Relevance of multifractal textures in static images

In the latest years, multifractal analysis has been applied to image analysis. The multifractal framework takes advantage of multiscaling properties of images to decompose them as a collection of different fractal components, each one associated to a singularity exponent (an exponent characterizing the way in which that part of the image evolves under changes in scale). One of those components, characterized by the least possible exponent, seems to be the most informative about the whole image. Very recently it has been proposed an algorithm to reconstruct the image from this component, just using physical information conveyed by it. In this paper, we will show that the same algorithm can be used to assess the relevance of the other fractal parts of the image

Microcanonical processing methodology for ECG and intracardial potential: application to atrial fibrillation

Cardiac diseases are the principal cause of human morbidity and mortality in the western world. The electric potential of the heart is a highly complex signal emerging as a result of nontrivial flow conduction, hierarchical structuring and multiple regulation mechanisms. Its proper accurate analysis becomes of crucial importance in order to detect and treat arrhythmias or other abnormal dynamics that could lead to life-threatening conditions. To achieve this, advanced nonlinear processing methods are needed: one example here is the case of recent advances in the Microcanonical Multiscale Formalism. The aim of the present paper is to recapitulate those advances and extend the analyses performed, specially looking at the case of atrial fibrillation. We show that both ECG and intracardial potential signals can be described in a model-free way as a fast dynamics combined with a slow dynamics. Sharp differences in the key parameters of the fast dynamics appear in different regimes of transition between atrial fibrillation and healthy cases. Therefore, this type of analysis could be used for automated early warning, also in the treatment of atrial fibrillation particularly to guide radiofrequency ablation procedures.Comment: Transactions on Mass-Data Analysis of Images and Signals 4, 1 (2012). Accepte

Edge-preserving smoothing of high-resolution images with a partial multifractal reconstruction scheme

International audienceThe new generation of satellites leads to the arrival of very high-resolution images which offer a new quality of detailed information about the Earth's surface. However, the exploitation of such data becomes more complicated and less efficient as a consequence of the great heterogeneity of the objects displayed. In this paper, we address the problem of edge-preserving smoothing of high-resolution satellite images. We introduce a novel approach as a preprocessing step for feature extraction and/or image segmentation. The method is derived from the multifractal formalism proposed for image compression. This process consists in smoothing heterogeneous areas while preserving the main edges of the image. It is performed in 2 steps: 1) a multifractal decomposition scheme allows to extract the most informative subset of the image, which consists essentially in the edges of the objects; 2) a propagation scheme performed over this subset allows to reconstruct an approximation of the original image with a more uniform distribution of luminance. The strategy we adopt is described in the following and some results are presented for Spot acquisitions with spatial resolution of 20 x 20 m2

ANALYSE MULTI FRACTALE DES ÉCHOS RADAR PAR LA MÉTHODE DES MAXIMUMS DES MODULES DE LA TRANSFORMÉE EN ONDELETTE (MMTO) 2D POUR LES SITES DE BORDEAUX (FRANCE), SÉTIF (ALGÉRIE) : APPLICATION À L'ÉLIMINATION DES ÉCHOS PARASITES

International audienceIn this work, the 2D-WTMM multifractal approach was applied to analysis the radar echoes, and to identify the unwanted echoes coming from terrestrial surface. With this intention, we considered radar images taken from two areas where different climates and relief prevail. We showed that almost Anaprops are characterized by a monofractal spectrum contrary to the echoes of precipitations which present a multifractal character. Moreover, we showed that the Holder coefficient and the combination of the spectrum mode and density of skeleton per pixel present robust factors to discriminate between the two types of echoes. Indeed, the unwanted echoes are practically eliminated at 98 per cent whereas the echoes of precipitation are almost preserved at 98,2 per cent. Also, we showed that the error between the measured intensity on the ground and the estimated intensity after treatment of the unwanted echoes does not exceed 5% for the Sétif site. Because the computation time is three minutes, the radar images can be processed in real-time.Dans le présent travail, l'approche MMTO-2d est appliquée pour l'analyse multi fractale des échos radar et l'identification des échos parasites en provenance de la surface terrestre. Pour ce faire, nous avons considéré des images radar prises dans deux régions où prévalent des climats et des reliefs différents. Il s'agit des sites de Sétif (Algérie) et Bordeaux (France). Nous avons montré que la plupart des Anaprops sont caractérisés par un spectre monofractal contrairement aux échos de précipitations qui présentent un caractère multi fractal. En outre, nous avons montré que le coefficient d'Holder ou la combinaison mode du spectre et densité de squelette par pixel se présentent comme des facteurs robustes de discrimination entre les deux types d'échos. En effet, les échos parasites sont pratiquement éliminés à 98% alors que les échos de précipitation sont quasiment conservés à 98,2%. Aussi, nous avons montré que l'erreur entre l'intensité mesurée au sol et estimée après traitement des échos parasites ne dépasse pas 5% pour le site de Sétif. Etant donné que le temps de traitement est égal à trois minutes, les images radar peuvent être traitées en temps réel

Algebraic Topology

The chapter provides an introduction to the basic concepts of Algebraic Topology with an emphasis on motivation from applications in the physical sciences. It finishes with a brief review of computational work in algebraic topology, including persistent homology.Comment: This manuscript will be published as Chapter 5 in Wiley's textbook \emph{Mathematical Tools for Physicists}, 2nd edition, edited by Michael Grinfeld from the University of Strathclyd

An optimized algorithm for the evaluation of local singularity exponents in digital signals

International audienceRecent works show that the determination of singularity exponents in images can be useful to assess their information content, and in some cases they can cast additional information about underlying physical processes. However, the concept of singularity exponent is associated to differential calculus and thus cannot be easily translated to a digital context, even using wavelets. In this work we show that a recently patented algorithm allows obtaining precise, meaningful values of singularity exponents at every point in the image by the use of a discretized combinatorial mask, which is an extension of a particular wavelet basis. This mask is defined under the hypothesis that singularity exponents are a measure not only of the degree of regularity of the image, but also of the reconstructibility of a signal from their points