Wavelet Operators and Multiplicative Observation Models—Application to SAR Image Time-Series Analysis

Abstract

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