Weiying Zhao

Charles-Alban Deledalle

Loïc Denis

Henri Maître

Jean-Marie Nicolas

J.-M Nicolas

F Tupin

Ltci

Ratio-based Multi-temporal SAR Images Denoising: RABASAR

Keywords: Multi-temporal SAR series, ratio image, superimage, SAR, speckle reduction

In this paper, we propose a fast and efficient multitemporal despeckling method. The key idea of the proposed approach is the use of the ratio image, provided by the ratio between an image and the temporal mean of the stack. This ratio image is easier to denoise than a single image thanks to its improved stationarity. Besides, temporally stable thin structures are well preserved thanks to the multi-temporal mean.

The proposed approach can be divided into three steps: 1) estimation of a "super-image" by temporal averaging and possibly spatial denoising; 2) denoising of the ratio between the noisy image of interest and the "super-image"; 3) computation of the denoised image by re-multiplying the denoised ratio by the "super-image".

Because of the improved spatial stationarity of the ratio images, denoising these ratio images with a speckle-reduction method is more effective than denoising images from the original multi-temporal stack. The amount of data that is jointly processed is also reduced compared to other methods through the use of the "super-image" that sums up the temporal stack. The comparison with several state-of-the-art reference methods shows better results numerically (peak signal-noise-ratio, structure similarity index) as well as visually on simulated and SAR time series. The proposed ratio-based denoising framework successfully extends single-image SAR denoising methods to time series by exploiting the persistence of many geometrical structures.

I. INTRODUCTION

S YNTHETIC aperture radar (SAR) imaging is widely used in the monitoring of land surfaces, disasters or the environment, due to its all-time acquisition capability, its sensitivity to geometric structures, its penetration characteristics, etc. However, the system-inherent speckle noise visually corrupts the appearance of images and severely diminishes the analysis and interpretation of SAR images. Therefore, a preliminary speckle reduction step is often necessary for the successful exploitation of SAR images.

The recent and unprecedented availability of long time series with Sentinel-1 constellation has opened new ways for SAR speckle reduction. This highly redundant information offers a new paradigm. Taking inspiration from temporal multi-looking and residual noise analysis we propose in this paper a ratio-based denoising approach (first presented in [START_REF] Zhao | RABASAR: A fast ratio based multi-temporal SAR despeckling[END_REF]), dedicated to long multi-temporal stacks.

The main problems of SAR image despeckling are the spatial resolution preservation, the edges and textures restoration, and the preservation of point-like targets. Spatial multilooking is a common way to reduce speckle fluctuations in a single SAR image, at the cost of a significant spatial resolution loss [START_REF] Argenti | A tutorial on speckle reduction in Synthetic Aperture Radar images[END_REF]. To effectively estimate the noise-free reflectivity as well as preserve the spatial resolution, many single-channel SAR speckle reduction methods have been proposed during the past decades. Detailed introductions of the methods are given by Touzi et al. [START_REF] Touzi | A review of speckle filtering in the context of estimation theory[END_REF], Argenti et al. [START_REF] Argenti | A tutorial on speckle reduction in Synthetic Aperture Radar images[END_REF] and Deledalle et al. [START_REF] Deledalle | Exploiting patch similarity for SAR image processing: the nonlocal paradigm[END_REF]. These methods mainly belong to four categories: Bayesian methods in the spatial domain (Lee filter [START_REF] Lee | Speckle analysis and smoothing of Synthetic Aperture Radar images[END_REF], Lee refined filter [START_REF]Refined filtering of image noise using local statistics[END_REF]), Bayesian methods in a transformed domain [START_REF] Meer | Multiresolution adaptive image smoothing[END_REF], selection-based filtering (IDAN [START_REF] Vasile | Intensity-driven adaptive-neighborhood technique for polarimetric and interferometric SAR parameters estimation[END_REF], PPB [START_REF] Deledalle | Iterative weighted maximum likelihood denoising with probabilistic patch-based weights[END_REF] and NL-SAR [START_REF] Deledalle | NL-SAR: A unified nonlocal framework for resolution-preserving (Pol)(In) SAR denoising[END_REF]) and sparse-based approaches [START_REF] Foucher | SAR image filtering via learned dictionaries and sparse representations[END_REF]. Many single-SAR-image denoising methods are designed by combining information in different domains (the spatial domain and a transformed domain), with different estimation criteria and various statistical models of speckle and radar reflectivities [START_REF] Argenti | A tutorial on speckle reduction in Synthetic Aperture Radar images[END_REF]. Several state-of-the-art single-image despeckling methods perform a weighted average of selected surrounding pixels values to estimate the speckle-free values. If the surrounding pixels are not well selected or if their associated weights are badly chosen, an estimation bias occurs which is generally visible in the form of a spatial resolution loss (over-smoothing and spreading of thin structures). The accurate selection of pixels is very challenging in single-look SAR images given the large fluctuations due to speckle phenomenon. Even when using state-of-the-art spatial denoising approaches (such as SAR-BM3D [START_REF] Parrilli | A nonlocal SAR image denoising algorithm based on LLMMSE wavelet shrinkage[END_REF] or NL-SAR [START_REF] Deledalle | NL-SAR: A unified nonlocal framework for resolution-preserving (Pol)(In) SAR denoising[END_REF]), the smallest and least contrasted structures can be damaged or some noticeable speckle fluctuations can remain after filtering.

Recently, convolutional neural networks have shown a high capability of denoising data affected by additive white Gaussian noise (AWGN) [START_REF] Zhang | Beyond a gaussian denoiser: Residual learning of deep CNN for image denoising[END_REF]. Application to SAR images has also been proposed, either through an homomorphic approach [START_REF] Chierchia | SAR image despeckling through convolutional neural networks[END_REF] or directly applying gamma distribution based methods [START_REF] Wang | SAR image despeckling using a convolutional neural network[END_REF]. Unlike traditional SAR image denoising approaches, these methods predict the noise-free value through the estimation of the speckle component. These recent techniques reach comparable despeckling results w.r.t. state-of-the-art approaches, both in terms of signal to noise ratio (simulated data) and in terms of spatial feature preservation. However, the training of these networks is time consuming and requires to building a training set with pairs of speckle-free / speckled SAR images.

With the launch of the latest generation of SAR satellites (Cosmo-SkyMed, TerraSAR-X, ALOS-2, Sentinel-1, etc.) [START_REF] Moreira | A tutorial on synthetic aperture radar[END_REF], more and more SAR images, with shorter revisit time or higher resolution, are now available. With multi-temporal images, both spatial and temporal information can be combined in the denoising process, which gives the potential of reaching better speckle reduction results than is achievable when processing a single image. This is the path followed by several kinds of multi-temporal denoising methods, proposing temporal weighted average [START_REF] Lee | Speckle reduction in multipolarization, multifrequency SAR imagery[END_REF], [START_REF] Quegan | Multitemporal ERS SAR analysis applied to forest mapping[END_REF], [START_REF] Quegan | Filtering of multichannel SAR images[END_REF], temporal weighted average in a transformed domain (M-TSF [START_REF] Coltuc | Time-space filtering of multitemporal SAR images[END_REF] and MSAR-BM3D [START_REF] Chierchia | Multitemporal SAR image despeckling based on block-matching and collaborative filtering[END_REF]), change-detection-aware multi-temporal average (Lê's method [START_REF] Lê | Adaptive multitemporal SAR image filtering based on the change detection matrix[END_REF], [START_REF] Lê | Change detection matrix for multitemporal filtering and change analysis of SAR and PolSAR image time series[END_REF] and 2SPPB [START_REF] Su | Two-step multitemporal nonlocal means for Synthetic Aperture Radar images[END_REF]), or filtering using three-dimensional adaptive neighborhoods [START_REF] Ciuc | Adaptiveneighborhood speckle removal in multitemporal Synthetic Aperture Radar images[END_REF], [START_REF] Lobry | Multitemporal SAR image decomposition into strong scatterers, background, and speckle[END_REF]. A drawback of such approaches is the increased computational complexity with longer time series.

Multi-temporal denoising methods take advantage of the increasing availability of SAR image time-series to solve the spatial denoising problems, for the benefit of a better spatial resolution preservation. Most of multi-temporal denoising methods process the whole time series in order to produce a denoised image at a given date. In this paper, we take a different approach by forming a summary of the multitemporal series (which we call a "super-image", the speckle in that image being strongly reduced), and by using only this "super-image" in addition to the speckle-corrupted image (rather than the whole time series) to obtain a denoised image at any given date.

This ratio-based multi-temporal denoising method fully exploits the significant information of the multi-temporal stack through the "super-image". After forming the ratio image between the noisy image and the "super-image", the proposed method takes advantage of the state-of-the-art single-image speckle reduction methods to denoise the ratio image [START_REF] Zhao | RABASAR: A fast ratio based multi-temporal SAR despeckling[END_REF]. We consider multi-temporal images acquired on the same orbit (i.e., either all ascending orbits, or all descending orbits), with similar incidence angles, and which have been accurately registered [START_REF] Nicolas | A first comparison of Cosmo-Skymed and TerraSAR-X data over Chamonix Mont-Blanc test-site[END_REF]. For the sake of reproducible research, we have released an open-source code at https://www.math.u-bordeaux. fr/ ∼ cdeledal/rabasar.php.

The contributions of this paper are the following:

• we introduce a generic ratio-based multi-temporal denoising framework, called RABASAR, to process a single SAR image using a multi-temporal stack; • we design a denoiser, called RuLoG, adapted to the statistical distribution of the ratio image formed between a speckle-corrupted image and a "super-image" with reduced speckle, by extending the MuLoG method proposed in [START_REF] Deledalle | MuLoG, or how to apply Gaussian denoisers to multi-channel SAR speckle reduction?[END_REF]; • we demonstrate the interest of the proposed method on simulated and SAR time series. The remainder of this paper is organized as follows. In Sec. II, we introduce the general framework of the proposed method. Section III presents different ways to compute the "super-image". Then the filtering of the ratio image is described in Sec. IV. Experimental results are presented in Sec. V. Finally, some conclusions and perspectives are drawn in Sec. VI.

II. PRINCIPLE OF THE PROPOSED METHOD

The temporal averaging (also called temporal multi-looking) of SAR time series produces an image with reduced speckle and a preserved spatial resolution. We call such a reducedspeckle image a "super-image". In this paper, we propose to exploit this super-image to build a ratio-based denoising framework.

The proposed method is composed of three main steps, illustrated in Fig. 1: 1) In the first step, a super-image is calculated from a time series of spatially registered and radiometrically calibrated SAR images. Averaging temporal intensity samples is the simplest way to obtain an image with reduced speckle. This corresponds to the maximum likelihood estimation of the reflectivity when speckle is considered temporally decorrelated and temporal fluctuations of the reflectivity are neglected (observations at a given pixel are then independent and identically distributed). In practice, changes impacting some regions of the image may occur at some dates. Rather than averaging all dates, it can be beneficial to select only relevant dates when forming the "super-image", as done for instance in [START_REF] Su | Two-step multitemporal nonlocal means for Synthetic Aperture Radar images[END_REF], [START_REF] Lê | Change detection matrix for multitemporal filtering and change analysis of SAR and PolSAR image time series[END_REF], [START_REF] Su | NORCAMA: Change analysis in SAR time series by likelihood ratio change matrix clustering[END_REF]. Moreover, because speckle is temporally correlated (by spatially varying correlation levels), some amount of spatial denoising helps to produce a "super-image" where speckle is reduced everywhere. Computation of the super-image, denoted ûm in the sequel, will be presented in Sec. III.

2) In the second step, the super-image is used to form the ratio τ t between the image v t at time t and the super-image, at each spatial location s:

τ t (s) = v t (s) ûm (s) (1)
We refer to τ t as the "ratio image" at time t. It contains the residual speckle noise between the two images, and the radiometric shifts when changes occur. When the length of the time series increases and in the absence of change, the superimage ûm converges to the reflectivity u t . The ratio image τ t then tends to pure speckle (i.e., a collection of independent identically distributed gamma random variables with unitary mean and the same number of looks as the original image). In contrast, when changes occur in the time series, these changes impact the super-image which then differs from the reflectivity u t of the image at time t. Processing the ratio image τ t is necessary to correctly recover the reflectivity u t . Both the noisy image v t and the super-image ûm suffer from speckle (the speckle in the super-image, though, is strongly reduced).

The processing of the ratio image requires to account for the specific noise statistics arising from this ratio of speckled images. A denoiser dedicated to the statistics of the ratioimage is derived in Sec. IV.

3) In the third step, the filtered image is recovered by multiplying the denoised ratio image with the original superimage.

III. COMPUTATION OF THE SUPER-IMAGE

The super-image can be computed from a time series by different ways. First, different Hölder means (such as the arithmetic or the geometric means) could be chosen; then, these means may be applied either on the intensity data or on the amplitude data. We may expect from these choices to enhance different pieces of information [START_REF] Quin | MIMOSA: An automatic change detection method for SAR time series[END_REF]. In this paper, registered and radiometrically corrected intensity SAR images are used. We propose to use the arithmetic mean for its good properties [START_REF] Quin | MIMOSA: An automatic change detection method for SAR time series[END_REF], in particular in terms of modeling the statistics of the super-image, with the option of using binary weights to discard the intensity at some dates when a change occurred.

A. Statistics of SAR images

We briefly recall in this section the statistics of fully developed speckle. Under Goodman's hypothesis [START_REF] Goodman | Speckle phenomena in optics: theory and applications[END_REF], fully developed intensity speckle follows a gamma distribution G(u, L) depending on the number of looks L and the mean reflectivity u (the aftersought signal of interest) of the scene:

p v v u, L = L uΓ(L) Lv u L-1 e -Lv u (2)
Speckle in SAR images acts like a multiplicative noise and the speckle model can be expressed as [START_REF] Lee | Speckle analysis and smoothing of Synthetic Aperture Radar images[END_REF]:

v = u w (3)
where w follows a gamma distribution G(1, L), the expectation is E[v] = u and the variance is Var[v] = u 2 /L. With the increase of the number of looks L, the variance decreases. The proportionality between the variance and u 2 reflects that noise is signal-dependent and multiplicative.

B. Arithmetic mean

Given a time series of T intensity values {v t (s)} T t=1 indexed by time t, the arithmetic mean is calculated at location s by:

ûAM m (s) = 1 T T t=1 v t (s) 1 ≤ t ≤ T (4)
Theoretically, with (i) no change in the time series,

u 1 (s) = u 2 (s) = • • • = u T (s)
, and (ii) T large enough and speckle sufficiently decorrelated from one image to an other, averaging the temporal intensity data is a simple yet effective approach to reduce the speckle [START_REF] Oliver | Understanding Synthetic Aperture Radar images[END_REF]. When speckle is fully decorrelated, the arithmetic mean ûAM m (s) corresponds to the maximum likelihood estimation of u(s) and the multi-look image ûAM m follows a gamma distribution G(u, L T ). In practice, because of temporal correlations of the speckle, the resulting equivalent number of looks (ENL) may be less than this theoretical value (L T ), especially in case of images in interferometric configuration. The associated ENL will be denoted by L m in the following, and its estimation is discussed in Sec. III-D.

When changes occur in the time series, the arithmetic mean ûAM m (s) no longer matches the scene reflectivity at time t. We can reduce this discrepancy between the temporal average and the reflectivity at date t by averaging only the unchanged temporal samples, as described in the next paragraph.

C. Binary weighted arithmetic mean

Instead of computing the super-image from the time series without considering the date t of the image under study, another option is to compute a dedicated super-image that is closer to the reflectivity u t . Only samples with similar and stable reflectivities can be considered in the average. To detect these samples, we suggest using a patch-based selection based on the generalized likelihood ratio (GLR) test. The GLR test between two independent observed intensity values v 1 and v 2 (with same number of looks L) confronts the following two hypotheses: the null hypothesis corresponding to a common reflectivity value u 12 accounting for the observed intensities v 1 and v 2 , and the alternative hypothesis where a different reflectivity is considered for each observation (u 1 for v 1 and u 2 for v 2 ) to account for a change in the reflectivity between the observations. The generalized likelihood ratio (GLR) test corresponding to this hypothesis test is given by [START_REF] Deledalle | How to compare noisy patches? patch similarity beyond gaussian noise[END_REF]:

GLR(v 1 , v 2 ) = v 1 v 2 (v 1 + v 2 ) 2 , (5)
where constant terms are omitted. Taking the log and extending the comparison to the image patches centered at location s in the images at date t and t lead to [START_REF] Deledalle | Iterative weighted maximum likelihood denoising with probabilistic patch-based weights[END_REF], [START_REF] Su | Two-step multitemporal nonlocal means for Synthetic Aperture Radar images[END_REF]:

d t,t (s) = δ log v t (s + δ) v t (s + δ) + v t (s + δ) v t (s + δ) (6)
where v t (s + δ) is the value in the noisy patch at date t. The sum is taken over all pixel shifts δ such that pixels with index s + δ are located inside a patch centered on s (small square window). Then, a binary weight w t,t (s), expressing whether there are temporal changes or not, is computed as:

w t,t (s) = 1, if d t,t (s) < σ 0, otherwise (7)
where σ is a threshold chosen as σ = quantile(d t,t (s), α) under the null hypothesis and is estimated with Monte Carlo simulations using gamma distributed data and α = 0.92 as proposed in [START_REF] Su | Two-step multitemporal nonlocal means for Synthetic Aperture Radar images[END_REF]. Then, the binary weighted arithmetic mean (denoted by BWAM in the sequel) is calculated at date t by:

ûBWAMt m (s) = 1 T t =1 w t,t (s) T t =1 w t,t (s)v t (s) (8)
where the notation BWAM t is used to highlight that the superimage depends on the targeted date t. For each pixel s and each date t of the stack, the weights w t,t (s) select the dates t for which no significant change occurred with respect to the image at date t. Thus, the binary weighted mean ûBWAMt m (s) provides a more faithful estimation of u t (s). This improvement is obtained at the cost of an increased complexity since the superimage is date specific: the stack has to be processed for each date t, while a single super-image is computed once and for all with the (unweighted) arithmetic mean. The interest of using such a temporal mean will be evaluated in Sec. V.

In practice, the selection performed by the binary weights affects the associated ENL L m of the super-image. In the next paragraph, we describe how L m is estimated.

D. ENL estimation

The super-image may have a spatially varying ENL because of temporally changing areas, because of spatially-varying coherence, or because of the use of locally-computed binary weights. To robustly estimate L m in the super-image calculated with medium or high resolution temporal SAR images, the log-cumulant method [START_REF] Tison | A new statistical model for markovian classification of urban areas in high-resolution SAR images[END_REF] is used within sliding windows. Empirical expressions for the local first and second order logcumulant estimators are (for N samples):

k1 (s) = 1 N δ log ûm (s + δ) (9)
and k2 (s

) = 1 N δ [log ûm (s + δ) -k1 (s)] 2 (10)
The sums are computed over all pixel shifts δ such that pixels with index s + δ are located inside a square window with N -pixels centered on s (in practice, we use N = 30 × 30 pixels). Assuming all samples in the local window are iid, we can obtain a local estimation of the ENL by inverting the following relationship (theoretical expression):

k2 (s) = ψ(1, Lm (s)) ( 11
)
where ψ is the first-order Polygamma function [START_REF] Tison | A new statistical model for markovian classification of urban areas in high-resolution SAR images[END_REF]. Note that the traditional estimation method (by means of the ratio

E[û m ] 2 /Var[û m ]
), the moment estimation method or the maximum likelihood (ML) estimation method could also be used for the ENL estimation [START_REF] Nicolas | Introduction to second kind statistics: Application of logmoments and log-cumulants to the analysis of radar image distributions[END_REF]. Nevertheless, samples within local windows centered on s are usually not iid, and the subsequent statistic Lm (s) underestimates L m (s). For this reason, once the ENLs have been locally estimated in all sliding windows, we consider a global ENL for the whole super-image obtained by Lm = quantile( Lm (s), α) where α = 0.98 (almost the maximum). While it may be argued that the ENL varies spatially in the image, we found on several images obtained by different sensors that using a constant ENL value leads to satisfying results in the subsequent denoising steps.

E. Denoising the super-image

When the number of dates T is not large enough or when the temporal correlation of speckle is too strong, the super-image obtained by (weighted) averaging suffers from significant remaining speckle fluctuations. A spatial filtering step is then necessary to improve the quality of the superimage ûm . At this step, any speckle reduction method can be used. In this paper, we use MuLoG-BM3D [START_REF] Deledalle | MuLoG, or how to apply Gaussian denoisers to multi-channel SAR speckle reduction?[END_REF] to perform this spatial filtering. MuLoG-BM3D will be discussed in more details in Sec. IV-B where we describe how it can be extended to process ratio images.

In summary, 4 super-images can be computed: the arithmetic mean image (according to Eq. ( 4)), the binary weighted arithmetic mean image (according to Eq. ( 8)), the denoised arithmetic mean image (DAM) and the denoised binary weighted arithmetic mean image (DBWAM). The impact of the method used to build the super-image on the output of RABASAR is evaluated in Sec. V. In the following, these four super-images will be indifferently denoted as ûm . After temporal averaging and spatial filtering, the remaining noise in the super-image follows a distribution without known closed-form. It can nonetheless be approximated by a gamma distribution G(u m , L m ) where L m is a constant parameter that can be re-estimated from the data in order to fit at best the empirical distribution as described in Sec. III-D.

IV. RATIO IMAGE DENOISING

The ratio image τ t formed at date t contains both the fluctuations due to the speckle component of the noisy image v t (pure speckle) and some structures that appear due to the discrepancy between the super-image ûm and the reflectivity u t (changes of reflectivity). Compared to the noisy image v t , the ratio image τ t is spatially far more homogeneous so that denoising the ratio image is advantageous.

A. Statistical analysis of the ratio

Denoising a ratio image requires a denoising method adapted to the statistics of the ratio between the SAR image v t at date t and the super-image ûm . Despite the temporal averaging (and possibly, the spatial filtering) of the superimage, remaining fluctuations are present in ûm . The statistical distribution of the ratio τ t = v t /û m therefore differs from the gamma distribution G(u t /u m , L) followed by the more ideal ratio v t /u m . Note that, for the sake of notation simplicity, we drop in this section the dependency with s and each formula must be understood pixelwise. The Mellin framework developed in [START_REF] Nicolas | Introduction to second kind statistics: Application of logmoments and log-cumulants to the analysis of radar image distributions[END_REF] provides an efficient way to derive the pdf of τ t . Indeed, the pdf of the product of two independent random variables is the Mellin convolution of the pdf of the two variables. The details of the computation are given in Appendix. It is shown that the ratio τ t = v t /û m follows a Fisher pdf F(ρ t , L, L m ) with the three parameters ρ t = u t /u m , L and L m :

p τ τ t ρ t , L, L m ∝ τ L-1 t ρ L t • L m + L τ t ρ t -L-Lm , (12)
where the normalization constant depends solely on the number of looks L and L m . Fisher random variables have a multiplicative behavior, τ t = ρ t w t , where w t follows a Fisher distribution F(1, L, L m ). We have

E [τ t ] = ρ t Lm Lm-1 , for L m > 1, and
Var [τ t ] = ρ 2 t L 2 m (Lm+L-1) L(Lm-2)(Lm-1) 2 , for L m > 2 .
The proportionality between Var [τ t ] and ρ 2 t reveals again that noise is signal-dependent.

B. Denoising of the ratio image: RuLoG algorithm

To denoise the ratio images, we need a method that can account for the Fisher distribution that arises when forming the ratio between gamma-distributed random variables with differing number of looks. To this end, we describe in this paragraph the MuLoG framework [START_REF] Deledalle | MuLoG, or how to apply Gaussian denoisers to multi-channel SAR speckle reduction?[END_REF] and how it can be extended in order to apply general-purpose Gaussian denoisers to Fisher-distributed noise.

MuLoG first stabilizes the variance by applying a logarithm transform to the image. We denote by y the log of the ratio image τ t at date t (the dependency to t is dropped to simplify the notations) and by x = log ρ t = log u t /u m the log of the ratio of the reflectivities. Thanks to the log transform, the multiplicative noise in the ratio image is mapped to an additive signal-independent noise with expectation

E[y] = x-log L Lm + Ψ(L) -Ψ(L m ) and variance Var[y] = ψ(1, L) + ψ(1, L m )
where Ψ(•) denotes the digamma function. This shows that, like for gamma-distributed noise, taking the log of the ratio image stabilizes the variance of the Fisher-distributed noise.

Unlike standard homomorphic approaches, and inspired by [START_REF] Bioucas-Dias | Multiplicative noise removal using variable splitting and constrained optimization[END_REF], MuLoG considers next the exact distribution of logtransformed data. Specifically, the n-pixels image x is obtained by maximum a posteriori estimation expressed as the solution of an optimization problem of the form:

x ∈ arg min x∈R n [-log p y (y|x) + R(x)] (13)
where the first term -log p y (y|x) is the exact likelihood of the log-transformed ratio y, and the second term R(x) = -log p x (x) is a prior enforcing some spatial regularity on the solution. MuLoG solves Problem [START_REF] Zhang | Beyond a gaussian denoiser: Residual learning of deep CNN for image denoising[END_REF] iteratively by the alternating direction method of multipliers (ADMM) algorithm that repeats, for β > 0, the updates ẑ ← arg min

z∈R n β 2 z -x + d 2 + R(z) (14)
d ← d + ẑ -x (15)
x ← arg min

x∈R n β 2 x -ẑ -d 2 -log p y (y|x) (16)
In practice, six iterations are considered with β = 1 + 2/L + 2/L m as suggested in [START_REF] Deledalle | MuLoG, or how to apply Gaussian denoisers to multi-channel SAR speckle reduction?[END_REF].

Regarding Problem [START_REF] Moreira | A tutorial on synthetic aperture radar[END_REF], in the case of gamma-distributed speckle considered in the original MuLoG method [START_REF] Deledalle | MuLoG, or how to apply Gaussian denoisers to multi-channel SAR speckle reduction?[END_REF], the likelihood term was corresponding to a Fisher-Tippett distribution. As discussed in the previous section, the distribution of ratio-images is not gamma-distributed but follows a Fisher distribution. Therefore the optimization of ( 16) has to be modified compared to the original MuLoG algorithm. By applying a change of variable, it can be shown that, after the log transform, Fisher-distributed random variables follow a z-Fisher pdf [START_REF] Fisher | On a distribution yielding the error functions of several well known statistics[END_REF] given by: p r y(s) x(s), L, L m ∝ exp L y(s) -x(s)

• (L m + L exp[y(s) -x(s)]) -L-Lm (17)
which leads to:

-log p y (y|x) = Cst + N s=1 Lx(s) + (L + L m ) log (L m + L exp[y(s) -x(s)]) . (18)
Injecting this expression in [START_REF] Moreira | A tutorial on synthetic aperture radar[END_REF] leads to solve N separable convex problems whose solutions can be obtained by a few iterations of Newton's method defined by:

x(s) ← x(s) - β(x(s) -ẑ(s) -d(s)) + L (1 -c(s)) β + Lc(s)(1 -L Lm+L c(s)) (19)
where c(s) = (L m + L) exp[y(s) -x(s)]/{L m + L exp[y(s)x(s)]}. Interestingly, when L m → ∞, the z-Fisher pdf tends to the Fisher-Tippet pdf, and taking the limit in [START_REF] Quegan | Filtering of multichannel SAR images[END_REF] leads to the original MuLoG algorithm [START_REF] Deledalle | MuLoG, or how to apply Gaussian denoisers to multi-channel SAR speckle reduction?[END_REF].

Regarding Problem [START_REF] Chierchia | SAR image despeckling through convolutional neural networks[END_REF], MuLoG adopts the strategy of plug-and-play ADMM (see for instance [START_REF] Chan | Plug-and-play ADMM for image restoration: Fixed point convergence and applications[END_REF]) which consists in replacing the minimization problem involving R(x) by the solution of a denoiser adapted to additive white Gaussian noise with variance 1/β. In this paper, we consider using BM3D [START_REF] Dabov | Image denoising by sparse 3-D transform-domain collaborative filtering[END_REF], an algorithm based on patch similarity and three-dimensional wavelet shrinkage, and reaching remarkable results with fast computation. We refer to this method for ratio image denoising as Ratio adaptation of MuLoG (RuLoG), see Algorithm 1.

After obtaining the estimated noise-free value ρt (s) = exp(x(s)) of the ratio, we obtain the denoised image ût through: d ← d + ẑ -x update x with a few Newton iterations:

ût (s) = ûm (s) • ρt (s) (20
10:
for from 1 to maxNewtoniter Step 1: computation of the super-image (AM or BWAM)

11: c ← (L m + L) exp[y -x]/{L m + L exp[y -x]} 12: x ← x - β(x -ẑ -d) + L (1 -c) β + Lc(1 -L Lm+L c)
1: ûm ← compute super image(y, t) 2: Lm ← estimate looks ûm

Step 2 (optional): denoising of the super-image

3: ûm ← MuLoG BM3D(û m , L = Lm ) 4: Lm ← estimate looks ûm
Step 3-4: denoising of the ratio image

5: τ t ← v t / ûm 6: ρt ← RuLoG BM3D(τ t , L, L m = Lm )
Step 5: computation of the restored image

7: ût ← ûm • ρt 8: return ût
The application of this pipeline -RABASAR -is illustrated Fig. 2. A typical filtering result on a ratio image is presented on Fig. 2(d) along with the original image, super-image and the ratio image. The pseudo-code is given in Algorithm 2.

V. EXPERIMENTAL RESULTS

To evaluate the performances of the proposed method, different experiments have been conducted on simulated The influence of different super-images on RABASAR denoising results are commented in Sec. V-C. Then, RABASAR is compared to some selected methods from the literature: UTA [START_REF] Lee | Speckle reduction in multipolarization, multifrequency SAR imagery[END_REF], [START_REF] Quegan | Multitemporal ERS SAR analysis applied to forest mapping[END_REF], NLTF [START_REF] Chierchia | Multitemporal SAR image despeckling based on block-matching and collaborative filtering[END_REF], 2SPPB [START_REF] Su | Two-step multitemporal nonlocal means for Synthetic Aperture Radar images[END_REF] and MSAR-BM3D [START_REF] Chierchia | Multitemporal SAR image despeckling based on block-matching and collaborative filtering[END_REF] (Sec. V-D).

A. Data presentation 1) Sentinel-1 and TerraSAR-X images: The proposed methods are tested on a time series of 69 descending Sentinel-1 IW SLC Level-1 images acquired from December 24, 2014 to May 6, 2017 with VV polarization over Saclay area, south of Paris1 (see Fig. 2(a) and(b)). Saclay plateau is mostly an agricultural area with pieces of forests and dispersed academic buildings. In the last five years, many new buildings and infrastructures have been under development.

In addition, 26 single-look TerraSAR-X images (13 images are sensed in 2009 and the other 13 images in 2011) acquired over Saint-Gervais-les-Bains, south-east of Geneva, are used (project DLR-MTH0232). These images are taken over a highly mountainous countryside, with a narrow inhabited valley concentrating many human artifacts (roads, bridges, dams) (see Fig. 3).

In this paper we are dealing with images acquired in interferometric configuration (same orbit and incidence angle). In this case, the registration step can be done accurately using the sensor parameters provided by the space agencies. Besides Sentinel-1 and TerraSAR-X images are radiometrically calibrated.

It often happens that SAR pixels are not spatially independent because of a slight over-sampling creating a spatial correlation. This spatial correlation should be taken into account during the denoising. However, most of the denoising methods are based on the hypothesis that the speckle noise is white. When applied directly to images with correlated noise, reduced performances may be expected from these methods. Therefore, we recommend to perform a spatial decorrelation before despeckling. In this paper, the noisy TerraSAR-X images are decorrelated using the method proposed in [START_REF] Abergel | Sidelobes suppression and strong targets extraction in single look complex SAR images with several applications[END_REF] and the Sentinel-1 images are decorrelated by resampling because of its special acquisition model (the beam both steering in range direction and steering from backward to forward in azimuth direction). All the SAR images are co-registered using subpixel image registration applied on the single look complex data [START_REF] Nicolas | A first comparison of Cosmo-Skymed and TerraSAR-X data over Chamonix Mont-Blanc test-site[END_REF].

2) Simulated data: Simulated SAR images are obtained according to Eq. ( 3), by multiplying a reflectivity map with a random gamma distributed noise.

Many simulations are based on reflectivity maps obtained from optical images. However, SAR images exhibit strong and persistent scatterers, especially in urban areas which can hardly be simulated using optical images. Therefore, we propose to use the arithmetic mean image of long time-series of SAR images, considered as a noise free image (a reflectivity map u) to create realistic simulations of SAR images. This map u is multiplied by a gamma distributed noise w t providing an image v t = uw t of the series. These simulated data correspond to the case without change and are an ideal case (pure iid samples of noise). Images of Fig. 2(b), Fig. 3(b) and 4(a) are used for the simulated case without changes.

Concerning changes in the temporal series, to deal with changes in a realistic way, temporal sequences have been introduced over various areas: forests, farmlands, building areas, etc. as shown in Fig. 4(b). The introduced temporal changes have been chosen according to changes observed in SAR time series and have the temporal profiles shown in Fig. 6.

B. Evaluation method

Measurement of speckle reduction performances is a challenging task, especially when noise-free data are unavailable. Visually checking the despeckling results is an immediate and important way for quality evaluation, but it lacks objectivity. To overcome this limitation, we use the peak signal-noiseratio (PSNR) and structure similarity (SSIM) indexes, even though they suffer from some limitations that have been well discussed in the literature. 1) PSNR: Peak signal-noise-ratio: The PSNR is a commonly used approach to evaluate the quality of restoration results. We consider the PSNR expressed on amplitude images

u A = √ u and ûA = √ û: PSNR = 10 • log 10 |u A | 2 max E[(u A (s) -ûA (s)) 2 ] (21)
where |u A | max is the maximum amplitude value in the noise free data, E[•] represents the spatial average and ûA is the denoised amplitude value.

2) MSSIM: Mean structure similarity index: To evaluate the preservation of image features, the SSIM index [START_REF] Wang | Image quality assessment: from error visibility to structural similarity[END_REF] (structural similarity index measurement) is often preferred to PSNR. From the SSIM, we derive the mean structural similarity index measurement (MSSIM) values which provide a comprehensive measure over the whole image:

MSSIM = 1 N N i=1 2 • E[u A ] • E[û A ] + α 1 E[u 2 A ] + E[û 2 A ] + α 1 2 • Cov[u A , ûA ] + α 2 Var[u A ] + Var[û A ] + α 2
where u A and ûA are noise free and denoised amplitude patches, Cov[•, •] is the measure of covariance, α 1 and α 2 are suitable constants, and N is the number of local windows in the image.

C. Which super-image gives the best denoising?

We presented in Sec. III-E different ways to compute the super-image (AM, BWAM and their spatially denoised versions). This section presents quantitative (Fig. 5) and qualitative (Fig. 7) denoising results obtained when using different super-images, and considering different time series lengths T .

In the binary weight computation (methods BWAM and DBWAM), a window of size 7×7 is used in Eq. ( 6), as suggested by the analysis in [START_REF] Buades | A non-local algorithm for image denoising[END_REF].

1) Using simulated radar images: Simulated images allow to evaluate quantitatively the performances obtained with each of the four different super-images by measuring averaged PSNR and MSSIM values (50 different noise realization are computed in order to compute the average PSNR and MSSIM values). The number of images T in the time series varies from 5 to 95.

For temporal images without reflectivity changes (Fig. 5(ab)), using denoised super-images provides better PSNR and MSSIM values when using a small number of images, but this benefit disappears with longer stacks of images (more than 60). When using denoised super-images, the obtained PSNR/MSSIM values is notably impacted by the value of the ENL.

In the case of changing areas, we performed restorations of 50 speckle realizations in the four different change scenarios discussed in paragraph V-A2. We report in Fig. 6 the mean value and 1σ confidence interval obtained by RABASAR. It can be observed that, in the case of fluctuations typically observed in farmland and forest areas (Fig. 6 (a) and (b)), both RABASAR-AM and RABASAR-BWAM perform well and produce an estimate with no significant bias (the reflectivity varies up to a factor 2 in our simulations). Because a smaller number of dates is selected in the case of RABASAR-BWAM, the estimation variance is very slightly larger. In Fig. 6 (c) and (d) we consider the case of much larger changes that typically occur in urban areas: an appearing large reflectivity or a temporarily large reflectivity. In this case, RABASAR-AM leads to a bias of the same order as the estimation standard deviation. The origin of this bias, observed in Fig. 6(c) for t ≥ 29, can be ascribed to the difference between the true reflectivity and the mean intensity. This difference is too small to be compensated when denoising the ratio. When t < 29, the difference is much larger and no bias can be observed (after denoising, the ratio image correctly compensates for the misfit of the super-image value). By selecting only similar dates, RABASAR-BWAM is more robust to this phenomenon: the bias is negligible compared to the estimation standard deviation. In the case of an appearing, then disappearing structure (Fig. 6(d)), no significant bias is visible with either method because the super-image with RABASAR-AM differs significantly from the true reflectivity and that difference is correctly recovered when denoising the ratio image (the signalto-noise ratio is sufficient so that it be recovered).

2) Using Sentinel-1 images: The temporal series of 69 Sentinel-1 images on the Saclay area is used to test the method on SAR images. Figure 7 can be used to visually assess the efficiency of speckle reduction when using different "superimages". RABASAR provides satisfying denoising results with each of the four different super-images. The use of an additional spatial filtering step to form the super-image seems beneficial in terms of restoration quality: the obtained images are smoother. When using AM and DAM, small areas with low values were sometimes smoothed, leading to the apparition of new points in the denoised results (Fig. 7(a-b) red rectangular). This phenomenon is obvious for impulsive and abrupt changes in building areas. Using BWAM and DBWAM reduces this problem (Fig. 7(c-d)). In some changing parts of the image, using BWAM, however, leads to poor filtering results because very few similar dates could be combined when computing the super-image.

3) Computation time: The computation time of the algorithm depends on the adopted RABASAR version (type of super-image -binary weighted or not-, ENL estimation method and choice of the spatial denoising).

When running the experiments on a time serie of 69 SAR images of size 512 × 768 (Saclay) in the MATLAB environment on a computer (4 cores, Intel(R) Core(TM) i7-7600U CPU @ 2.80GHz), the averaged elapsed time of RABASAR-DBWAM is given in Table I.

As can bee seen, the main part of RABASAR computing time is due to the denoising step. If both the super-image and the ratio image are denoised, the time is multiplied by 2. The similarity and weight computation represents only 1% of the total time. Nevertheless, when using binary weighted arithmetic mean, the super-image has to be computed again for each image to process, whereas the super-image computation is done only once for AM or DAM versions of RABASAR.

D. Denoising performances of RABASAR compared to existing methods

The proposed method is compared with state-of-the-art multi-temporal denoising methods, both on simulated and SAR images. Numerical and visual results are provided when comparing RABASAR with the chosen methods: UTA [START_REF] Quegan | Multitemporal ERS SAR analysis applied to forest mapping[END_REF], NLTF [START_REF] Chierchia | Multitemporal SAR image despeckling based on block-matching and collaborative filtering[END_REF], 2SPPB [START_REF] Su | Two-step multitemporal nonlocal means for Synthetic Aperture Radar images[END_REF] and MSAR-BM3D [START_REF] Chierchia | Multitemporal SAR image despeckling based on block-matching and collaborative filtering[END_REF].

1) Quantitative comparison: As in the previous sections, to quantitatively compare the filtering performance of these methods, averaged PSNR and MSSIM are computed. Only RABASAR-AM and RABASAR-DAM performances are presented in the curves of Fig. 8.

In this simulation, we see from Fig. 8, that RABASAR generally provides better PSNR than other filtering methods. Table II presents some results about PSNR and MSSIM results. It shows that RABASAR-DAM provides the best results in this ideal situation when there is no change in the temporal series.

2) Sentinel-1 and TerraSAR-X image denoising: This section presents and discusses the results obtained when denoising Sentinel-1 and TerraSAR-X images. Since the noise free images are not available, the denoised results and the residual noise (ratio of the original image by the denoised image) are visually evaluated. The residual noise should correspond to pure gamma-distributed noise samples of mean 1. Residual structures, homogeneous areas, and radiometric variations correspond to the following perturbations: the destruction of structures, the absence of filtering, and bias introduction.

First, compared to MuLoG-BM3D applied on a single image, RABASAR-DAM provides a much better result, preserving fine strucures and isolated objects, see Fig. 11. This experiment shows that the exploitation of the super-image and the ratio image, which is much more stationary than the original noisy image, helps preserving the original resolution.

Figure 9 presents different filtering results on a Sentinel-1 time serie of Saclay. Compared to 2SPPB and MSAR-BM3D methods, RABASAR-DAM and RABASAR-DBWAM provide better denoising results since they take both the ratio results and the texture characteristics into account. Some noisy areas can be observed in the RABASAR-DBWAM results due to the temporal samples selection (for some pixels, only few similar temporal samples can be found to compute the binary weighted arithmetic mean). 2SPPB method does not give good results for seasonal changing farmland areas, and shows an obvious bias in the residual noise (see for instance the blue circle area in Fig. 9(e)).

Since MSAR-BM3D method detects the bright points in advance and prohibits any denoising around these points [START_REF] Chierchia | Multitemporal SAR image despeckling based on block-matching and collaborative filtering[END_REF], building areas in the residual noise are homogeneous (Fig. 9(f) red circle area) showing that no filtering has been applied in these areas (Fig. 9(b) red circle area).

This also explains why MSAR-BM3D has lower PSNR values. In addition, some of the textures in MSAR-BM3D results are over smoothed, such as the blue circle area in Fig. 9(b).

Similar phenomena can be observed on the TerraSAR-X images presented Fig. 10. Whereas Fig. 10

VI. CONCLUSION AND FUTURE WORK

This paper has proposed a ratio-based multi-temporal denoising framework. During the restoration of each SAR image, it exploits the temporal information through a super-image. The use of different strategies to compute the super-images has been analyzed. RABASAR can provide better PSNR and MSSIM values when using a spatially denoised super-image. With the increase of the number of images in the time series, the differences of using different super-images decrease. When there are changes in the time series, using a binary weighted arithmetic mean can also provide good results. Based on the processing of time-series corrupted by simulated speckle noise, actual Sentinel-1 stacks and TerraSAR-X stacks, the qualitative and quantitative comparisons with UTA, NLTF, MSAR-BM3D and 2SPPB methods showed the potential of RABASAR to better preserve structures in multi-temporal SAR images while efficiently removing speckle. Besides, the super-image can be easily updated when a new data becomes available so as to process new images on-line.

Future work will be devoted to the updating framework, specially for the "re-computation" of the super-image and to the further processing of denoised time series.

Fig. 1 :

 1 Fig. 1: Summary of RABASAR method for speckle reduction of SAR time series and associated sections of the paper.

  Fig. 2: (a) Sentinel-1 noisy image, (b) Arithmetic mean image, (c) Ratio image, (d) Denoised result on ratio image. Appearing (dark areas in (d)) and disappearing buildings (clear areas) are located in the middle of the image.

Fig. 3 :Fig. 4 :

 34 TerraSAR-X series: (a) one image of the temporal series, (b) arithmetic mean of the 26 images. Images used for data simulation (a) optical image used as a noise free image, (b) arithmetic mean of Sentinel-1 images with the location of different land-cover types used to simulate temporal changes (green: farmland changes, yellow: forest changes, red: appearing building, blue: appearing then disappearing building. and SAR images. Recall that an open-source code of RABASAR is available at https://www.math.u-bordeaux.fr/ ∼ cdeledal/rabasar.php.

Fig. 5 :

 5 RABASAR performances obtained with different super-images, in the absence of change, as measured with the PSNR (a) and the MSSIM (b). The solid line represent the mean value computed over 50 different noise realizations. For each index box, the bottom and top edges indicate the 25th and 75th percentiles, respectively. The outliers are plotted individually using the '+' symbol.

Fig. 6 :

 6 Fig. 6: Evaluation of RABASAR estimators in 4 different change scenarios: typical temporal variations of reflectivities in (a) farmland areas; (b) forest areas; (c) urban areas when a new building appears; (d) urban areas with an appearing then disappearing structure. The mean and 1σ confidence interval are displayed for RABASAR-AM (top curve) and RABASAR-BWAM (bottom curve).

Fig. 7 :

 7 Fig. 7: RABASAR denoising results on Sentinel-1 images over the Saclay area (middle row) and corresponding ratio-images (lower row) based on the use of: (a) AM, (b) DAM, (c) BWAM, (d) DBWAM. 69 Sentinel-1 images are used. The zoom of the red box is enlarged in the first row together with the original noisy image to ease the comparison.

Fig. 8 :Fig. 9 :

 89 Fig. 8: Comparison of different temporal denoising methods: UTA, NLTF, 2SPPB, MSAR-BM3D, RABASAR-AM and RABASAR-DAM. Averaged PSNR (upper row) and MSSIM (lower row) as a function of the number of images for two data sets: (a) simulated time serie without changes using Fig. 4(a)), (b) realistic simulated SAR data without changes using Fig. 4(b)). Each experiment is repeated 50 times, with the same noise free value multiplied by different gamma distribution noise. The line features shows the mean value of PSNR (top) and MSSIM (bottom)over the 50 experiments.

10 :

 10 (c) and (d) visually provide satisfying results and homogeneous residual noise images, both 2SPPB and MSAR-BM3D smoothed out some Denoising results of TerraSAR-X images over Saint-Gervais (using Rulog). The original noisy image is available in Fig. 4(f). (a) 2SPPB, (b) MSAR-BM3D, (c) RABASAR-DAM, (d) RABASAR-DBWAM, (e-h) residual noise images (ratio of noisy image and denoised results). 26 TerraSAR-X images are used. textures (Fig. 10 blue circle areas), or keep unchanged some noisy areas.

Fig. 11 :

 11 Fig. 11: Noisy image (a) and denoising results (b-c) of MuLoG-BM3D and RABASAR-DAM. 64 resampled Sentinel-1 images are used to calculate the arithmetic mean image.

)

  Algorithm 1 Extension of MuLoG to ratio images (RuLoG) Input: ratio image τ t number of looks L of the numerator number of looks Lm of the denominator

	Output: Denoised ratio image ρt
	1: maxADMMiter ← 6	(typical value)
	2: maxNewtoniter ← 10	(typical value)
	3: d ← 0
	4: y ← log(τ t ) 5: x ← log(τ t ) + log( L/ Lm ) + ψ( Lm ) -ψ( L)
	6: β ← 1 + 2 L + 2 Lm
	7: for k from 1 to maxADMMiter 8: ẑ ← BM3D denoise(x -d, noise std = 1/	√	β)
	9:

  Input: T co-registered SAR images {v t } T t=1 , targeted date t, input number of looks L.

	13:	end
	14: end
	15: return exp(x)
	Algorithm 2 Multi-temporal speckle reduction (RABASAR)

Output: Image with reduced speckle ût .

TABLE I :

 I AVERAGED COMPUTATION TIME OVER TEN RUNS OF THE DIFFERENT STEPS FOR A TIME SERIE OF 69 IMAGES WITH SIZE 512 × 768 PIXELS

	Step		Averaged time
	BWAM super-image (step 1)	0.74 sec	(1.0%)
	ENL estimation	(step 1 and 2) 0.18 sec	(0.3%)
	MuLog-BM3D	(step 2)	34.3 sec	(50.5%)
	RuLog-BM3D	(step 4)	32.7 sec	(48.2%)
	→ ADMM itself	(step 2 & 4)	1.90 sec	(2.8%)
	→ BM3D itself	(step 2 & 4)	65.1 sec	(95.9%)
	Total time		67.9 sec.

TABLE II :

 II

	Sentinel-1 Evaluation	UTA NLTF 2SPPB MSAR		RABASAR
		methods				-BM3D	-AM -DAM -BWAM -DBWAM
	without	PSNR	25.37 19.49	25.16	21.90	27.34	28.61	26.72	28.15
	changes	MSSIM	0.84	0.82	0.83	0.67	0.86	0.89	0.85	0.88
	With few images in the stack, MSAR-BM3D and 2SPPB also
	provide competitive PSNR and MSSIM. However, with the
	increase of the number of images, MSAR-BM3D curve does
	not rise as fast as 2SPPB. When using less than 4 images,
	MSAR-BM3D provides the best MSSIM values. With an
	increasing number of images, UTA PSNR and MSSIM values
	keep increasing.

NUMERICAL RESULTS PROVIDED BY DIFFERENT METHODS WITH 32 IMAGES IN THE SERIES. EVALUATION IS MADE BY PSNR AND MSSIM. FOR PSNR AND MSSIM, LARGER VALUES EXPRESS BETTER DENOISING RESULTS. THE MEAN VALUES OF 20 TIMES TEST ARE USED.

All the Sentinel-1 images can be downloaded from Copernicus Open Access Hub (https://scihub.copernicus.eu/dhus/).

ACKNOWLEDGMENT

The authors would like to thank the CNES (French Space Agency, project DAJ/AR/IB-2016-10117102) and the China Scholarship Council for their fundings.

DISTRIBUTION OF THE RATIO IMAGE

In this appendix we present a simple way of deriving the distribution (pdf) followed by the ratio τ t = vt ûm . The pdf of the ratio of 2 gamma-distributed random variables has been established in [START_REF] Cramer | Mathematical methods of statistics[END_REF], and used in [START_REF] Touzi | A statistical and geometrical edge detector for SAR images[END_REF] for edge detection in SAR images. Following [START_REF] Epstein | Some applications of the Mellin transform in statistics[END_REF] [START_REF] Nicolas | A Fisher-MAP filter for SAR image processing[END_REF], the Mellin framework allows a straightforward derivation.

We assume that the numerator v t follows a gamma distribution G(u t , L), the denominator ûm follows a gamma distribution G(u m , L m ), and that v t and ûm are independent. We will denote by × the Mellin convolution, and H(u) the homothetic pdf defined by H(u)(ν) = 1 u δ( ν u -1). We have the following relationship between a gamma distributed pdf of mean 1 and a gamma pdf of mean u t [START_REF] Nicolas | Introduction to second kind statistics: Application of logmoments and log-cumulants to the analysis of radar image distributions[END_REF]:

In the same way, we have the following relation for the pdf of ûm :

The variable û-1 m follows an inverse gamma distribution GI given by:

Multiplying the variables v t and 1 ûm , we thus have the Mellin convolution of H ut um , G(1, L) and GI(1, L m ) which leads


-from different genomes -with polymorphism (SNPs and large structural polymorphism) -with no close reference genome (most of the time)

Reference guided genome assembly in metagenomic samples

Objectives :

Assemble an genome of interest -from metagenomic reads, -using a remote reference genome -detecting stuctural polymorphism Existing methods :

Successful assembly of a bacterial genome in one circular contig Discovery of unknown phage structural variants

Degnan 2008 [START_REF] Degnan | Diverse Phage-Encoded Toxins in a Protective Insect Endosymbiont[END_REF] -7 known variants known, differing by a ~5kb virulence cassette

Results :

-3 new phage variants discovered in 5 samples -Coabundant phage successfully assembled in 3 samples

+

Assembly is reference-free

-Assembly first

De novo metagenomic assembly + contig taxonomic assignation

Ex : MegaHit[1] + Blast

Mapping first

Assembly of reads Mapped on reference

Ex : BWA+Minia [START_REF] Chikhi | Space-efficient and exact de Bruijn graph representation based on a Bloom filter[END_REF] Hybrid stategy

Reference guided assembly Local assembly of the regions diverging from the reference Ex : MitoBim [START_REF] Hahn | Reconstructing mitochondrial genomes directly from genomic nextgeneration sequencing reads -A baiting and iterative mapping approach[END_REF] Assembly

  Cervin Guyomar, Wesley Delage, Fabrice Legeai, Christophe Mougel, Jean-Christophe Simon, Claire Lemaitre To cite this version: Cervin Guyomar, Wesley Delage, Fabrice Legeai, Christophe Mougel, Jean-Christophe Simon, et al.. Reference guided genome assembly in metagenomic samples. RECOMB 2018 -22nd International Conference on Research in Computational Molecular Biology, Apr 2018, Paris, France. pp.1. halpolymorphism) An hybrid approach : -Mapping reads and assembly of conserved regions -Gapfilling of diverging regions with all reads Reads mapping on the remote reference (BWA) Assembly (Minia) Output : super-contigs (fasta) + gfa format MindTheGap algorithm -De Bruijn graph assembly starting from a contig end kmer -Search target kmers in the contig graph 2 solutions between contig 1 and contig 2 1 solution between contig 1 and contig 3 Metagenomic readset All the reads Results in the pea aphid holobiont Metagenomics = A mixture of reads :

→

  Need for a tool dedicated to guided assembly in metagenomic context MindTheGap results -One-contig for 30 samples -Genome length close to the real reference -On average 7 times faster than Megahit Assembly of Buchnera aphidicola from 42 pea aphid metagenomic samples [5] using Buchnera genome from another species Assembly of the phage APSE from 42 pea aphid metagenomic samples In developpment on GitHub https://github.com/GATB/MindTheGap/tree/contig_dev


Thi Minh Vu

Hang

Analyzing influences of Revenue Management on customers' perceived fairness, price acceptance and switching intention in the Service industry

Revenue Management (RM), also known as "Yield Management", has been applied widely across industries around the world. Although helping to maximize short term profit of firms, this pricing strategy has caused the price differentiation for not only different customers by market segments but also for the same customer over the time due to the demand variance.

This research aims to examine impacts of the price differentiation for the same customer and the same service on customers' perceived fairness and price acceptance. Instead of fairness of prices, the fairness perception examined here relates to the RM pricing practice based on market demand. Moreover, in this RM practice, since prices vary over the time, customers probably tend to more often find and select the most suitable price within available service providers. This motivates the research to test if there is a direct effect of price differentiation on switching intention of customers.

As looking into this price differentiation for a certain customer, this research considers both advantageous price variances (price decreases) and disadvantageous ones (price increases). Furthermore, the contrast between advantageous price variances and disadvantageous ones as well as interactive effect of these types of price variances on perceived fairness, price acceptance, switching intention are first examined in this study. Specifically, there are several aspects relating to price differentiation will be examined. Firstly, some characteristics of price differentiation include: 1-intensity of price variances (from minor to major variances); 2-speed of price variances (from slow to fast variances) defined as temporal distance between two points of prices; 3-the regularity and predictability of price variances. Secondly, the role of explaining price differences on three dependent variables (perceived fairness, price acceptance and witching intention) is simultaneously tested in this research.

For a certain customer consuming the service, the previous price they last paid can be used as their reference price to evaluate price variance. This price does not remain stable, it varies over the time causing price differences. This research will vary the intensity and speed of price increases and price decreases to discover which level of price variances is considered as fair and acceptable. Particularly, this research will in turn take intensity and speed of price variances in to account in relation to types of price variances: advantageous variances (price decreases), disadvantageous ones (price increases) or two-way price variances (as customers encounter both price decreases and price increases in a certain period). In case of two-way price variance practice, it is planned to test the role of advantageous price variances on mitigating negative effect of disadvantageous ones on price acceptance as well as switching intention.

Regarding the third characteristic of price variances, customers probably prefer the pricing practice with regular and recurrent price variances rather than the practice in which prices vary irregularly according to random variance of demand. Since price changes are regular and recurrent, it is likely to be easier for customers to get familiar with and adapt to this pricing practice. Thus, the regular and recurrent pricing practice is hypothesized to have higher perceived fairness, higher price acceptance and lower switching intention. Obviously, the regularity and recurrence of price variances are associated with predictability of price variances. However, such predictability is not only linked to regularity and recurrence, it is more determined by the extent of information about price variances disclosed by service sellers. Through the predictability based on disclosed information, customers can adapt to price fluctuations and make decisions in advance to take advantages of price decrease or avoid losses from price increases. Thus, the impact of predictability of price variances based on given information is planned to be investigated. Furthermore, when customers confuse and wonder the reason of their price difference, in this RM pricing practice, it is assumed that giving them explanation about price increases as well as price decreases makes sense in improving their perceived fairness, price acceptance and switching intention.

Finally, the moderating roles of three factors in the relationships between two investigated aspects of RM pricing practice and customers' perceived fairness, price acceptance and switching intention will be taken in to account, consisting of customers' familiarity with demand-based RM pricing, types of services and types of service industries. Besides two remaining factors, there is a need to investigate the role of types of services because in the same service industry, a service could be either hedonic or utilitarian one depending to consumption contexts, for ex: (an airline ticket for travelling is hedonic service but another for participation in an important meeting is a utilitarian one). It is assumed that types of services are directly linked to customers' price sensitivity and thus, linked to their perceived fairness and switching intention. H1.4: For two-way price variance practice in which the sum of price increases (X 1 ) is larger than the sum of price decreases (Y 1 ), the intensity of difference between the sum of price increases and the sum of price decreases (X 1 -Y 1 ) negatively impacts perceived fairness (a) and price acceptance (b), positively impacts switching intention (c). H1.5: For two-way price variance practice in which the sum of price increases (X 2 ) is smaller than the sum of price decreases (Y 2 ), the intensity of difference between the sum of price decreases and the sum of price increases (Y 2 -X 2 ) negatively impacts perceived fairness (a); positively impacts price acceptance (b) and negatively impacts switching intention (c). H1.6: For two-way price variance practice in which the sum of price decreases approximates the sum of price increases, average intensity of price variances negatively impacts perceived fairness (a) but no longer impacts price acceptance (b) and switching intention (c). H2.4: For two-way price variance practice in which the average speed of price increases (X 3 ) is faster than the average speed of price decreases (Y 3 ), the difference between the average speed of price decreases and the average speed of price increases (Y 3 -X 3 ) negatively impacts perceived fairness (a), price acceptance (b) and positively impacts switching intention (c). H2.5: For two-way price variance practice in which the average speed of price increases (X 4 ) is slower than the average speed of price increases (Y 4 ), the difference between the average speed of price increases and the average speed of price decreases (X 4 -Y 4 ) negatively impacts perceived fairness (a); positively impacts price acceptance (b) and negatively impacts switching intention (c).

Hypotheses about

Hypotheses about the regularity and predictability of price variances:

H3.1: The two-way price variance practice which is regular and recurrent leads to higher perceived fairness (a), higher price acceptance (b) and lower switching intention (c) than the irregular two-way price variance practice does.

1 :

 1 Intensity of price increases negatively impacts perceived fairness (a) and price acceptance (b), positively impacts switching intention (c). H1.2: Intensity of price decreases negatively impacts perceived fairness (a), positively impacts price acceptance (b); negatively impacts switching intention (c). H1.3: Intensity of price variances negatively impacts perceived fairness (a); with the same intensity, intensity of price decreases is perceived less unfair than intensity of price increases (b).

H2. 1 :

 1 speed of price variances: speed of price increases negatively impacts perceived fairness (a), price acceptance (b) and positively impacts switching intention (c). H2.2: speed of price decreases negatively impacts perceived fairness (a), positively impacts price acceptance (b) and negatively impacts switching intention (c). H2.3: Speed of price variances negatively impacts perceived fairness (a); with the same speed, speed of price decreases is perceived as less unfair than speed of price increases (b).

H3. 2 : 1 :

 21 level of predictability of price variances positively impacts perceived fairness (a), price acceptance (b) and negatively impact switching intention (c). 4. Hypotheses about explaining reasons for price variances H4.Presenting price decreases as discounts due to promotion program leads to higher perceived fairness (a) and price acceptance (b) and lower switching intention (c) associated with price decreases H4.2: Showing customers that additional revenue from price increases is used to expand supply and reduce congestion leads to higher perceived fairness (a) and price acceptance (b) and lower switching intention (c) associated with price increases 5. Hypotheses about moderating role of familiarity, types of services and types of industries


Eric Mottay

Pierre Rigail

Robert Braunschweig

Christophe Pierre

Sébastien Ermeneux

Clément Rossignol

Jean-Michel Rampnoux

Stefan A Dilhaire
email: stefan.dilhaire@u-bordeaux1.fr

Sebastien Ermeneux

Clement Rossignol

Heterodyne Optical Sampling for Picosecond Ultrasonics and Nanoscale Heat Transfer

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Introduction

Pump-probe techniques are widely used to measure events on time scales much shorter than the resolution of electronic detectors, and are applied in such diverse fields as ultrafast spectroscopy, photo-acoustics, TeraHertz imaging, etc. In ultrafast photoacoustics measurements for instance, a pump beam launches in the sample acoustic waves, which are detected by a second, temporally shifted probe beam. Typical detection methods rely on very small changes in the reflection coefficient of the sample surface, requiring an averaging of the signal to improve the signal to noise ratio. Traditional pump-probe methods use a mechanical delay line to shift the two pulses in the time domain, where each measurement point corresponds to a single mechanical position of the delay line. Although very efficient for small measurement ranges, extending this method in the hundreds of picoseconds or nanosecond lead to a very long acquisition time, and unpractical length for the delay line.

We present a new, compact detection system, using a compact dual-oscillator ultrafast laser system, specifically designed for pump-probe measurements over time scales as long 20 ns, with a sub-picosecond resolution. This system does not use any mechanical delay line, and allows for extremely fast acquisition time.

Experimental setup

The system uses a new, compact diode-pumped ultrafast laser, incorporating two femtosecond oscillators, frequency locked to each other, with a small difference (1 kHz) between the repetition frequency of the two lasers. Each pulse is therefore temporally shifted from the previous one without any mechanical adjustment. Since the repetition rate of the laser is very high (50 MHz), a complete measurement can be done in a very short time (1ms).Since there is no need for delay line control, pump beam modulation, lock-in amplifier detection, the detection system is extremely simple, and only requires a high speed detector and averaging system. The pump and probe beam wavelength are slightly shifted in order to eliminate the residual pump signal on the detector. Averaging 10000 times the signal takes only 10 s allows to reach a signal to noise ratio close to 10 -6 .

Results

The first result shown in figure is the photothermal response of a 250 nm Tungsten film deposited on a silicon substrate. The signal exhibits a typical fast behavior coming from the electronic response of the material followed by a slow thermal decay. Superimposed with thermal background appear acoustic echoes showing the longitudinal acoustic wave reflected back and forth in the Tungsten film. The time of flight between successive echoes allows the determination of mechanical properties of the film while the thermal decay can be used to identify the thermal properties of nanomaterials. Once a point measurement is done on can scan the whole surface of interest of the material and reconstruct the film of the material response. Thus figure 1 is one of the pixel responses of the photothermal and photoacoustic movie. Frame rate is about 1 Tera images per second during 20 ns. Acquisition time of the movie is a few ten of minutes. Figure 3 shows a frame of the photothermal and photoacoustic movie. One can easily see the thermal contribution to the signal giving rise, through thermoelastic coupling, to surface waves in the film.

Conclusion

We have presented a new dual-source based optical sampling system designed for nanothermal applications which permits the imaging of the thermal and acoustic response of nano materials with a subpicosecond resolution. It is a breakthrough in nanomaterial thermal transport and ultrasonics experimental studies.

  Figure 2

Figure 3

 3 Figure 3