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Wavelet Operators and Multiplicative Observation Models - Application to Change-Enhanced Regularization of SAR Image Time Series Abdourrahmane Atto, Emmanuel Trouvé, Jean-Marie Nicolas, Thu Trang Le

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Abdourrahmane Atto, Emmanuel Trouvé, Jean-Marie Nicolas, Thu Trang Le. Wavelet Operators and Multiplicative Observation Models - Application to Change-Enhanced Regularization of SAR Image Time Series. 2016. hal-00950823v3

HAL Id: hal-00950823 https://hal.archives-ouvertes.fr/hal-00950823v3 Preprint submitted on 26 Jan 2016

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Abdourrahmane M. Atto1,∗, Emmanuel Trouve1, Jean-Marie Nicolas2, Thu-Trang L^e1

Abstract—This paper first provides statistical prop- I. Introduction - Motivation erties of wavelet operators when the observation model IGHLY resolved data such as Synthetic Aperture can be seen as the product of a deterministic piece- Radar (SAR) image time series issued from new wise regular function (signal) and a stationary random H field (noise). This multiplicative observation model is generation sensors show minute details. Indeed, the evo- analyzed in two standard frameworks by considering lution of SAR imaging systems is such that in less than 2 either (1) a direct wavelet transform of the model decades: or (2) a log-transform of the model prior to wavelet • high resolution sensors can achieve metric resolution, decomposition. The paper shows that, in Framework providing richer spatial information than the deca- (1), wavelet coefficients of the time series are affected by metric data issued from ERS or ENVISAT missions. intricate correlation structures which affect the signal singularities. Framework (2) is shown to be associated • the earth coverage has increased: recent satellites such with a multiplicative (or geometric) wavelet transform as TerraSAR-X and Sentinel-1A repeat their cycle in and the multiplicative interactions between wavelets a dozen of days. and the model highlight both sparsity of signal changes The increase of those spatial and temporal resolutions near singularities (dominant coefficients) and decorre- makes information extraction tricky from highly resolved lation of speckle wavelet coefficients. The paper then SAR image time series. This compels us re-considering derives that, for time series of synthetic aperture radar data, geometric wavelets represent a more intuitive and data features and representations in order to simplify relevant framework for the analysis of smooth earth data processing. fields observed in the presence of speckle. From this The paper presents a parsimonious framework for analysis, the paper proposes a fast-and-concise geomet- the analysis of huge data associated with multiplica- ric wavelet based method for joint change detection and tive type interactions. These data are observed in many regularization of synthetic aperture radar image time situations, for instance when acquiring signals from series. In this method, geometric wavelet details are radar/sonar/ultrasonic waves [1]/[2]/[3],[4], when analyz- first computed with respect to the temporal axis in or- ing seasonality from meteorology data [5] or when focus- der to derive generalized-ratio change-images from the time series. The changes are then enhanced and speckle ing on proportionality in economy data [6] and political is attenuated by using spatial bloc sigmoid shrinkage. sciences [7]. We focus speci cally on SAR systems, a Finally, a regularized time series is reconstructed from challenging imagery domain with huge amount of data the sigmoid shrunken change-images. An application a ected by multiplicative type interactions. of this method highlights the relevancy of the method From the literature, analysis of SAR image time se- for change detection and regularization of SENTINEL- ries has been mainly performed on short-length image 1A dual-polarimetric image time series over Chamonix- sequences. This is the consequence of SAR data cost (very Mont-Blanc test site. high), long satellite revisit time and short satellite lifetime, Index Terms—Wavelets ; Geometric convolution ;Syn- among other issues. Literature concerns both theoretical thetic Aperture Radar ; Image Time Series Analysis. and application guided methods for: 1 LISTIC, EA 3703, UniversiteSavoie Mont Blanc - Universite identifying appropriate statistics/similarity measures Grenoble Alpes, France • 2 LTCI, CNRS UMR 5141, Telecom ParisTech, 46 rue Barrault, [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], etc.; 75013 Paris, France • detecting and analyzing speci c features, for instance ∗ [email protected] urban areas expansion [8], [18], [19], glaciers dynamics ||||||||{ The work was supported by PHOENIX ANR-15-CE23-0012 grant [13], [20], [21], snow cover mapping [22], sea clutter of the French National Agency of Research. analysis [23], forest mapping [24], earthquake mon- 2

itoring [8], sea ice motion analysis [25], coastline propose a joint ltering and change detection of high detection [26], soil erosion [27], etc.; resolution SAR image time series. Section VI concludes

• regularizing SAR data for speckle reduction and fea- the work. ture enhancement [19], [24], [26], [28], [29]. Most of these methods yield computationally greedy algo- II. Statistical properties of additive wavelet rithms because they have been built for the sole sake of transforms on multiplicative observation models performance over short-length image sequences. A. Problem formulation For long-time sequences such as those expected with the A multiplicative observation model involving future Sentinel constellation, a direct application of these strictly positive interactions of a piecewise regular methods is not an option: this direct application may be deterministic function f and a random process X can be unthinkable due to computational cost and unnecessary written as: for performance/robustness. Indeed, dense/long temporal y = fX = f + f(X − 1). (1) sampling results in redundant information on the time axis so that a purely temporal analysis may be sucient for In the model given by Eq. (1), function f is observed in a monitoring of most large scale earth structures. multiplicative signal-independent-noise X or, equivalently, The issue raised by new generation SAR sensors is thus in an additive signal-dependent-noise f(X−1). We assume that denotes a stationary sequence of revisiting these methods with the sake of adapting them X = (X[k])k∈Z to long and dense temporal image samples. Among the positive (strictly) real random variables. references provided above, we consider hereafter wavelet The (standard) wavelet transform operates on Eq. (1) in based approaches derived in [8], [13] for change detection a way such that (linearity with respect to `+' operation) and in [29],[30] for image regularization. Wy = Wf + Wf(X − 1). (♣) For change detection, [8] computes a log-ratio change measure and applies a wavelet transform to this log-ratio Question: assuming sparsity of W on f, what are the measure in order to emphasize di erent levels of changes. statistical properties of the noisy observation Wy? In contrast, [13] computes the wavelet transform of images In a noisy environment, the useful sparsity is strongly prior to change detection by using probabilistic pixel linked to the noise properties since noise a ects the non- features. zero coecients, and thus a ects the quality of the approx- For image regularization, [29] and [30] propose wavelet imation that can be obtained by considering those non- shrinkages by using: 1) a parametric Bayesian approach zero coecients. Noise being Wf(X − 1) in model (♣), [29] and 2) a non-parametric sigmoid based approach [30]. the issue is then the statistical properties of this quantity. The wavelet transform applies on the spatial axes for both The following rst recalls basics on wavelet transforms parametric and non-parametric methods, so as to be more (Section II-B). Then Section II-C provides the statistical robust to speckle. Despite the somewhat di erent strategy, properties of wavelet coecients of the noise involved in parametric and non-parametric approaches can be shown transform (♣). equivalent up to a probabilistic prior speci cation. The present paper revisit [8], [13], [29] and [30] for deriv- B. Basics on wavelet based transforms ing a joint and intuitive framework for change detection In the following, we are interested in multi-scale de- and regularization. The main contributions provided by composition schemes involving, up to a normalization con- the paper are enumerated through the following paper stant, some paraunitary lters (H0, H1) associated with a organization. Section II provides statistical properties of wavelet decomposition, see [31], among other references. standard (additive) wavelet transforms on a multiplicative A one-level wavelet decomposition involves splitting observation model. It highlights the non-stationarities of [32] a given functional space 2 , de ned as Wj,n ⊂ L (R) wavelet coecients when the decomposition applies di- the closure of the space spanned into {τ2jkWj,n : k ∈ Z} rectly on the multiplicative interactions. A multiplicative direct sums of subspaces , spanned (Wj+1,2n+)∈{0,1} wavelet de nition from algebraic inference is described respectively by , where {τ2j+1kWj+1,2n+ : k ∈ Z} in Section III. Its statistical properties on multiplicative ∈{0,1} τkf : t 7− f(t − k). The splitting of Wj,n follows from observation models are discussed in the same section. This decimated arithmetic convolution operations: wavelet transform is shown to be associated with sta- → tionary and decorrelated noise coecients when focusing Wj+1,2n+(t) = h[`]Wj,n(t − 2`). (2) `∈ on homogeneous radiometry sections. Section IV provides XZ bloc sigmoid shrinkage functions for change information for ∈ {0, 1}, where h denotes the impulse response of enhancement. Section V then exploits both shrinkage and the scaling lter (when = 0) or the wavelet lter (when decorrelation induced by the multiplicative wavelets to = 1). 3

The consequence of Eq. (2) is that a function g having • The autocorrelation function of Y,RY[k, `] = 2 coecients c = (c[`]) ∈ ` ( ) on {τ j Wj,n : k ∈ }: [Y[k]Y[`]] satis es, by taking into account Eq. (6): `∈Z Z 2 k Z E R R (8) g = c[`]τ2j`Wj,n ∈ Wj,n Y[k, `] = f[k]f[`] ( X[k − `] − 1) . `∈ XZ can be expanded1 [31] in terms of Remark 1: Eqs. (7) and (8) above highlight that the additive signal-dependent noise Y is non-stationary in g = c0[`]τ2j+1`Wj+1,2n + c1[`]τ2j+1`Wj+1,2n+1 general, except some few cases, for instance when f is `∈ `∈ constant. XZ XZ ∈W ∈W j+1,2n j+1,2n+1 Let us now analyze the wavelet coecients of Y. Denote where| its coecients{z }c | = {z(c [`]) }on by + the coecients of on subband . We have `∈ Cj,n Y Wj,n , for Z , {τ2j+1kWj+1,2n+ : k ∈ Z}∈{0,1} ∈ {0, 1} + j j (9) satisfy Cj,n[k] = hj,n[`]f[` − 2 k](X[` − 2 k] − 1). `∈ c[k] = h[`]c[` − 2k]. (3) XZ `∈ It follows that XZ Starting the decomposition from a function f ∈ W0,0, + j (10) ECj,n[k] = (µ0 − 1) hj,n[`]f[` − 2 k] `∈Z f = c[`]τ`W0,0, X and the autocorrelation function R+ `∈Z [k, `] = X h i j,n the subband coecients of then follow from + + of + is: Wj,n f E Cj,n[k]Cj,n[`] Cj,n j cj,n[k] = hj,n[`]c[` − 2 k] (4) R+ j,n[k, `] = hj,n[p]hj,n[q]× `∈ XZ p∈Z q∈Z X X j j where the Fourier transform Hj,n of hj,n is (see [33, Eq. f[p − 2 k]f[q − 2 `]× (26)]): R [p − q − 2j(k − `)] − 1 . (11) " j # X j/2 `−1 Hj,n(ω) = 2 H (2 ω) . (5) From Eq. (11), we derive that + is non-stationary in ` Cj,n `=1 j j Y general due to the presence of the term f[p − 2 k]f[q − 2 `] Eq. (4) can be used in practice for computing discrete in Eq. (11) and this, even if µ0 = 1 in Eq. (10). wavelet transforms from sample observations (terminolo- Remark 2 (Non-stationarity of C+ for exponential gies of `discrete wavelet transform' when n ∈ {0, 1}, `dis- j,n type function f): Assume that µ = 1 and function f crete wavelet packet transform' when j , 0 n ∈ {0, 1, . . . , 2 −1} satis es (exponential type functions), `adapted discrete wavelet packets' for a suitable selec- f[k]f[`] = f[k + `] where does not reduce to the constant 1. In this case, tion of -indices). Some splitting schemes involving non- f n we derive decimation (factor 2j in Eq. (4)) are also available and j π yield the concept of frames and the notion of stationary + f[−2 (k + `)] 2 i2j(k−`)ω R [k,`]= γX0(ω)|Gj,n(ω)| e dω(12) wavelet transforms [34]. The reader can refer to the general j,n 2π Z−π literature on wavelets for more details on wavelet trans- where Gj,n = F ∗ Hj,n and F is the Fourier transform of forms. . The non-stationarity of + is then due to the term f Cj,n f[−2j(k + `)] in Eq. (12) above. C. Stochasticity properties of the additive wavelet co- More generally, even when assuming that µ0 = 1, it is ecients easy to check that most standard functions f lead to the non-stationarity of + . In particular, linear functions of In model (♣), noise is associated with a random se- Cj,n quence having the form type f[k] = f0 × k (for certain k in a nite set) have a term in which cannot be simpli ed in R+ . High k` j,n[k, `] Y[k] = f[k](X[k] − 1). (6) order polynomial functions have bivariate monomial terms involving λ η in R+ . Functions of type sin cos Since we have assumed that are stationary with k ` j,n[k, `] , (X[k])k∈Z and autocorrelation function R satisfy f[k]f[`] = g1[k + `] + g2[k − `] and in this case, EX[k] = µ0 X[k, `] = R then: the contribution of g1 implies non-stationarity as in the E [X[k]X[`]] , X[k − `], exponential case given above, etc. • The mean of Y[k] is An appealing case of stationarity sequence + corre- Cj,n (7) EY[k] = f[k](µ0 − 1). sponds to a constant function f associated with a random sequence X with unit mean: 1 2 Equalities hold in L (R) sense in these expansions. 4

−3 Remark 3 (Stationarity): When µ0 = 1 and f is a • a \small" value is a value close to 1 (10 and constant function: , then + and 3 have the same signi cance in terms of absolute f[k] = f0 ECj,n[k] = 0 10 furthermore, we derive R+ R+ R+ proportion, j,n[k, `] = j,n[k − `] = j,n[m] with: • a missing value must be replaced by 1, • shrinkage forces to 1, the coecients that are close to 2 π 1. f j R+ 0 2 i2 mω d (13) j,n[m] = γX0 (ω) |Hj,n(ω)| e ω The multiplicative algebra implies de ning the support 2π −π Z of the sequence x as the sub-sequence composed with where γ 0 denotes the spectrum of the random sequence X elements that are di erent from 1. We will thus use X0 = X − 1. the standard terminologies of nite/in nite supports with R −imω γX0 (ω) = ( X[m] − 1) e . respect to the above remark. When such a sequence x is m∈Z in nite, we will assume that log log X (x) = (( x[k])k∈Z) ∈ This case of a constant function observed in a multi- 2 f ` (Z). plicative noise represents homogeneous area observation in When considering a scalar sequence (impulse response practical SAR applications. This case is the sole favorable of a lter for instance) where for h = (h[`])`∈Z h[`] ∈ R scenario for standard additive wavelets when the challenge every ` ∈ Z, then we will keep the standard terminology is to simplify the multiplicative model fX. related to support de nition from non-zero elements (non- null real numbers). Due to the non-stationarity of + in general (except Cj,n The multiplicative convolution de ned below is based few cases such as that of Remark 3), modeling or es- on this binary operation (notation x × y , xy for x, y ∈ timating additive wavelet coecients of a multiplicative + a R ) and real scalar power operations (notation a∧x , x model is not an easy task. The following shows that mul- + for x ∈ R and a ∈ R). tiplicative implementations of wavelets highlight desirable De nition 1 (Multiplicative convolution): Let stochasticity properties for simplifying model . h = fX denote the impulse response of a digital lter. (h[`])`∈Z We de ne the multiplicative convolution of and on III. multiplicative wavelet implementation - x h the vector space + as: Statistical properties on multiplicative (R , ×, ∧) h[k−`] observation models y[k] = x > h[k] , (x[`]) A multiplicative wavelet transform (multiplicative lin- `∈Z Y h[`] earity where W distributes over `×' operation), when = (x[k − `]) , h > x[k]. (14) `∈ applied on model given by Eq. (1), must satisfy: YZ One can remark that, in contrast to the standard convo- Wy = (Wf) × (WX) . (♠) lution operation on (R, +, ×) sequences, discrete sequence This transform is derived hereafter from multiplicative h plays a non-commutative scalar role with respect to x convolution operator. since the external operation `power' used in Eq. (14) is Note that performing a geometric wavelet decompo- not commutative. This justi es the second , in Eq. (14): sition satisfying model amounts to apply a log- (♠) the equality x > h = h > x applies index-wise on the transform on the input data, perform a standard wavelet multiplicative convolution, given that the scalar sequence transform and apply an exponential transform on the h operates to the power of elements of x, by de nition. wavelet coecients of this standard transform. We con- 2 If h ∈ ` (Z), then x > h[k] exists and is nite for sider hereafter the description of such operations by di- 2 almost every k since we have assumed that log(x) ∈ ` (Z). rectly embedding wavelet operators in a multiplicative Depending on h, Eq. (14) makes the computation of algebra with binary internal multiplication and external multiplicative approximations and details of the input power operation. data x possible. An example of multiplicative approximation is obtained A. Multiplicative (geometric) convolution by the so-called geometric mean of a nite sequence

The binary operation considered in the following is the {x1, x2, . . . , xN}: multiplication (× symbol) over positive real numbers +. N R √ 1/N N (15) Consider a data sequence , with + y = x1x2 ··· xN = x` . x = (x[`])`∈Z x[`] ∈ R `=1 for every ` ∈ Z. Since this sequence represents a multi- Y plicative phenomenon, then This geometric mean is associated with an N-length Haar- type approximation lter • \zero" or \nothing" or \no change" corresponds to

the identity element \1" h0[k] = ν for k = 1, 2, . . . , N. (16) 5

Multiplicative approximations computed by using the l- j + 1. As in the standard additive formulation given in ter h0 (low pass lter) will thus be called geometric Section II-B (see Eq. (3)), di erent wavelet decomposi- approximations. Filter h0 can be associated with a Haar- tion schemes (orthogonal wavelets, stationary wavelets, type detail lter: adapted wavelet packets, etc.) and perfect reconstructions can be obtained from Eqs. (20) and (21) respectively. h [k] = (−1)k−1ν for every k = 1, 2, . . . , N (17) 1 This geometric transform is nothing but the formaliza- which performs geometric di erencing (ratio involving tion of \log transform of data before wavelets and exp several consecutive elements), where constant ν > 0 is transform of coecients after wavelets" in terms of an xed so as to impose paraunitarity for the corresponding algebraic inference where implementation implies √ pair of lters ( for standard wavelet lters ν = 2/2 Haar • executing environment (×, ∧) for every call of envi- when ). For the sake of standardizing terminology, N = 2 ronment (+, ×) and the multiplicative convolution of Eq. (14) will be called • replacing calls of `0s' by `1s' (decimation corresponds geometric convolution whatever the lter used and the to replacing one coecient out of two by the number same holds true for the wavelet transform de ned below. 1). In the following, we will address the statistical properties B. Multiplicative (geometric) wavelet decomposition of the coecients issued from Eq. (20). In the following, we consider the same paraunitary wavelet lters 2 2 as in Section II-B. (h0, h1) ∈ ` (Z)×` (Z) Let C. Statistical properties of the geometric wavelet transform on Eq. (1) h[k] = h[−k]. The geometric wavelet decomposition × of Eq. De ne the wavelet decomposition of x with respect to W × the geometric convolution (geometric wavelet decompo- (20) distributes over the product fX: W [fX] = × × × sition) by: (W f)(W X). Thus, in model (♠), with W = W de ned by Eq. (20), noise contribution is × where c [k] = x h [2k], (18) W X 1,0 > 0 we have assumed that is a stationary X = (X[k])k∈Z (19) × c1,1[k] = x > h1[2k] unit-mean random sequence. Assuming sparsity of W on , the focus of this section is establishing the statistical and, recursively, for ∈ {0, 1} (wavelet packet splitting f × formalism described in [32]): properties of W X. The geometric wavelet coecients of the decomposition × cj+1,2n+[k] = cj,n h[2k]. (20) of on subspace will be denoted (we > X Wj,n (Cj,n)j,n In the decomposition given by Eq. (20) above, sequence assume that this stochastic sequence is well de ned in the following). Note that if c represents: Cj+1,2n+[k] = Cj,n > h[2k] j+1,2n+ where is a stationary sequence, then geometric approximation of when , Cj,n Cj+1,2n+ • cj,n = 0 is also stationary. Since is assumed to be geometric di erencing (details) of when . C0,0 = X • cj,n = 1 stationary, we derive that all geometric wavelet sequences The level j = 0 coecients represent the input sequence . x C are stationary for j 0 and n ∈ {0, 1, . . . , 2j − 1}. As in the standard case, the above wavelet packet splitting j,n > Let Y = log X. We assume hereafter that Y is a second- is associated to a wavelet decomposition when subspace order random process, continuous in quadratic mean. Let splitting concerns only approximations (c ) . j,0 j>1 log × . Note that and are stationary Dj,n = Cj,n Y Dj,n Proposition 1 (Geometric wavelet reconstruction): sequences. Assume that EY[k] = 0 for every k ∈ Z. Then for every . We have: EDj,n[k] = 0 k ∈ Z Let R [m] = R [k−`] = [Y[k]Y[`]] be the autocorre- c [k] = (c h [k]) × (c h [k]) , (21) Y Y E j,n j+1,2n > 0 j+1,2n+1 > 1 lation function of Y, where the rst equality above holds where true for any pair (k, `) ∈ Z × Z such that m = ±|k − `|. u[k] if = 0, u[2k + ] = (22) Proposition 2 below derives the autocorrelation function 1 if = 1. R of the log-scaled geometric wavelet coecient . Dj,n Dj,n We assume that h[p − 2k]h[q − 2`]RD [p, q] Proof: The proof is a direct consequence of the q∈Z j,n exists for every j 0 and n ∈ {0, 1, . . . , 2j − 1}. expansion of the right hand side of Eq. (21), by taking >P into account Eq. (20) and the paraunitary condition which Proposition 2 (Autocorrelation Function of Dj,n): imposes h[`]h[` − 2k] = δ[k]. Assume that R has a spectrum (power spectral density) `∈Z Y

PropositionP 1 represents the reconstruction of the level- −imω γY(ω) = RY[m]e j-wavelet-coecients from the coecients located at level m∈ XZ 6 and that is bounded. Denote by , the spectrum Eq. (27), the equivalent lter of this sequence can be γY γDj,n of Dj,n: rewritten in the form: j−1 2 R −imω (23) Haar 2 sinc(2 ω) γDj,n (ω) = Dj,n [m]e . j (28) Hj,0 (ω) = 2 −1 , m∈ sinc(2 ω) XZ We have, for j 0, n ∈ {0, 1, . . . , 2j−1} and ∈ {0, 1}: where sinc denotes the cardinal sine function, sincω = > Haar π sin ω/ω. The autocorrelation R of the corresponding 1 2 Dj,0 R [m]= H (ω) γ (ω)e2imωdω, (24) geometric wavelet coecients is then: Dj+1,2n+ c Dj,n 2π −π Z 2 2j πsinc(2j−1ω) where γD = γY. Haar j 0,0 R [m]= γY(ω)cos 2 mω dω. (29) Dj,0 π sinc(2−1ω) Proof: See Appendix A. Z0 Proposition 4 (Limit Autocorrelation Function): By taking into account that sequence Dj,n issues from a lter bank (H ) (low-pass when = 0 and high- lim RHaar (30) ` `=1,2,...,j ` D [m] = γY(0)δ[m] j + j,0 pass when ` = 1) and has the equivalent representation given by Eq. (5), we derive recursively from Eq. (24): Proof: See→ Appendix∞ C. π j Proposition 4 highlights an asymptotic decorrelation R 1 2 2 imω d (25) Dj,n [m] = |Hj,n(ω)| γY(ω)e ω. 2π −π property with j. This property can be extended by con- Z sidering di erent paraunitary lters. For instance, when Eq. (25) governs the behavior of the autocorrelation of considering the N-length Haar-type approximation lter D . From this equation, decorrelating geometric wavelet j,n h0 and detail lter h1 given by Eqs. (16) and (17), the coecients involves selecting wavelet lters such that equivalent wavelet lter is quantity j sin `−2 2 π 2 j (2 Nω) 1 2 (31) cos j d (26) |Hj,n(ω)| = 2 −1 . |Hj,n(ω)| γY(ω) 2 mω ω sin(2 (ω + `π)) 2π `=1 Z−π Y It follows that the corresponding autocorrelation R is behaves approximately like Dirac δ[m]. This is strongly Dj,n linked to the shape of and can be achieved either by: γY R Dj,n [m] (i) choosing a sequence of wavelet lters such that func- j π j sin `−2 2 2 2 (2 Nω) j tion |Hj,n(ω)| γY(ω) is approximately constant or cos d = −1 γY(ω) 2 mω ω, π sin(2 (ω + `π)) (ii) seeking asymptotic decorrelation with j (provided 0 `=1 Z Y that it applies). j 2 π sin −j+`−2 1 (2 Nω) ω cos d Item is parametric in the sense that it relates to = −j−1 π γY j mω ω (i) π sin(2 ω + ` ) 2 0 `=1 2 adapted wavelet selection for decorrelating Y. Item (ii) Z Y (non-parametric) exploits properties of recursive convolu- which tends to γY(0)δ[m] when j tends to ininity, for the tions. For instance, if we consider the Haar wavelet lters approximation path (n = 0). (used below for illustrations), we can derive: This decorrelation property can also be extended by considering di erent paths, lters and wavelet packet Proposition 3 (Haar equivalent wavelet lter se- splitting schemes, as done in [33] for additive noise and quence Haar): A sequence has equivalent Hj,n (h` )`=1,2,...,j arithmetic wavelet transforms. lter: j IV. Change detection: parsimony of the Haar 2 j 2 `−2 π H (ω) = 2 cos 2 ω + ` . (27) j,n 2 signal-versus-noise separation makes relevant `=1 Y basic dissimilarity operators Proof: See Appendix B. A. Change information perceived from arithmetic and In the usual wavelet splitting scheme, only approxima- geometric di erencing tion coecients are decomposed again (the shift parameter From now on, we will use the terminologies of Arith- n ∈ {0, 1}). This implies ltering sequences with the form metic Discrete Wavelet Transform (ADWT) and Geo-   metric Discrete Wavelet Transform (GDWT) to point out, respectively, the additive and multiplicative imple- h , h ,..., h , h  0 0 0 j+1  mentations given by Eq. (4) and Eq. (20). j times j+1∈{0,1} When analyzing the multiplicative interactions in y at decomposition| level{z j + 1}. Consider a j-length approxi- given by Eq. (1), Section II has shown that ADWT coef- mation sequence Haar of Haar type. Then from cients will be non-stationary in general whereas GDWT h0 `=1,2,...,j 7

Fig. 1. Pixel time series with 4 change dates, its noisy speckled version, as well as absolute change information from arithmetic di erencing (corresponds to Haar level-1 ADWT details) and ratioing (geometric di erencing, corresponds to Haar level-1 non sub-sampled GDWT details). The ratio-data have been re-scaled logarithmically so as to make comparison on a single display possible.

Fig. 2. Pixel time series with 4 change dates, its noisy speckled version, as well as absolute level 1 and 2 ratioing (non sub-sampled GDWT). coecients are stationary and, in addition, GDWT has a change evaluation. The `main di erence' between these noise decorrelation property (see Section III). basic arithmetic and the geometric di erencing operators Let us consider the level j = 1 details obtained by using on the observation model of Eq. (1) is illustrated in Figure Haar lters with N = 2 in Eq. (17) (one vanishing moment 1. wavelet). These details are proportional to: As it can be seen in Figure 1, change information

• yk − yk−1 for ADWT (arithmetic di erencing of can be retrieved without e ort with the basic geometric (R, +) elements), di erencing (sparsity of change information, in addition • yk/yk−1 for GDWT (geometric di erencing with noise decorrelation) whereas a non-intuitive post- + ratioing of (R , ×) elements). processing needs to be performed for observing the same These di erencing operators are the basic ones used⇒ in changes for the arithmetic di erencing, due to strong 8 correlations induced by f(X − 1). Some examples of level- apply through sigmoid shrinkage functions [30]. These 1 generalized wavelet based ratioing (geometric wavelet functions have the following form: di erencing) are given below. sgn(x)(|x| − t)+ δt,θ,λ(x) = , (34) • Case of a biorthogonal wavelet with 2 vanishing mo- −ζ(θ) |x| −1 1 + e ( λ ) ments: y0.35y0.35 where k−1 k+1 . (32) 10 sin θ y0.70 ζ(θ) = (35) k 2 cos θ − sin θ Case of a box spline wavelet with 2 vanishing mo- • with sgn(x) = 1 (resp. -1) if x > 0 (resp. x < 0) and, ments: (resp. 0) if (resp. ). (x)+ = x x > 0 x < 0 y0.6875y0.21875y0.03125 k−1 k−2 k−3 (33) Note that since the wavelet transform is performed with 0.6875 0.21875 0.03125 . yk yk+1 yk+2 respect to the time axis, a geometric wavelet based-change- image contains: Depending on the sharpness of the change transitions, it might be relevant to consider multi-level changes. For • either a bidate change information (level j = 1 detail instance, the transitions between temporal observations of coecients when using a lter h with 2 non-zero coecients such as Haar lters) Figure 1 being linear (non-instantaneous), level j = 2 Haar geometric details are shown to discriminate well change • or a multidate change information when: transitions of this observation in Figure 2. – j > 2, whatever the lter used, provided that the In the rest of the paper, we consider only the geometric lter has at least 2 non-zero coecients, wavelet framework for and easy change enhancement on – j > 1, when the lter used has more than 2 non- the time axis (sparsity of the geometric temporal details zero coecients (see for instance Eqs. (32) and in decorrelated noise). (33)). For highlighting the multitemporal changes in their spatio- temporal context, the above sigmoid shrinkage function B. Sigmoid enhancement of change information will be applied hereafter on spatial blocks of wavelet based

Consider the synthetic image time series P = temporally di erenced data. For a pixel intensity Zm,q(k) given by Figure 3-[Row 1], where pertaining to a log-scaled change-image, the shrinkage (Pm,q(tk))k=1,2,3,4 m, q refer to spatial variables and denotes proposed is de ned as: (1 6 m, q 6 2048) tk the time variable. Figure 3-[Row 2] provides change infor- sgn (Zm,q(k))(|Zm,q(k)| − t)+ (36) mation (binary masks) between the di erent dates, with δt,θ,λ(Zm,q(k)) = ! ||VZ (k)|| −ζ(θ) m,q 2 −1 denoting changes in-between dates and λ M(tk, tk+1) tk 1 + e t and M◦(t , t , t , t ) the total amount of changes. k+1 1 2 3 4 where V is a vector with the form V = When applying a geometric wavelet transform Zm,q(k) Zm,q(k) {Z (k), m = m − , . . . , m + , q = q − ν , . . . , q + q } W×[P (•)] with respect to the time axis solely m,q 0 0 0 0 m,q and , ν are natural numbers chosen suciently small (in practice, this assumes an accurate image registration), 0 0 (spatial neighborhood of the detail pixel (Zm,q(k)), with then the detail subbands2 C× [P ](•) j,n m,q || · || denoting the `2 norm. This penalized shrinkage then 16m,q62048 2 displayed as images in Figure 3-[Row 3] provide spatio- consists in: temporal multiscale change information. These subbands • forcing to zero all temporal log-scaled geometric are hereafter called change-images. As expected wavelet change-image pixel with spatial neighborhood (consequence of Section III) , these spatio-temporal norm smaller than the rst threshold t, geometric change-images show both sparsity of change • attenuating temporal log-scaled geometric wavelet information (changes are rare and signi cant when change-image pixel with large spatial neighborhood present) and stationarity/decorrelation for speckle norm thanks to an attenuation degree θ and a second noise in homogeneous areas with no temporal change threshold λ. information. Change information processing is thus spatio-temporal The change enhancement proposed below involves using due to the presence of variable k (geometric temporal a spatio-temporal block shrinkage for smoothly penalizing change-image) and the variations of spatial variables m, q. weak changes in pixel intensities. This shrinkage will

2Since J = 2 for this example, we have, in an orthogonal GDWT, C. Quantitative change evaluation 3 multiscale subbands due to decimation steps (2 subbands at level In [8], changes are analyzed by using shrinking arith- j = 1 and 1 subband at level j = J = 2). However, we consider displaying all subbands (no decimation) excepted the border ones in metic wavelet coecients of (standard) log-ratio images order to highlight di erent change information. (we recall that the standard log-ratio operator is described 9

I(t1) I(t2) I(t3) I(t4)

◦ M(t1, t2) M(t2, t3) M(t3, t4) M (t1, t2, t3, t4)

× × × × C1,1[I](t2) C1,1[I](t3) C1,1[I](t4) C2,1[I](t2)

AWaveShrink × AWaveShrink × AWaveShrink × AWaveShrink × C1,1[I](t2) C1,1[I](t3) C1,1[I](t4) C2,1[I](t2)

SigShrink × SigShrink × SigShrink × SigShrink × C1,1[P](t2) C1,1[P](t3) C1,1[P](t4) C2,1[P](t2)

Fig. 3. Row 1: large, small and tiny elliptical structures with di erent shapes and overlaps, observed in synthetic speckle noise. Row 2: the binary date-to-date and the total M◦ change maps (true changes). Row 3: Geometric wavelet change-images of time series given in Row 1. Row 4: [AWaveShrink] arithmetic wavelet based regularization for the change-images given in Row 3. Row 5: [SigShrink] direct bloc sigmoid shrinkage from Eq. (36) (without additional arithmetic wavelet transform) for the change-images given in Row 3. 10

minimal to maximal change-image values, see Figure 4) measurements illustrate the advantages and limitations of both approaches:

• for more than 20% of false positives (high toleration of false positives!), then AWaveShrink is slightly preferable than SigShrink,

• for less than 20% of false positives, the SigShrink probability of detection is higher than that of AWaveShrink, . . . , for example, at 5% of false positives, AWaveShrink yields 60% of true positives whereas SigShrink yields 80% of true positives. Thus, for change information enhancement, a direct bloc sigmoid shrinkage (SigShrink) is preferable than an arithmetic wavelet based regularization (AWaveShrink), espe- cially when we have no a priori on the sizes and the types of changes (case of glacier surface monitoring addressed hereafter).

Fig. 4. ROC curves for: 1) AWaveShrink (arithmetic wavelet regular- V. Geometric wavelets for joint change ization of geometric wavelet details) and 2) SigShrink (bloc sigmoid detection and regularization of polarimetric shrinkage of geometric wavelet details). The ROC curves have been ◦ SAR image time series computed on the total amount of changes M (t1, t2, t3, t4) occurring in the time series of images given by Figure 3. A. Bloc sigmoid shrinkage of polarimetry vectors/matrices as the level 1 geometric Haar operator in the formalism We consider a PolSAR scattering/covariance image time presented in this paper). First, the approach of [8] can be series P = Puv (k), where: extended by considering, not only the standard log-ratio m,q , stands for Horizontal / operator, but also generalized log-ratio operators (several • (u, v) ∈ {H, V} × {H, V} H/V Vertical respectively and levels of geometric Haar decomposition for instance). This refer to (spatial, time) variables, with extension, consisting of an Arithmetic Wavelet transform • ([m, q], k) 1 6 m M, 1 q Q and 1 k K. and Shrinkage of geometric wavelet details C× [P] will 6 6 6 6 6 j,n We have uv uv uv where denotes be referred as AWaveShrink × in the following Pm,q(k) = Im,q(k)Θm,q(k) I Cj,n[P] tests. Change penalization from AWaveShrink is provided moduli and Θ stands for unit-norm complex exponential in Figure 3-[Row 4], when the shrinkage is performed by phase terms. The temporal geometric wavelet transform is chosen to apply on uv : the transform is performed using sigmoid based functions, for the sake of unbiased Im,q(•) to decompose series uv with respect to the time comparison (note that hard and soft shrinkages are partic- Im,q(k) variable solely. Terms uv are stored and added ular cases of sigmoid shrinkage functions). AWaveShrink k Θm,q(k) change regularization appears suitable mainly for large- after regularization of moduli time series I. size abrupt changes whereas small target change informa- Section IV has shown that spatio-temporal block shrink- tion tends to be blurred by the arithmetic wavelet based age of geometric change-images makes change enhance- regularization. ment possible. For polarimetry images, the geometric We then apply the block Sigmoid Shrinkage (notation wavelet transform is chosen to be separable with respect SigShrink × ) given by Eq. (36) directly on the to polarimetry channels whereas the shrinkage of Eq. (36) Cj,n[P] change-images of Figure 3-[Row 3]. This SigShrink oper- can be either: ator yields change-images of Figure 3-[Row 5]. As it can • scalar be seen in Figure 3-[Row 5], a direct sigmoid shrinkage sgn uv uv uv (Zm,q(k))(|Zm,q(k)| − t)+ δ (Z (k)) = (37) less impacts the sizes of small structures because it does t,θ,λ m,q   VZuv (k) m,q 2 not involve the smoothing e ect intrinsic to wavelet based −ζ(θ) λ −1 regularization (compare Figure 3-[Row 5] with the change 1 + e masks of Figure 3-[Row 2]). where uv is a pixel moduli pertaining to a log- Zm,q(k) Finally, a comparison based on Receiver Operating scaled PolSAR change-image. p Characteristic (ROC, probability of detection versus prob- • or vectorial where neighborhood V consists of ` ability of false alarm for threshold values ranging from norms of PolSAR covariance moduli vector/matrix 11

change-images: Sentinel constellation of the European Space Agency

uv uv (ESA) is a source of highly resolved spatio-temporal data. sgn(Z (k))(|Z (k)| − t)+ δ (Zuv (k)) = m,q m,q (38) The data considered in this section corresponds to an area t,θ,λ m,q −ζ(θ)U(Z (k),λ) 1 + e m,q covering the glaciers Mer de Glace and Argentiere, in the where mountainous Chamonix-Mont-Blanc site, in France.

  Since the launch of Sentinel-1A in April 2014, a time V Zuv (k) series of PolSAR data over this test site has been collected:  ( m,q )(u,v)∈{H,V}2   p 2  U(Zm,q(k), λ) =  − 1 the test dataset is described in Figure 5 (images are avail-  λ    able free of charge from ESA repository, co-registration has been made thanks to a xed corner re ector). This The time series regularization principle is then to time series, denoted P, is composed of 11 dual PolSAR IW use shrunken geometric wavelet change-images for recon- level-1 Single Look Complex (SLC) SAR images acquired structing time series with sharp pixel change transitions. in descending pass from November 15, 2014 to March 15, This is the joint parsimonious change evaluation and 2015 with 12 days sampling period. A sample image, P2, time series regularization proposed in this paper. We will is displayed in Figure 5 with a Pauli color rendering in use the following parameters for block sigmoid shrinkage: order to enhance dual-polarimetry information. Di erent types of changes can occur on this glacier p = 1, parameter t0 is the universal threshold of [35], site due to the long period of observation: for instance θ = π/5 and λ ∈ {λ1, λ2}, where λ1 = t0, λ2 = 2t0. The snow fall, snow accumulation in speci c areas, serac falls, sigmoid shrinkage operator is denoted Sλ. avalanches, human activities, etc. It is worth noticing Note that when 2J PolSAR image samples are available, then, by restricting the wavelet transform to the time that a pixel-per-pixel and date-per-date search is possible, see for instance [36]. However, this is with very high axis and by performing a level J decomposition, we have computational cost, in comparison with the geometric to take into account the levels j = 1, 2, . . . , J change- temporal wavelet shrinkage proposed below. Speci cally, images, with 2J−j change-images at decomposition level we consider both scalar sigmoid shrinkage (polarimetry j 6 J (decimation in order to reject redundant change information). channels are considered independently for building VZ in The overall computational complexity depends on 2 Eq. (36)) and vector sigmoid shrinkage (VZ is a sequence main factors and remains reasonable since it relies only of `p-norms of PolSAR channels) for comparison purpose. on basic operations (does not involve curves tting, it- Change information from geometric wavelets: erative optimization procedures or maximum likelihood Due to the limited size of the paper, only one geometric solutions): wavelet change-image is displayed in Figure 6-Top. As • applying a temporal wavelet transform (M×Q×O(K) expected, the details look stochastic, except in few areas. for the orthogonal transform and M × Q × O(K2) for Some areas where signi cant changes appear in Figure 6- non-decimated/stationary versions of the transform) Top are indicated on the photographic map of Figure 6- on the logarithms of each moduli of the input time Bottom: series and using an inverse wavelet transform (same • a serac fall area on the Argentiereglacier (blue-line), complexity as the decomposition); • an accumulation area near the glacier of Bossons • applying a pixelwise shrinkage function involving (yellow-dashed), sums and exponentiations on a small spatial change- • the borders of glacier Mer de Glace (magenta-dotted). image pixel neighborhood (3 ). ×3 Changes detected on the borders of Mer de glace glacier Note also that the method is highly parallelizable since can be due to co-registration errors. However, since Ar- the sole recursion is linked to a single axis: the temporal gentiereglacier borders do not respond equivalently, this axis concerned by the wavelet transform. suspicious behavior needs to be confronted with ground truth because these change responses can reveal other B. Application to Sentinel-1A dual-polarimetric SAR phenomena such as glacier and moraine constriction. image time series Change enhancement: The geometric temporal wavelet shrinkage for both Scalar sigmoid shrinkage (polarimetry channels are con- change information enhancement and regularization aims sidered independently) of Figure 6-Top yields the change- at simplifying the analysis of long time series of SAR image given by Figure 7-Top whereas vector sigmoid images. Indeed, the challenge in exploiting such huge data shrinkage leads to the change-image of Figure 7-Bottom. is in dimensionality handling and requires methods that One can notice that the latter enhances more accurately have very low computational load. polarimetry change information than the former. 12

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Appendix A j Proof of Proposition 2 Haar −j/2 −iω (44) Hj,n (ω) = 2 1 + (1 − 2`)e . By considering the log of × denoted by `=1 Cj+1,2n+ Y Thus, Dj+1,2n+, we are concerned by an additive combinations of log × . Dj,n = Cj,n j Haar 2 cos `−1 (45) The autocorrelation functions Hj,n (ω) = 1 + (1 − 2`) (2 ω) . `=1 R [k, `] = D [k]D [`] Y Dj+1,2n+ E j+1,2n+ j+1,2n+ `−1 The proof follows by noting that (1 − 2`) cos(2 ω) = and `−1 cos(2 ω + `π) after some straightforward simpli ca- R tions by using trigonometry double angle properties. Dj,n [k, `] = EDj,n[k]Dj,n[`] 14

P = {P(t1), P(t2),... P(t11)} Data: SAR, Single Look Complex t1 = 2014 − 11 − 15 Revisit time: 12 days t11 = 2015 − 03 − 15 Orbit pass: descending Dual PolSAR IW level-1 Resolution: 3.5 × 20 m2

P2 = P(t2) with t2 = 2014 − 11 − 27

Fig. 5. Sentinel-1A dual PolSAR image of the Chamonix-Mont-Blanc test site.

Appendix C In this respect, we derive

Proof of Proposition 4 Haar R [m] ||γY|| From a change of variable in Eq. (29), we obtain Dj,0 6 so that, from the Lebesgue dominated convergence theo- 2jπ 2 ∞ Haar 1 sinc(ω/2) ω R [m] = γY( ) cos mω dω. rem, Dj,0 sinc j+1 j π 0 (ω/2 ) 2 Z 1 + First, we observe that: lim RHaar[m] = γ (0) (sinc(ω/2))2 cos mω dω Dj,0 Y j j + π 2 π 2 ! 0 ∞ Haar 1 sinc(ω/2) Z R [m] ||γY|| × dω . Proposition 4 then follows by noting that Dj,0 6 sinc j+1 → ∞ π 0 (ω/2 ) Z + and, furthermore, we∞ have (sinc(ω/2))2 cos mω dω = πδ[m] 0 ∞ 1 + sinc(ω/2) 2 Z dω = 1. π sinc(ω/2j+1) Z0 ∞ 15

D3,1[P]/

Airborne photography

Fig. 6. Top: 1 sample geometric change-image of PolSAR time series P described in Figure 5. Bottom: airborne photography [ c RGD 73-74] showing Chamonix urban valley (red dash-dotted), glaciers (Argentiere, Mer de Glace, Bossons) and localization of signi cant changes. 16

Ss D3,1[P]/

Sv D3,1[P]/

Fig. 7. Scalar (Top, Ss ) and vector (Bottom, Sv ) sigmoid shrinkages of the geometric change-image given in Figure 6. D3,1 D3,1 D3,1 17

s s Pb2 = Pb (t2) t2 = 2014 − 11 − 27

v v Pb2 = Pb (t2) t2 = 2014 − 11 − 27

Fig. 8. Scalar (Top) and vector (Bottom) geometric wavelet regularization of Sentinel-1A PolSAR time series P described in Figure 5.