SAR-to-optical translator networks are especially used to overcome the lack of optical images under cloudy conditions. Those translations being used for downstream tasks, they require the reconstruction of reliable patterns with respect to the underlying objects. In this paper, we propose a novel training strategy to account for land-cover complexity through a conjoint Translation-Classification Loss (TCL). The proposed loss evaluates the classifiability of translated images with a pre-trained land-cover classifier by assessing the reliability of its predictions and the relevance of its extracted hidden features. This new loss is applied to nine translators from the literature and to a tenth architecture introduced in the paper. Experiments show that applying the TCL not only improves the credibility of structures, patterns and textures but it also allows for better class discrimination and transitions while avoiding unreliable hallucinated artifacts produced by standard losses in adversarial approaches.

Detecting slope instabilities on Synthetic Aperture Radar (SAR) interferograms using deep learning approaches presents several challenges. This detection task suffers from the lack of transparency of deep networks, the complexity of the input data (i.e. complex values, sensitivity to distortions and presence of counterfactuals) and the complexity of the target phenomena (i.e. the variable velocities and the complex underground processes). In this paper, we propose a new framework called ”Explainability Constrained-claSsifier for Pairing the detection and the Localization of moving Areas on INterferograms” (ECSPLAIN), to generate decision, localization and segmentation maps from a single but explainable classifier network. It consists of training a classifier to detect whether an instability is located in the patch or not, and to explain its decision with a Class Activation Map (CAM) that matches the actual location of the instability. Therefore by using a single classifier network, the framework can pair the detection and the localization of moving areas. Four CAMs are investigated for the training of the ECSPLAIN framework. Experiments on the ISSLIDE dataset show that our proposal achieves better explainability than standarda posterioriCAMs with more than 0.20 points of improvements in terms of Dice and IoU scores. It also allows competitive performance with segmentation-only networks with only 0.04 points of difference in terms of Dice and IoU scores. Thus, the proposed method is competitive with the most efficient methods while being lighter, faster, and delivering a decision based on a human-like reasoning process. Finally, the ECSPLAIN framework is applied to enrich the ISSLIDE dataset, discovering more than 470 manually validated slope instabilities over the Alps.

Snow is a vital environmental parameter that holds significance across various disciplines, such as hydrology, meteorology, and natural disaster management. With the increasing accessibility of snow products derived from Synthetic Aperture Radar (SAR) and optical data, like Sentinel-1 wet snow and Sentinel-2 total snow, users have benefited from improved snow mapping and monitoring. However, snow mapping in the mountainous areas remains challenging due to the difficulty of obtaining reliable ground truth data on steep mountain terrain. In this study, we introduce a deep semantic segmentation framework, SACUNet, specifically designed for wet snow detection from SAR image time series in mountainous environments. To address the lack of ground truth, we constructed a high-confidence training and validation database through a rigorous decision-fusion process combining multi-temporal Sentinel-1 wet snow detections with Sentinel-2 total snow maps. We also propose two complementary metrics, the Conditional Agreement Rate (CAR) and the Wet Snow Intersection over Union (WSIoU), to quantify the robustness and consistency of the fusion procedure, therefore ensuring the reliability of training labels in the absence of in-situ data. SACUNet integrates advanced techniques like: (i) Depthwise Separable Convolution, which captures cross-channel dependencies and adapts feature representations, and (ii) Atrous Separable Convolution, which further refines and consolidates the learned features, into the U-Net architecture. The proposed framework has been successfully employed to monitor wet snow in the Mont-Blanc massif, using a time series of 69 Sentinel-1 images acquired from 05 July 2020, to 29 August 2021. SACUNet demonstrates remarkable accuracy in wet snow detection, with an Overall Accuracy of 97%, Precision of 94%, Recall of 97%, Intersection over Union at 92%, and an F1-Score reaching 96%. Validation against meteorological records from four alpine stations confirmed that SACUNet effectively tracks seasonal wet snow dynamics, suppresses false detections during cold periods, and captures realistic high-altitude melt events. Moreover, the model trained in Mont-Blanc generalized successfully to the Vanoise massif, demonstrating its transferability to other alpine regions. Beyond quantitative accuracy, SACUNet enables the spatio-temporal analysis of wet snow evolution, offering insights into its extent, frequency, and seasonal progression across elevation bands. These findings highlight the framework’s potential as an operational tool for large-scale wet snow monitoring in mountainous environments.

We consider the problem of classifying a pair of Synthetic Aperture Radar (SAR) images by proposing an explainable and frugal algorithm that integrates a set of divergences. The approach relies on a statistical framework that takes standard probability distributions into account for modelling SAR data. Then, by learning a combination of parameterized Renyi divergences and their parameters from the data, we are able to classify the pair of images with fewer parameters than regular machine learning approaches while also allowing an interpretation of the results related to the priors used. Experiments on real multi-class data demonstrate the virtues of the suggested method when compared to both Random Forest and Convolutional Neural Networks (CNN) classifiers, showing its resilience to disturbances such as polluted labels and variations in the percentage of training data.

SAR-to-optical remote sensing translator neural networks are mostly trained on flat areas, avoiding SAR geometrical distortion issues in steeply sloped areas. Their degraded performance under such topology severely limits the ability to detect disasters such as landslides in cloud covered areas. In this paper, we first propose a new SAR-DEM-optical dataset in mountainous regions to improve the performance of SAR-to-optical image translators under these extreme conditions. Then we upgrade SARDINet (SAR Distorted Image translator Network) model previously developed for urban areas, to take a Digital Elevation Model (DEM) together with the SAR image as input and perform translation in a natural mountain environment. Several fusion strategies are explored to efficiently merge SAR and DEM images: late fusion, early fusion and an intermediate fusion based on balanced separable convolutions. These approaches show improvements in distorted regions compared to the original SARDINet and two standard adversarial networks - Pix2pix and CycleGAN.

Due to the high data demand of machine learning algorithms, multiple datasets are emerging in remote sensing. But these datasets are costly and time consuming to annotate especially for change detection or natural phenomena monitoring. In particular, early warning systems on slow-moving disasters are lacking of training datasets as they require both geomorphological and SAR interferometry expertise. In this paper, (i) we propose a novel InSAR dataset for Slow SLIding area DEtection (ISSLIDE) with machine learning algorithms. The latter consists of manually annotated patches of generated interferograms over slow moving areas. (ii) We implement the segmentation of ISSLIDE interferograms with classical deep learning approaches. FCN, DeepLabV3 and U-Net-like architectures are explored to serve as baseline for future works. To the best of our knowledge, this is the first dataset adapted to machine learning and targeting slow sliding area detection.

This article proposes a new learning paradigm based on the concept of concordant gradients for ensemble learning strategies. In this paradigm, learners update their weights if and only if the gradients of their cost functions are mutually concordant in a sense given by paper. The objective of the proposed concordant optimization framework is robustness against uncertainties by postponing to a later epoch, the consideration of examples associated with discordant directions during a training phase. Concordance constrained collaboration is shown to be relevant, especially in intricate classification issues where exclusive class labeling involves information bias due to correlated disturbances affecting almost all training examples. The first learning paradigm applies on a gradient descent strategy based on allied agents, subjected to concordance checking before moving forward in training epochs. The second learning paradigm is related to multivariate dense neural matrix fusion, where the fusion operator is itself a learnable neural operator. In addition to these paradigms, this article proposes a new categorical probability transform to enrich the existing collection and propose an alternative scenario for integrating penalized SoftMax information. Finally, this article assesses the relevance of the above contributions with respect to several deep learning frameworks and a collaborative classification involving dependent classes.

The article proposes a plural learning framework combining the ingredients found in a tribunal for the derivation of a more generalized artificial intelligence (GAI) when starting from a specialized set of convolutional neural networks (CNNs). This framework involves at least two different training stages called, respectively, specialization and generalization. In the specialization stage, any CNN considered in a given set learns to predict independently of other elements of the set. In the second stage called generalization, an integration network learns to predict from assessment measures fed by downstream specialized CNNs. The assessment measures considered are categorical softmax probabilities and learning to judge from these assessments relies on independent CNNs. Generalization proof of concepts is provided in terms of multimodel, multimodal, and distributed schemes. The multimodel framework is such that different CNN models operating on the same modality cooperate for decision purpose. The multimodal framework implies specializations of CNN with respect to different input modalities. The distributed framework proposed is associated with assessment exchanges: it such that the aggregation aims at determining relevant joint assessments for mapping a given input to a single or a multiple output category. The performance of these aggregation frameworks is shown to be outstanding for both standard and extreme classification issues.

We propose a new approach for wet snow extent mapping in Synthetic Aperture Radar (SAR) images by using a convolutional neural network (CNN) designed to learn with respect to snowpack outputs from the state-of-the-art snow model Crocus. The CNN was trained to classify the wet snow conditions based on features extracted from the SAR images, using both the VV,VH channel and the ratio between these channels and those of a reference image in summer. One of the key points of this work is the comprehensive comparison we have made between the performance of the CNN method and other advanced statistical methods. We found that the CNN was able to achieve good accuracy in wet snow classification, and giving a complementary vision of the solutions obtained by other machine learning algorithms such as the Random Forest classifier. The results of this study demonstrate the potential of using CNNs and SAR images for wet snow classification and highlight the importance of using physical information model for training machine learning models in snow state identification, a domain where collecting ground truth is intricate due to the complexity of the snowpack moisture measurement systems.

In this paper, two SAR image time series acquired by PAZ and Sentinel-1 satellites in 2020 (29 and 60 images respectively) are used to investigate surface changes of different ice/snow-covered areas in the Mont-Blanc massif. The evolution of the backscatter coefficient and several statistical parameters in both X and C band SAR images is analyzed on ice aprons, on valley glacier accumulation and ablation areas, and on ice-free areas. Dry and wet snow changes are observed and correlated with meteorological data (temperature at 4 different elevations and snow height) acquired by a weather station.

Snow holds a significant importance as a fundamental environmental factor in multiple domains. Obtaining accurate ground truth data for snow mapping in mountainous areas presents a significant challenge. To address this issue, this paper presents a deep semantic learning framework for the segmentation of Sentinel-1 images for wet snow detection in mountainous areas. Firstly, we propose to create a deep leaning database based on snow products derived from Sentinel-1 and Sentinel-2 data. Afterward, we introduce a deep convolutional neural network called ReXcepUnet, which combines the U-Net architecture and the powerful Xception backbone. Finally, the proposed framework has been successfully applied to monitor wet snow in the Mont Blanc massif, yielding high accuracy results. The ReXcepUnet model demonstrates a good performance in wet snow detection, particularly in high-relief regions like the Mont Blanc massif.

Multimodal approaches for Earth Observations suffer from both the lack of interpretability of SAR images and the high sensitivity to meteorological conditions of optical images. Translation methods were implemented to solve them for specific tasks and areas. But these implementations lack of generalizability as they do not include samples with challenging characteristics. Firstly, this paper sums up the main problems that a general SAR to optical image translator should overcome. Then, a SAR Distorted Image to optical translator Network (SARDINet) alternating knowledgeable channel-wise spatial convolutions and cross-channel convolutions is implemented. It aims at solving a problem of major concern in remote sensing: translating layover disturbed SAR images into disturbance-free optical ones. SARDINet is trained through a classical and an adversarial framework and compared to cGAN and cycleGAN from the literature. Experimental results prove that adversarial approaches are more qualitative but worsen quantitative results.

Deep Neural Networks (DNNs) are getting increasing attention to deal with land cover classification relying on Satellite Image Time Series (SITS). Though high performances can be achieved, the rationale of a prediction yielded by a DNN often remains unclear. An architecture expressing predictions with respect to input channels is thus proposed in this paper. It relies on convolutional layers and an attention mechanism weighting the importance of each channel in the final classification decision. The correlation between channels is taken into account to set up shared kernels and lower model complexity. Experiments based on Sentinel-2 SITS show promising results.

This brief addresses understandability of modern machine learning networks with respect to the statistical properties of their convolution layers. It proposes a set of tools for categorizing a convolution layer in terms of kernel property (meanlet, differencelet, or distrotlet) or kernel sequence property (frame spectra and intralayer correlation matrix). These tools are expected to be relevant for determining the generalization capabilities of a convolutional neural network. In particular, this brief highlights that the less frequency penalizing network among AlexNet, GoogleNet, RESNET101, and VGG19 is the more relevant one in terms of solutions for low-level ice-sheet feature enhancement.

This paper aims to propose two main contributions in the field of multivariate data analysis through the possibility theory. The first proposition is the definition of a generalized family of multivariate elliptical possibility distributions. These distributions have been derived from a consistent probability-possibility transformation over the family of so-called elliptical probability distributions. The second contribution proposed by this paper is the definition of two divergence measures between possibilistic distributions. We prove that a symmetric version of the Kullback-Leibler divergence guarantees all divergence properties when related to the space of possibility distributions. We further derive analytical expressions of the latter divergence and of the Hellinger divergence for certain possibility distributions pertaining to the elliptical family proposed, especially the normal multivariate possibility divergence in two dimensions. Finally, this paper provides an illustration of the developed possibilistic tools in an application of bi-band change detection between optical satellite images.

High resolution in synthetic aperture radar (SAR) leads to new physical characterizations of scatterers which are anisotropic and dispersive. These behaviors present an interesting source of diversity for target detection schemes. Unfortunately, such characteristics have been integrated and have been naturally lost in monovariate single-look SAR images. Modeling this behavior as nonstationarity, wavelet analysis has been successful in retrieving this information. However, the sharp-edge of the used wavelet functions introduces undesired high side-lobes for the strong scatterers present in the images. In this paper, a new family of parameterized wavelets, designed specifically to reduce those side lobes in the SAR image decomposition, is proposed. Target detection schemes are then explored using this spectro-angular diversity and it can be shown that in high-resolution SAR images, the non-Gaussian and robust framework leads to better results.

A time series issued from modern synthetic aperture radar satellite imaging sensors is a huge dataset composed by many hundreds of million pixels when observing large-scale earth structures such as big forests or glaciers. A concise monitoring of these large scale structures for anomaly spotting thus requires loading and analyzing huge spatio/polarimetric multi-temporal image series. The contributions of the present paper for the sake of parsimonious analysis of such huge datasets are associated with a framework having two main processing stages. The first stage is the derivation of an index called geometric multi-wavelet total variation for fast and robust anomaly spotting. This index is useful for identifying significant abnormal patterns appearing as geo-spatial non-stationarities in multi-wavelet total variation map. The second stage consists in the proposal of a concise asymmetric multi-date change information matrix on regions associated with significant multi-wavelet total variations. This stage is necessary for a fine characterization of change impacts on existing geo-spatial structures. Experimental tests based on Sentinel-1 data show relevant results on a wide Amazonian forest surrounding the Franco-Brazilian Oyapock Bridge.

In this paper, we propose new detectors for Change Detection between two multivariate images. The data is supposed to fol-Iowa Compound Gaussian distribution. By using Likelihood Ratio Test (LRT) and Generalised LRT (GLRT) approaches, we derive our detectors. The CFAR behaviour has been studied and the simulations show that they outperform the classic Gaussian Detector when the data is highly heterogeneous.

In this paper we present a new similarity measure between possibility distributions based on the Kullback-Leibler (KL) divergence in the domain of real numbers. The possibility distributions are obtained thanks to the DFMP probability-possibility transformation [1] lying on the principle that a possibility measure can encode a family of probability measures. We consider here two particular possibility distributions built from parameter estimation of the Weibull and Rayleigh probability laws. The analytical expression of the KL divergence for the two considered possibility distributions are given, allowing a simple computation which depends on the parameters of the possibility distribution obtained. This new similarity measure is compared to the existing KL divergence for probability distributions in a context of change detection over simulated images as they provide a ground-truth of the changes required to evaluate the rate of true detection against false alarm.

Regularized Tyler Estimator's (RTE) have raised attention over the past years due to their attractive performance over a wide range of noise distributions and their natural robustness to outliers. Developing adaptive methods for the selection of the regularisation parameter α is currently an active topic of research. Indeed, the bias-performance compromise of RTEs highly depends on the considered application. Thus, finding a generic rule that is optimal for every criterion and/or data configurations is not straightforward. This issue is addressed in this paper for undersampled configurations (number of samples lower than the dimension of the data). The paper proposes a new regularisation parameter selection based on a subspace reduction approach. The performance of this method is investigated in terms of estimation accuracy and for adaptive detection purposes, both on simulation and real data.

Usually, in radar imaging, the scatterers are supposed to respond the same way regardless of the angle from which they are viewed and have the same properties within the emitted spectral bandwidth. Nevertheless, new capacities in SAR imaging (large bandwidth, large angular extent) make this assumption obsolete. An original application of the Linear Time-Frequency Distributions (LTFD) in SAR imaging allows to highlight the spectral and angular diversities of these reflectors. This methodology allows to transform a monovariate SAR image onto multivariate SAR image. Robust detection schemes in Gaussian or non-Gaussian background (Adaptive Matched Filter (AMF), Adaptive Normalized Matched Filter (ANMF), Anomaly Kelly Detector) associated with classical or robust Covariance Matrix Estimates (Sample Covariance Matrix (SCM), M-estimators) can then be applied exploiting these diversities. The combined two-methodologies show their very good performance for target detection.

This paper first provides statistical properties of wavelet operators when the observation model can be seen as the product of a deterministic piecewise regular function (signal) and a stationary random field (noise). This multiplicative observation model is analyzed in two standard frameworks by considering either: 1) a direct wavelet transform of the model; or 2) a log-transform of the model prior to wavelet decomposition. The paper shows that, in Framework 1, wavelet coefficients of the time series are affected by intricate correlation structures which blur signal singularities. Framework 2 is shown to be associated with a multiplicative (or geometric) wavelet transform, and the multiplicative interactions between wavelets and the model highlight both sparsity of signal changes near singularities (dominant coefficients) and decorrelation of speckle wavelet coefficients. This paper then derives that, for time series of synthetic aperture radar data, geometric wavelets represent a more intuitive and relevant framework for the analysis of smooth earth fields observed in the presence of speckle. From this analysis, this paper proposes a fast-and-concise geometric-wavelet-based method for joint change detection and regularization of synthetic aperture radar image time series. In this method, geometric wavelet details are first computed with respect to the temporal axis in order to derive generalized-ratio change images from the time series. The changes are then enhanced, and speckle is attenuated by using spatial block sigmoid shrinkage. Finally, a regularized time series is reconstructed from the sigmoid shrunken change images. Some applications highlight relevancy of the method for the analysis of SENTINEL-1A and TerraSAR-X image time series over Chamonix Mont Blanc.

This paper presents a new model called Autoregressive Fractional Brownian Field (ARFBF) for analyzing textures which contain stationary and non-stationary components. The paper also proposes two estimation methods for the parameter of an isotropic fractional Brownian field based on Wavelet Packet (WP) spectrum: the Log-Regression on Diagonal WP spectrum (Log-RDWP) and the Log-Regression on Polar representation of WP spectrum (Log-RPWP). The Log-RPWP method provides a better estimation performance for small size images. We show the interest of ARFBF model and Log-RPWP for characterizing High-Resolution Transmission Electron Microscopy (HRTEM) images.

This paper presents a method for analyzing SAR image time series and provides initial change detection results on a time series of 11 descending Interferometric Wide Swath (IW) Level-1 Single Look Complex (SLC) Sentinel-1 SAR images over Chamonix-Mont-Blanc, France. This method is based on the Change Detection Matrix (CDM) which identifies the presence of changes in the time series. It provides a useful information to gather homogeneous samples for spatio-temporal speckle filtering and to obtain a map of change dynamics in order to reveal the temporal evolution.

The paper addresses random field time series analysis and simulation. The analysis constrains a spatial isotropic fractional Brownian field to a dynamic temporal behavior from separable time varying Hurst parameters. The constrained dynamic applies by embedding the wavelet packet spectrum of the input random field into different spectra associated with the same random family (exponential spectrum decay). The paper highlights the relevance of the approach for representing and simulating isotropic light source and cloud dynamics.

This paper presents a generalized non-stationary and fractional model for texture synthesis. The model is based on convolution and modulation operations of fractional Brownian fields and its associated spectral representation contains many poles with unit norm. Synthesized textures generated from this model can exhibit several non-trivial fringes which can be visualized in natural textures such those involved in high resolution transmission electron microscopy.

This paper proposes an approach for temporal adaptive filtering of Polarimetric Synthetic Aperture Radar (PolSAR) image time series by integrating a change detection technique. The filtering strategy is based on the detection of changed and unchanged areas derived by applying an appropriate similarity test. A time series including 7 descending fine-quad polarization RADARSAT2 images acquired from January 29, 2009 to Jun 22, 2009 over Chamonix-MontBlanc test-site which includes different kinds of change is used to validate the proposed method.

This letter presents an adaptive filtering approach of synthetic aperture radar (SAR) image times series based on the analysis of the temporal evolution. First, change detection matrices (CDMs) containing information on changed and unchanged pixels are constructed for each spatial position over the time series by implementing coefficient of variation (CV) cross tests. Afterward, the CDM provides for each pixel in each image an adaptive spatiotemporal neighborhood, which is used to derive the filtered value. The proposed approach is illustrated on a time series of 25 ascending TerraSAR-X images acquired from November 6, 2009 to September 25, 2011 over the Chamonix-Mont-Blanc test-site, which includes different kinds of change, such as parking occupation, glacier surface evolution, etc.

The paper provides a spatio-temporal change detection framework for the analysis of image time series. In this framework, the detection of changes in time is addressed at the image level by using a matrix of cross-dissimilarities computed upon wavelet and curvelet image features. This makes possible identifying the acquisitions of interest: the acquisitions that exhibit singular behavior with respect to their neighborhood in the time series, and those that are representatives of some stationary behavior. These acquisitions of interest are compared at the pixel level to detect spatial changes characterizing the evolution of the time series. Experiments carried out over European Remote Sensing (ERS) and TerraSAR-X time series highlight the relevancy of the approach for analyzing synthetic aperture radar image time series.

This brief derives a 2-D spectrum estimator from some recent results on the statistical properties of wavelet packet coefficients of random processes. It provides an analysis of the bias of this estimator with respect to the wavelet order. This brief also discusses the performance of this wavelet-based estimator, in comparison with the conventional 2-D Fourier-based spectrum estimator on texture analysis and content-based image retrieval. It highlights the effectiveness of the wavelet-based spectrum estimation.

The paper addresses multi-channel complex image filtering. It provides regularization cost functions associated to non-conventional vector and matrix iv norms for promoting geometry properties. The approach is shown to be efficient for filtering PolSAR images.

The paper addresses the analysis and interpretation of second order random processes by using the wavelet packet transform. It is shown that statistical properties of the wavelet packet coefficients are specific to the filtering sequences characterizing wavelet packet paths. These statistical properties also depend on the wavelet order and the form of the cumulants of the input random process. The analysis performed points out the wavelet packet paths for which stationarization, decorrelation and higher order dependency reduction are effective among the coefficients associated with these paths. This analysis also highlights the presence of singular wavelet packet paths: the paths such that stationarization does not occur and those for which dependency reduction is not expected through successive decompositions. The focus of the paper is on understanding the role played by the parameters that govern stationarization and dependency reduction in the wavelet packet domain. This is addressed with respect to semi-analytical cumulant expansions for modeling different types of nonstatonarity and correlation structures. The characterization obtained eases the interpretation of random signals and time series with respect to the statistical properties of their coefficients on the different wavelet packet paths.

The paper addresses content-based image retrieval from texture data bases, by using stochastic modeling in the wavelet domain. It pro poses an analysis of the key parameters involved in such a content based texture retrieval. These parameters are the wavelet order and the goodness-of-fit measure used to select the best family of distributions for modeling the subband wavelet coefficients. It is shown that taking suitable parameters into consideration makes it possible to attain high retrieval rates in content-based texture retrieval.

This paper provides asymptotic properties of the autocorrelation functions of the wavelet packet coefficients of a fractional Brownian motion. It also discusses the convergence speed to the limit autocorrelation function, when the input random process is either a fractional Brownian motion or a wide-sense stationary second-order random process. The analysis concerns some families of wavelet paraunitary filters that converge almost everywhere to the Shannon paraunitary filters. From this analysis, we derive wavelet packet based spectrum estimation for fractional Brownian motions and wide-sense stationary random processes. Experimental tests show good results for estimating the spectrum of1/fprocesses.

The paper presents a general framework for change detection in radar images, for an operational purpose and in the context of environmental monitoring. This framework is based on a processing which provides highly sparsifiable representations of data. This processing is called turbo-median and is a combination of the sample median robustness and the turbo principle for iteratively correcting errors. The turbo-median processing of a scene is an homogenized representation based on an iterative median and which consists in spreading the statistically more robust measurements of the scene under consideration over the size of the image representing this scene. It allows for reducing the change detection problem into the problem of detecting a signal, with unknown distribution, in additive noise.

This paper presents a new sigmoid-based wave shrink function. The shrinkage obtained via this function is particularly suitable to reduce noise without impacting significantly the statistical properties of the signal to be recovered. The proposed WaveShrink function depends on a parameter that makes it possible to control the attenuation degree imposed to the data, and thus, allows for a flexible shrinkage.