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Wavelet Operators and Multiplicative Observation Models 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 To cite this version: 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 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. 1 Wavelet Operators and Multiplicative Observation Models - Application to Change-Enhanced Regularization of SAR Image Time Series Abdourrahmane M. Atto1;∗, Emmanuel Trouve1, Jean-Marie Nicolas2, Thu-Trang Le^1 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])k2Z 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
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