reconstruction algorithm to estimate a denoised image from the feature set L ∞ .

• Experimental results over three image databases, under different levels of noise, show that the proposed method outperforms both classical denoising techniques as well as recent deep learning techniques in the field.

The rest of the paper is organised as follows: In section 2.1 we mathematically formulate the image acquisition problem and objective and briefly summarize the multifractal analysis of acquired samples. Here in subsection 2.3 we propose a technique for computing the quantity h(x) required for estimating the feature set L ∞ and in subsection 2.4 we propose a method for estimating a denoised image from L ∞ . In section 3 we present the experiment results and discuss them and finally conclude in section 4.

Proposed Method

Preliminaries & Objective

If f : Ω → R is the acquired noisy image defined on the image domain Ω ⊂ R and u : Ω → R its noise free version, the image formation process for fluoescence microscopy can be written as: [START_REF] Montagner | An unbiased risk estimator for image denoising in the presence of mixed poisson-gaussian noise[END_REF][START_REF] Luisier | Image denoising in mixed poisson-gaussian noise[END_REF]

f = τ (u) + w (1)
where τ ( Gaussian noise in f . [START_REF] Montagner | An unbiased risk estimator for image denoising in the presence of mixed poisson-gaussian noise[END_REF] (σ, ζ) were empirically determined keeping into account the fact that higher the σ value, higher the Gaussian noise. Similarly higher the ζ value, higher is the Poisson noise.

Noise is generated with different sets of values of (σ, ζ), over the ground truth, following the above equation. We intend to recover an image û ≈ u from the noisy fluorescent microscopy sample f .

Multifractal Analysis and Feature Extraction

As discussed earlier, natural scale-free systems exhibit self-similarity and multifractal behavior implying therefore that signals (or images) acquired through such systems will exhibit similar traits. If f (x) (where x is a vector representing the location of a pixel), which in our case is the acquired noisy image, is a signal with multifractal properties then it can be used to define a measure with multi-scale properties, [START_REF] Turiel | The multifractal structure of contrast changes in natural images: From sharp edges to textures[END_REF][START_REF] Turiel | Microcanonical multifractal formalism -a geometrical approach to multifractal systems: Part 1. singularity analysis[END_REF][START_REF] Turiel | Numerical methods for the estimation of multifractal singularity spectra on sampled data: A comparative study[END_REF][START_REF] Yahia | Description of turbulent dynamics in the interstellar medium: multifractal-microcanonical analysis -i. application to herschel observations of the musca filament[END_REF] µ(x, r) (x: pixel location, r scale) such that we have the limiting behaviour:

h(x) = lim r→0 log µ(x, r) log r (2)
which in turn implies that

µ(x, r) ≈ α(x)r h(x) (3)
where α(x) is a signal dependent constant. h(x) is known as the singularity exponents or SE's.

The transitions within a signal are well-recorded in h(x), [START_REF] Maji | Edges, transitions and criticality[END_REF] which are in turn independent of the measure µ, as long as it is multiscale. There are different ways to compute the h(x), once µ is defined properly. The most evident algorithm consists in noting that equation (3) implies that, by taking the logarithms on both sides:

log(µ(x, r)) = log(α(x)) + h(x) log(r) (r → 0) (4)
and then perform log-log regression for many different scales r. However, the accuracy of such method depends on the number of scales r chosen for the regression; it has been shown [START_REF] Yahia | Description of turbulent dynamics in the interstellar medium: multifractal-microcanonical analysis -i. application to herschel observations of the musca filament[END_REF] (and the references herein) that a most advanced methodology consists in evaluating h(x) by use of a local correlation measure. This is justified by the existence of a local reconstruction of the whole signal at the points where h(x) at the lowest values. [START_REF] Yahia | Description of turbulent dynamics in the interstellar medium: multifractal-microcanonical analysis -i. application to herschel observations of the musca filament[END_REF] We postpone this computation to section 2.3 below. For the moment, supposing that h(x) has been properly calculated, they are used to define a multifractal hierarchy of multiple fractals L h as level sets: [START_REF] Badri | Fast and accurate texture recognition with multilayer convolution and multifractal analysis[END_REF] from where it is possible to extract a particular set L ∞ consisting of components associated to the smallest

L h = {x | h(x) = h},
Algorithm 1 Input: h(x), Output: L ∞
Step 1: Compute the multifractal exponents h(x) from Algorithm 2.

Step 2: Define density

δ F∞ (x) = 1 if h ∞ -∆h ≤ h(x) ≤ h ∞ + ∆h 0 otherwise
Step 3: ∆h is conventionally fixed (0.2 is usually a good choice).

Step 4: δ L∞ (x) is the binary map that locates the L ∞ pixels.

possible value h ∞ defined as:

L ∞ = {x | h(x) = h ∞ = min(h(x))} (5)
The h ∞ 's only encode the pixels where it observes sharp transitions in the image i.e., the boundaries of objects inside the image, and maintains its stability under noise. This can be visualised in Fig. 1 (middle row). The procedure for computing the set L ∞ has been listed in algorithm 1.

Local correlation measure for the computation of h(x)

As explained in the last subsection, a local correlation measure can be used to evaluate the SEs h(x), instead of log-regression across the scales (which necessitates to define various multiscale versions of the original signal). The key observation lies in the existence of the set L ∞ , which allows a local reconstruction of the signal from the most singular exponents (see section 2.4 and [START_REF] Yahia | Description of turbulent dynamics in the interstellar medium: multifractal-microcanonical analysis -i. application to herschel observations of the musca filament[END_REF] ).

In this study, we propose a new local correlation measure, well adapted the noise reduction problem we have in sight.

In this paper we propose a method to compute h(x) at every pixel location x = (i, j) in the image taking into consideration their 4 neighbouring pixels: denoting f (i, j) the gray-level value at pixel x = (i, j), the 4 neighbouring pixel values form the set {f (i -1, j), f (i + 1, j), f (i, j -1), f (i, j + 1)}. In an image of size N × M (N, M : numbers of pixels in the x and y di- rections respectively), and assuming the whole image of surface 1, the minimum resolution is

r 0 = 1 √ N × M .
The complete procedure to compute h(x) at resolution r 0 is given in Algorithm 2 (where * denotes convolution and T stands for transposition). Note that at minimum fixed resolution r 0 , µ(x, r 0 ) depends only on x.

Algorithm 2 Input: Image f , Output: Singularity exponent h(x) at x = (i, j)

Step 1:

ϵ x = f ⋆ [1 0 -1]); ϵ y = f ⋆ -[1 0 -1] T , ϵ
x , ϵ y correspond to unit pixel displacements at maximum scale r 0 .

Step 2: e(x) = ϵ 2 x + ϵ 2 y .

Step 3: µ(x, r 0 ) = e(x).

Step 4: h(x) = log µ(x,r 0 ) log r 0 .

Reconstructing the Denoised Image

In this section, we propose a novel formulation to reconstruct a denoised image û ≈ u from the sparse feature set L ∞ . Such a reconstruction comes from the possibility on integrating gradients over the set L ∞ (see [START_REF] Badri | Fast and accurate texture recognition with multilayer convolution and multifractal analysis[END_REF][START_REF] Yahia | Description of turbulent dynamics in the interstellar medium: multifractal-microcanonical analysis -i. application to herschel observations of the musca filament[END_REF] ). However in this section we will introduce a new reconstruction algorithm which makes use of a L2-L1 minimization with sparse gradients instead of reconstruction in Fourier space as in. [START_REF] Badri | Fast and accurate texture recognition with multilayer convolution and multifractal analysis[END_REF][START_REF] Yahia | Description of turbulent dynamics in the interstellar medium: multifractal-microcanonical analysis -i. application to herschel observations of the musca filament[END_REF] This choice is motivated by our primary goal of noise reduction, for which sparse methods have been shown to be particularly effective ( [START_REF] Chartrand | Exact reconstruction of sparse signals via nonconvex minimization[END_REF][START_REF] Chartrand | Fast algorithms for nonconvex compressive sensing: Mri reconstruction from very few data[END_REF][START_REF] Gasso | Recovering sparse signals with a certain family of non-convex penalties and dc programing[END_REF] ).

Let f x , f y be the gradients of the noisy image f and δ L∞ be the binary map of the set L ∞ . We intend to recover u from the sparse gradient set v = { fx , fy }, where fx = f x •δ L∞ and fy = f y •δ L∞ .

Since L ∞ encodes the most meaningful pixels of the image, the gradients corresponding to L ∞ will be the most informative. A reconstruction from the set v will therefore lead to a proper estimation of the whole image and noise propagation will be automatically minimal since a lot of noisy pixels are dropped. Keeping this in mind, we therefore propose to minimize the following optimization problem:

û = argmin u 1 2 ∥∇u -v∥ 2 2 + λ∥∇u∥ 1 ( 6
)
where λ is a positive regularization parameter. With Half-Quadratic splitting, using an intermediate variable z and an additional positive regularization term β, we get:

û = argmin u,z 1 2 ∥∇u -v∥ 2 2 + λ{∥z∥ 1 + β 2 ∥∇u -z∥ 2 2 } (7)
which is further organised into independent sub-problems p 1 and p 2 :

p 1 : z (k+1) ← argmin z ∥z∥ 1 + β 2 ∥∇u (k) -z∥ 2 2 p 2 : u (k+1) ← argmin u ∥∇u -v∥ 2 2 + λ∥∇u -z (k+1) ∥ 2 2
Problem p 1 is in the proximal form argmin z ∥z∥ 1 + β 2 ∥x -z∥ 2 2 whose solution correspond to a soft thersholding shrinakge operator: 30

z = shrink lp (x, β) = max 0, |x| - |x| p-1 β |x| x (8)
Considering z as a 2-component vector, the solution to sub-problem p 1 is given as: Problem p 2 is in the regular quadratic form and can be solved using Euler-Lagrange. The final solution using FFT will be:

z (k+1) =            z (k+1) x = shrink l 1 ∇ x u (k) , β z (k+1) y = shrink l 1 ∇ y u (k) , β (9)
u (k+1) = F -1 F(div(v) + div(z (k+1) )) (1 + λ)∆ ( 10
)
where F denotes Fourier transform, ∆ = ∇ 2 the Laplace operator. The results of reconstruction, under different levels of noise, are shown in Fig. 1 (bottom row) and Fig. 2. We have used two types of data: Synthetic data and fluorescence microscopy acquisition data. For comparison purpose we have considered the four popular classical denoising techniques designed for removing MPG noise: SURE-LET, [START_REF] Luisier | Image denoising in mixed poisson-gaussian noise[END_REF] PG-URE, 5 VST+BM3D 15 and GAT+BM3D. [START_REF] Makitalo | Optimal inversion of the generalized anscombe transformation for poisson-gaussian noise[END_REF] We have also used two recent deep learning based self-supervised denoising methods that learns only from the input noisy image, thereby making them ideal for comparison. They are: Self2Self 20 and Nb2Nb. [START_REF] Huang | Neighbor2neighbor: Self-supervised denoising from single noisy images[END_REF] We have also used DnCNN 32 and Noise2Noise 33 models for comparison on Fluorescence Microscopy Denoising (FMD) dataset. [START_REF] Zhang | A poisson-gaussian denoising dataset with real fluorescence microscopy images[END_REF] The parameter settings for all these methods have been kept the same as suggested by their authors in the papers and in the codes. For the purpose of quantitative evaluation, we have considered the peak signal-to-noise ratio (PSNR) and structural similarity index (SSIM) metrics, , defined as follows:

PSNR = 10 log 10 u 2 max (∥û -u∥ 2 /(m × n)) (11)
where u max is the maximum intensity of the noise-free image u and m × n the size of the image u.

SSIM(u, û) = (2µ u µ û + c 1 )(2σ uû + c 2 ) (µ 2 u + µ 2 û + c 1 )(σ 2 u + σ 2 û + c 2 ) (12)
where µ u , µ û are the average of u and û; σ 2 u , σ 2 û are the variance of u and û; σ uû the covariance of u and û respectively; c 1 and c 2 are two normalizing parameters.

Synthetic Data

We have considered three image datasets: the Classic5, Set12 and BSD68 which consists of 5, 12 and 68 synthetic images respectively. We have generated multiple noisy versions of these datasets, by adding MPG noise of level {σ, ζ} = {10 -2.5 , 10 -3 }, {10 -1 , 10 -2 } and {10 -2 , 10 -1.5 }, over which we perform denoising. In Fig. 2, we present the visual comparison results of our technique with the competing methods. As can be seen from the excerpts (highlighted in green bounding boxes), SURE-LET is capable of retaining the details to a certain extent, but introduces artefacts in the denoised output. PG-URE, VST + BM3D and GAT+BM3D does better denoising than SURE-LET, but are mostly accompanied by over smoothing which leads to loss in details. For the cameraman image one can see that the background building behind the tripod, in the case of boat image the paint linings corresponding to the boat name and in the case of foreman image the black spots in the building behind are best reconstructed by the proposed method. The proposed technique is therefore able to retain the fine details of the image while rendering better visual quality than the others in terms of noise removal. This is also justified quantitatively by the results of the mean PSNR and mean SSIM, calculated over all the images of the datasets, under different levels of noise, as is shown in table. 1.

Fluorescence Microscopy Data

The dataset that we have used is the fluorescence microscopy denoising dataset (FMD). [START_REF] Zhang | A poisson-gaussian denoising dataset with real fluorescence microscopy images[END_REF] The vi- during training) and has therefore recovered the low noise also in its reconstruction. The proposed method (as well as PG-URE, SURE-LET, VST+BM3D, GAT + BM3D and DnCNN) cleans the background noise completely. However its PSNR and SSIM values will be lower because it mismatches with the low noisy ground truth. The ground truth of BPAE is therefore not suitable for quantitative analysis using PSNR and SSIM.

sual

Run Time

In the table below, we provide the running time for all the comparing methods for real data all of whose image size are 512 × 512 pixels. It is to be noted that PG-URE, SURE-LET, VST+BM3D, GAT+BM3D and Proposed methods are Matlab based methods and utilizes the CPU. We tested it on a standard core i5 8th Gen laptop with 8GB RAM. DnCNN and N2N are deep learning models

  •) ∼ P(•/ζ) represents the signal dependent noisy component with ζ ≥ 0 being the gain of the overall image acquisition system. w ∼ N (b, σ 2 ) represents the signal independent additive noise component with mean b and variance σ 2 . σ, ζ ≥ 0 determines the intensity of Poisson-

Fig 1 :

 1 Fig 1: Top: Noisy realizations. Middle: L ∞ computed over them and bottom: Proposed denoising results.

Fig 2 :

 2 Fig 2: Result over synthetic data corrupted with MPG noise. Top row: σ, ζ =10 -2.5 ,10 -3 Middle row: σ, ζ =10 -1 ,10 -2 and Bottom row: σ, ζ =10 -2 ,10 -1.5 .

Fig 3 :

 3 Fig 3: Result over FMD dataset: single-channel (gray) image of zebrafish embryo under confocal microscopy

Fig 4 :Fig 5 :

 45 Fig 4: Result over FMD dataset: single-channel (gray) image of mice brain under twophoton microscopy

  results of denoising are shown in fig.3, fig.4and fig.5respectively. The ground truth images shown in these figures are obtained (as described by the authors) image averaging over 50 realizations of the noisy image. In fig.3the denoising results of a single channel (gray) image of zebrafish embryo captured under confocal microscopy is shown. The image is of size 512×512 pixels. It can be observed from the zoomed section that the details of the embryo are best preserved in the output of the proposed method. Noise2Noise and VST+BM3D also tries to retain the details to a certain extent, but are not that accurate in comparison to the output of the proposed method. PG-URE and GAT+BM3D gives a blurred impression whereas SURE-LET and DnCNN gives oversmoothed results. Fig.4shows the denoising results of a single channel (gray) image of mice brain captured under two photon microscopy. The image is of size 512 × 512 pixels. Careful observation shows that the proposed method better reconstructs the details and sharpness of the white spots as well as the blob like structures inside the cells, than the Noise2Noise model which is slightly smoothened in its output. DnCNN, VST+BM3D and PG-URE also gives better denoising but has smoothened outputs compared to the proposed model and Noise2Noise. The rest gives over moothened outputs.Quantitative evaluation of fig.3and fig.4, as shown in table 3, also justifies the superior denoising results of the proposed method.

Fig. 5

 5 Fig.5shows the denoising results over a multi channel (color) image of bovine pulmonary

4 Conclusion

 4 This paper presents a denoising technique for recovering low resolution fluorescence microscopy samples. The denoising technique is motivated from the concept of multifractal feature extraction from an observed noisy sample, and subsequent reconstruction from the gradient information limited to those features. The feature extraction method records only the most informative pixels and drops the remaining pixels. A denoised image is then reconstructed from the gradients of this feature set by solving a half-quadratic based optimization problem. Experiments were performed on both synthetic image databases and real fluorescence microscopy data. Results, both qualitative and quantitative, have proved the superior denoising potential of the proposed technique over multiple competing techniques in the field.

Table 1 :

 1

			45 22.41 20.48 22.82 22.57 21.48 21.98 24.67
	BSD68	10 -2.5 10 -3 25.37 31.23 28.25 30.22 29.77 28.35 28.85 33.89 10 -1 10 -2 17.39 23.27 22.61 23.64 23.78 22.37 22.13 24.87 10 -2 10 -1.5 13.52 21.53 19.50 21.84 21.53 20.63 20.23 23.07
		Param	Mean SSIM
	Classic5	10 -2.5 10 -3 0.751 0.929 0.916 0.870 0.865 0.854 0.856 0.942 10 -1 10 -2 0.597 0.792 0.711 0.756 0.772 0.714 0.710 0.817 10 -2 10 -1.5 0.417 0.657 0.622 0.644 0.653 0.620 0.624 0.688
	Set12	10 -2.5 10 -3 0.768 0.934 0.924 0.918 0.905 0.893 0.896 0.954 10 -1 10 -2 0.588 0.784 0.695 0.771 0.770 0.707 0.701 0.811 10 -2 10 -1.5 0.437 0.662 0.547 0.654 0.645 0.631 0.635 0.694
	BSD68	10 -2.5 10 -3 0.748 0.929 0.891 0.905 0.911 0.898 0.904 0.943 10 -1 10 -2 0.568 0.765 0.700 0.755 0.748 0.702 0.701 0.798 10 -2 10 -1.5 0.444 0.654 0.588 0.648 0.632 0.611 0.621 0.688

Quantitative evaluation using PSNR (refer equation

(11)

) and SSIM (refer equation (

12

)) metric over standard image dataset. Best cases are highlighted in bold. Param Mean PSNR σ ζ Input PG-URE SURE VST+ GAT+ Self2Self Nb2Nb Proposed -LET BM3D BM3D Classic5 10 -2.5 10 -3 24.56 33.33 32.90 32.19 32.01 30.20 30.02 34.91 10 -1 10 -2 18.48 25.04 23.84 23.90 25.32 24.78 24.14 26.84 10 -2 10 -1.5 13.88 22.824 21.96 20.73 21.53 20.19 20.87 24.90 Set12 10 -2.5 10 -3 25.99 33.92 31.09 30.79 30.42 29.85 29.64 35.46 10 -1 10 -2 18.80 25.16 24.14 25.18 25.38 24.64 24.48 26.45 10 -2 10 -1.5 13.

Table 2 :

 2 Quantitative evaluation using PSNR (refer equation(11)) and SSIM (refer equation (12)) metric on FMD datasets. Best cases are highlighted in bold.

	Sample	Mean PSNR
		Raw PG-URE SURE VST+ GAT+ DnCNN N2N Proposed
		-LET BM3D BM3D
	Confocal Ze-	22.76 30.63 27.75 31.46 30.86 32.24 33.04 35.66
	brafish
	Twophoton	22.37 33.46 28.38 34.25 33.22 34.58 34.91 36.06
	Mice Brain
	Widefield	20.06 27.05 28.05 27.73 27.78 28.76 29.69 29.11
	BPAE
	Sample	Mean SSIM
	Confocal Ze-	0.427 0.834 0.856 0.883 0.888 0.898 0.911 0.947
	brafish
	Twophoton	0.278 0.907 0.798 0.915 0.908 0.922 0.924 0.949
	Mice Brain
	Widefield	0.759 0.885 0.927 0.892 0.911 0.914 0.968 0.944
	BPAE

Table 3 :

 3 Running time comparison ultilize the GPU, for which they are faster but computationally expensive in terms of resource utilization. Algorithm 1 always achieves real-time. Half quadratic splitting depends of the size of the input image. For a 10000 × 10000 pixel image, GPU implentation of Half quadratic splitting can be reduced down to 1 second cpu time on a high level desktop (we tested it on Nvidia Quadro NVS 290 with Cuda).

	Sample			Running Time (in seconds)
				CPU Running Time			GPU Running Time
		PG-URE SURE-LET VST+BM3D GAT+BM3D Proposed DnCNN	N2N
	Widefield	16.56	4.54	8.23	4.15	3.9	0.19	0.08
	BPAE

and

Disclosures

The authors have no relevant financial interests in the manuscript and no other potential conflicts of interest to disclose.

Code, Data, and Materials Availability

The datasets analysed during the current study are available in the CBSD68 repository, https: //paperswithcode.com/dataset/cbsd68 and FMD repository. [START_REF] Zhang | A poisson-gaussian denoising dataset with real fluorescence microscopy images[END_REF] using approaches from statistical physics; he is also studying sparse signal representations and filtering with specific application to signals in astrophysics in collaboration with the Astrophysics Laboratory of Bordeaux (analysis of turbulent properties in the interstellar medium). H.Yahia is involved in many national, European, and international contracts and has been supervising over 15 PhD. students. He has authored or coauthored more than 100 publications in international peer-reviewed top journals and conferences.

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