Combining local regularity estimation and total variation optimization for scale-free texture segmentation

Texture segmentation constitutes a standard image processing task, crucial to many applications. The present contribution focuses on the particular subset of scale-free textures and its originality resides in the combination of three key ingredients: First, texture characterization relies on the concept of local regularity ; Second, estimation of local regularity is based on new multiscale quantities referred to as wavelet leaders ; Third, segmentation from local regularity faces a fundamental bias variance trade-off: In nature, local regularity estimation shows high variability that impairs the detection of changes, while a posteriori smoothing of regularity estimates precludes from locating correctly changes. Instead, the present contribution proposes several variational problem formulations based on total variation and proximal resolutions that effectively circumvent this trade-off. Estimation and segmentation performance for the proposed procedures are quantified and compared on synthetic as well as on real-world textures

Time-Scale Block Bootstrap tests for non Gaussian finite variance self-similar processes with stationary increments

Scaling analysis is nowadays becoming a standard tool in statistical signal processing. It mostly consists of estimating scaling attributes which in turns are involved in standard tasks such as detection, identification or classification. Recently, we proposed that confidence interval or hypothesis test design for scaling analysis could be based on non parametric bootstrap approaches. We showed that such procedures are efficient to decide whether data are better modeled with Gaussian fractional Brownian motion or with multifractal processes. In the present contribution, we investigate the relevance of such bootstrap procedures to discriminate between non Gaussian finite variance self similar processes with stationary increments (such as Rosenblatt process) and multifractal processes. To do so, we introduce a new joint time-scale block based bootstrap scheme and make use of the most recent scaling analysis tools, based on wavelet leaders

Bootstrap for Multifractal Analysis

Multifractal analysis, which mainly consists in estimating scaling exponents, has become a popular tool for empirical data analysis. Although widely used in different applications, the statistical performance and the reliability of the estimation procedures are still poorly known. Notably, little is known about confidence intervals, though they are of first importance in applications. The present work investigates the potential uses of bootstrap for multifractal estimation: Can bootstrap improve current estimation procedures or be used to obtain reliable confidence intervals~? Comparing the statistical performance of different estimators, our major result is to show that bootstrap based procedures provide us both with accurate estimates and reliable confidence intervals. We believe that this brings substantial improvements to practical empirical multifractal analyses

Bootstrap tests for the time constancy of multifractal attributes

4 pages, 2 figures, 3 tableOn open and controversial issue in empirical data analysis is to decide whether scaling and multifractal properties observed in empirical data actually exist, or whether they are induced by intricate non stationarities. To contribute to answering this question, we propose a procedure aiming at testing the constancy along time of multifractal attributes estimated over adjacent non overlapping time windows. The procedure is based on non parametric bootstrap resampling and on wavelet Leader estimations for the multifractal parameters. It is shown, by means of numerical simulations on synthetic multifractal processes, that the proposed procedure is reliable and powerful for discriminating true scaling behavior against non stationarities. We end up with a practical procedure that can be applied to a single finite length observation of data with unknown statistical properties

Régularité locale pour l'analyse de texture : le mariage des coefficients dominants et de la minimisation proximale

National audienceDans cette contribution, nous revisitons la question de la caractérisation de textures d’images par analyse des fluctuations de régularité locale en combinant deux éléments nouveaux. D’une part, dans l’estimation de la régularité locale proprement dite, la quantité multirésolution utilisée était classiquement le coefficient d’ondelette, l’accroissement ou l’oscillation. Nous la remplaçons par les coefficients dominants, une déclinaison des coefficients d’ondelettes inventée dans le contexte de l’analyse multifractale et qui apportent robustesse et amélioration dans les performances d’estimation. D’autre part, le post-traitement des estimées est réalisé par la mise en place d’une procédure de segmentation en partition minimale résolue à l’aide d’outils proximaux, ce qui nous permet de gérer efficacement le grand volume de données impliqué. Nous générons des textures synthétiques gaussiennes caractérisées par un changement de régularité locale dans une zone dont la position et la surface sont tirées aléatoirement. Nous mettons en oeuvre systématiquement la combinaison de l’estimation par coefficients dominants et segmentation proximale, illustrons la qualité et l’intérêt de cette approche pour segmenter les textures et quantifions la qualité de la segmentation obtenue en fonction de l’amplitude de l’amplitude du changement de régularité locale et de la surface de la zone modifiée

Impact of Data Quantization on Empirical Multifractal Analysis.

Multifractal Analysis is nowadays commonly used in real-life data analyses and involved in standard signal processing tasks such as detection, identification or classification. In a number of situations, mostly in Image Processing, the data are available for the analyses only in (possibly severely) quantized versions. The present contribution aims at analyzing the robustness of standard multifractal estimation procedures against quantization. To this end, we analyze the behaviors and statistical performance of these procedures when applied to a large number of realizations of known synthetic multifractal processes subject to various quantization levels. Our study shows that immunity against quantization can be obtained by restricting the range of scales involved in multifractal parameter estimation to the largest ones. Comparing multifractal analyses based on different multiresolution quantities, increments, wavelet coefficients and leaders, we show that wavelets, thanks to their good frequency localization, bring robustness against quantization when increments do not. This study provides the practitioner with a clear guide line to perform multifractal analysis over quantized data

Learning grounded word meaning representations on similarity graphs

This paper introduces a novel approach to learn visually grounded meaning representations of words as low-dimensional node embeddings on an underlying graph hierarchy. The lower level of the hierarchy models modality-specific word representations through dedicated but communicating graphs, while the higher level puts these representations together on a single graph to learn a representation jointly from both modalities. The topology of each graph models similarity relations among words, and is estimated jointly with the graph embedding.The assumption underlying this model is that words sharing similar meaning correspond to communities in an underlying similarity graph in a low-dimensional space. We named this model Hierarchical Multi-Modal Similarity Graph Embedding (HM-SGE). Experimental results validate the ability of HM-SGE to simulate human similarity judgements and concept categorization, outperforming the state of the art.Peer ReviewedPostprint (published version

Bootstrap for log Wavelet Leaders Cumulant based Multifractal Analysis.

Multifractal analysis, which mostly consists of estimating scaling exponents related to the power law behaviors of the moments of wavelet coefficients, is becoming a popular tool for empirical data analysis. However, little is known about the statistical performance of such procedures. Notably, despite their being of major practical importance, no confidence intervals are available. Here, we choose to replace wavelet coefficients with wavelet Leaders and to use a log-cumulant based multifractal analysis. We investigate the potential use of bootstrap to derive confidence intervals for wavelet Leaders log-cumulant multifractal estimation procedures. From numerical simulations involving well-known and well-controlled synthetic multifractal processes, we obtain two results of major importance for practical multifractal analysis : we demonstrate that the use of Leaders instead of wavelet coefficients brings significant improvements in log-cumulant based multifractal estimation, we show that accurate bootstrap designed confidence intervals can be obtained for a single finite length time series

On an Iterative Method for Direction of Arrival Estimation using Multiple Frequencies

International audienceWe develop a method for the estimation of the location of sources from measurements at multiple frequencies, including wideband measurements, recorded by a linear array of sensors. We employ interpolation matrices to address unequal sampling at different frequencies and make use of the Kronecker theorem to cast the nonlinear least squares problem associated with direction of arrival estimation into an optimization problem in the space of sequences generating Hankel matrices of fixed rank.We then obtain approximate solutions to this problem using the alternating direction method of multipliers. The resulting algorithm is simple and easy to implement. We provide numerical simulations that illustrate its excellent practical performance, significantly outperforming subspace-based methods both at low and high signal-to-noise ratio

Optimization of a Fast Transform Structured as a Convolutional Tree

To reduce the dimension of large datasets, it is common to express each vector of this dataset using few atoms of a redundant dictionary. In order to select these atoms, many models and algorithms have been proposed, leading to state-of-the-art performances in many machine learning, signal and image processing applications. The classical sparsifying algorithms compute at each iteration matrix-vector multiplications where the matrix contains the atoms of the dictionary. As a consequence, the numerical complexity of the sparsifying algorithm is always proportional to the numerical complexity of the matrix-vector multiplication. In some applications, the matrix-vector multiplications can be computed using handcrafted fast transforms (such as the Fourier or the wavelet transforms). However, the complexity of the matrix-vector multiplications very often limits the capacities of the sparsifying algorithms. It is particularly the case when the transform is learned from the data. In order to avoid this limitation, we study a strategy to optimize convolutions of sparse kernels living on the edges of a tree. These convolutions define a fast transform (algorithmically similar to a fast wavelet transform) that can approximate atoms prescribed by the user. The optimization problem associated with the learning of these fast transforms is smooth but can be strongly non-convex. We propose in this paper a proximal alternating linearized minimization algorithm (PALMTREE) allowing Curvelet or Wavelet-packet transforms to be approximated with excellent performance. Empirically, the profile of the objective function associated with this optimization problem has a small number of critical values with large watershed. This confirms that the resulting fast transforms can be optimized efficiently, which opens many theoretical and applicative perspectives