Wavelet Operators and Multiplicative Observation Models - Application to Change-Enhanced Regularization of SAR Image Time Series

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 affect the 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. The 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, the 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 bloc sigmoid shrinkage. Finally, a regularized time series is reconstructed from the sigmoid shrunken change-images. An application of this method highlights the relevancy of the method for change detection and regularization of SENTINEL-1A dual-polarimetric image time series over Chamonix-Mont-Blanc test site

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

International audienceThis paper first provides statistical properties of wavelet operators when the observation model can be seen as the product of a deterministic piece-wise 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 decorre-lation of speckle wavelet coefficients. The 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, the 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

3D Wavelet Transformation for Visual Data Coding With Spatio and Temporal Scalability as Quality Artifacts: Current State Of The Art

Several techniques based on the three–dimensional (3-D) discrete cosine transform (DCT) have been proposed for visual data coding. These techniques fail to provide coding coupled with quality and resolution scalability, which is a significant drawback for contextual domains, such decease diagnosis, satellite image analysis. This paper gives an overview of several state-of-the-art 3-D wavelet coders that do meet these requirements and mainly investigates various types of compression techniques those exists, and putting it all together for a conclusion on further research scope

Iterative Bias Reduction Multivariate Smoothing in R: The ibr Package

In multivariate nonparametric analysis curse of dimensionality forces one to use large smoothing parameters. This leads to a biased smoother. Instead of focusing on optimally selecting the smoothing parameter, we fix it to some reasonably large value to ensure an over-smoothing of the data. The resulting base smoother has a small variance but a substantial bias. In this paper, we propose an R package named ibr to iteratively correct the initial bias of the (base) estimator by an estimate of the bias obtained by smoothing the residuals. After a brief description of iterated bias reduction smoothers, we examine the base smoothers implemented in the package: Nadaraya-Watson kernel smoothers, Duchon splines smoothers and their low rank counterparts. Then, we explain the stopping rules available in the package and their implementation. Finally we illustrate the package on two examples: a toy example in R2 and the original Los Angeles ozone dataset

A Course in Harmonic Analysis

These notes were written to accompany the courses Math 6461 and Math 6462 (Harmonic Analysis I and II) at Missouri University of Science & Technology during the 2018-2019 academic year. The goal of these notes is to provide an introduction to a range of topics and techniques in harmonic analysis, covering material that is interesting not only to students of pure mathematics, but also to those interested in applications in computer science, engineering, physics, and so on

Discrete Wavelet Transforms

The discrete wavelet transform (DWT) algorithms have a firm position in processing of signals in several areas of research and industry. As DWT provides both octave-scale frequency and spatial timing of the analyzed signal, it is constantly used to solve and treat more and more advanced problems. The present book: Discrete Wavelet Transforms: Algorithms and Applications reviews the recent progress in discrete wavelet transform algorithms and applications. The book covers a wide range of methods (e.g. lifting, shift invariance, multi-scale analysis) for constructing DWTs. The book chapters are organized into four major parts. Part I describes the progress in hardware implementations of the DWT algorithms. Applications include multitone modulation for ADSL and equalization techniques, a scalable architecture for FPGA-implementation, lifting based algorithm for VLSI implementation, comparison between DWT and FFT based OFDM and modified SPIHT codec. Part II addresses image processing algorithms such as multiresolution approach for edge detection, low bit rate image compression, low complexity implementation of CQF wavelets and compression of multi-component images. Part III focuses watermaking DWT algorithms. Finally, Part IV describes shift invariant DWTs, DC lossless property, DWT based analysis and estimation of colored noise and an application of the wavelet Galerkin method. The chapters of the present book consist of both tutorial and highly advanced material. Therefore, the book is intended to be a reference text for graduate students and researchers to obtain state-of-the-art knowledge on specific applications

On the Structure and Dynamics of Sheared and Rotating Turbulence: Direct Numerical Simulations and Wavelet Based Coherent Vortex Extraction

The influence of rotation on the structure and dynamics of sheared turbulence is investigated using a series of direct numerical simulations. Five cases are considered: turbulent shear flow without rotation, with moderate rotation, and with strong rotation, where the rotation configuration is either parallel or antiparallel. For moderate rotation rates an antiparallel configuration increases the growth of the turbulent kinetic energy, while the parallel case reduces the growth as compared to the nonrotating case. For strong rotation rates decay of the energy is observed, linear effects dominate the flow, and the vorticity probability density functions tend to become Gaussian. Visualizations of vorticity show that the inclination angle of the vortical structures depends on the rotation rate and orientation. Coherent vortex extraction, based on the orthogonal wavelet decomposition of vorticity, is applied to split the flow into coherent and incoherent parts. It was found that the coherent part preserves the vortical structures using only a few percent of the degrees of freedom. The incoherent part was found to be structureless and of mainly dissipative nature. With increasing rotation rates, the number of wavelet modes representing the coherent vortices decreases, indicating an increased coherency of the flow. Restarting the direct numerical simulation with the filtered fields confirms that the coherent component preserves the temporal dynamics of the total flow, while the incoherent component is of dissipative nature

A generalized wavelet extrema representation

The wavelet extrema representation originated by Stephane Mallat is a unique framework for low-level and intermediate-level (feature) processing. In this paper, we present a new form of wavelet extrema representation generalizing Mallat`s original work. The generalized wavelet extrema representation is a feature-based multiscale representation. For a particular choice of wavelet, our scheme can be interpreted as representing a signal or image by its edges, and peaks and valleys at multiple scales. Such a representation is shown to be stable -- the original signal or image can be reconstructed with very good quality. It is further shown that a signal or image can be modeled as piecewise monotonic, with all turning points between monotonic segments given by the wavelet extrema. A new projection operator is introduced to enforce piecewise inonotonicity of a signal in its reconstruction. This leads to an enhancement to previously developed algorithms in preventing artifacts in reconstructed signal