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2011 | Book

Function Spaces Vs. Scaling Functions: Tools for Image Classification Read chapter Read first chapter

Mathematical Image Processing

University of Orléans, France, March 29th - April 1st, 2010

Editor: Maïtine Bergounioux

Publisher: Springer Berlin Heidelberg

Book Series : Springer Proceedings in Mathematics

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

The contributions appearing in this volume are a snapshot of the different topics that were discussed during the Second Conference "Mathematics and Image Processing” held at the University of Orléans in 2010. They mainly concern, image reconstruction, texture extraction and image classification and involve a variety of different methods and applications. Therefore it was impossible to split the papers into generic groups which is why they are presented in alphabetic order. However they mainly concern: texture analysis (5 papers) with different techniques (variational analysis, wavelet and morphological component analysis, fractional Brownian fields), geometrical methods (2 papers ) for restoration and invariant feature detection, classification (with multifractal analysis), neurosciences imaging and analysis of Multi-Valued Images.

Table of Contents

Frontmatter
Function Spaces Vs. Scaling Functions: Tools for Image Classification
Abstract
We investigate the properties of several classes of new parameters issued from multifractal analysis and used in image analysis and classification. They share the following common characteristics: They are derived from local quantities based on wavelet coefficients; at each scale, l p averages of these local quantities are performed and exponents are deduced form a regression through the scales on a log–log plot. This yields scaling functions, which depend on the parameter p, and are used for model selection and classification. We expose possible variants, and their pros and cons. We relate the values taken by these scaling functions with the determination of the regularity of the image in some classes of function spaces, and we show that looking for richer criteria naturally leads to the introduction of new classes of function spaces. We will show which type of additional information this information yields for the initial image.
Stéphane Jaffard, Patrice Abry, Stéphane Roux
A Second Order Model for 3D-Texture Extraction
Abstract
In this paper we present the 3D-implementation of a second-order model for texture extraction that has been fully described in (Bergounioux and Piffet, Set Valued Anal. Variational Anal. 18(3–4):277–306 (2010)). Numerical experimentation has been performed for 2D-images. We generalize the discrete model to the 3D case. In particular we describe the whole discretization process. In addition, we add an algorithmic modification that improves texture extraction using a modified Hessian matrix. We end with numerical examples arising in biomedical imaging.
Maïtine Bergounioux, Minh Phuong Tran
Analysis of Texture Anisotropy Based on Some Gaussian Fields with Spectral Density
Abstract
In this paper, we describe a statistical framework for the analysis of anisotropy of image texture. This framework is based on the modeling of the image by two kinds of non-stationary anisotropic Gaussian field with stationary increments and spectral density: the extended fractional Brownian field (EFBF) and a specific Gaussian operator scaling field (GOSF), which both correspond to a generalization of the fractional Brownian field. In this framework, we tackle anisotropy analysis using some directional processes that are either defined as a restriction of the image on an oriented line or as a projection of the image along a direction. In the context of EFBF and GOSF, we specify links between the regularity of line and projection processes and model parameters, and explain how field anisotropy can be apprehended from the analysis of process regularity. Adapting generalized quadratic variations, we also define some estimators of the regularity of line and projection processes, and study their convergence to field model parameters. Estimators are also evaluated on simulated data, and applied for illustration to medical images of the breast and the bone.
Hermine Biermé, Frédéric J. P. Richard
Image Reconstruction Via Hypoelliptic Diffusion on the Bundle of Directions of the Plane
Abstract
In this paper we present a model of geometry of vision which generalizes one due to Petitot, Citti and Sarti. One of its main features is that the primary visual cortex V1 lifts the image from \({R}^{2}\) to the bundle of directions of the plane \(PT{\mathbb{R}}^{2} = {\mathbb{R}}^{2} \times{P}^{1}\). Neurons are grouped into orientation columns, each of them corresponding to a point of the bundle \(PT{\mathbb{R}}^{2}\).
In this model a corrupted image is reconstructed by minimizing the energy necessary for the activation of the orientation columns corresponding to regions in which the image is corrupted. The minimization process gives rise to an hypoelliptic heat equation on \(PT{\mathbb{R}}^{2}\). The hypoelliptic heat equation is studied using the generalized Fourier transform. It transforms the hypoelliptic equation into a 1-d heat equation with Mathieu potential, which one can solve numerically.
Preliminary examples of image reconstruction are hereby provided.
Ugo Boscain, Jean Duplaix, Jean-Paul Gauthier, Francesco Rossi
Projective Invariant Features Detection and the Registration Group
Abstract
Affine invariant scale space analyses have been introduced in the 1990s and largely used in shape recognition applications. Nowadays they are also used for the determination of points of interests. Nevertheless, a projective analysis is necessary if information is to be gathered from images taken of the same scene but under different points of view. In previous work, we replaced the projective analysis due to the heat equation by a flow of second order equations. The registration group allows to model the deformations of an image due to camera motion by a 6 parameter group. We will show that it allows also to view in a new light the projective analysis and the Affine Morphological Scale Space of Alvarez, Lions, Guichard and Morel (Arch. Ration. Mechan. 16:200–257 (1993)). Moreover, the registration group gives directly a first approach for a Projective Scale Invariant Feature Tracking algorithm.
Françoise Dibos
Morphological Component Analysis for Decomposing Dynamic Textures
Abstract
The research context of this work is dynamic texture analysis and characterization. A dynamic texture can be described as a time-varying phenomenon with a certain repetitiveness in both space and time.
Many dynamic textures can be modeled as a large scale propagating wavefront and local oscillating phenomena.
The Morphological Component Analysis approach with a well chosen dictionary is used to retrieve the components of dynamic textures. We define two new strategies for adaptive thresholding in the Morphological Component Analysis framework, which greatly reduce the computation time when applied on videos. These strategies are studied with different criteria. Finally, tests on real image sequences illustrate the efficiency of the proposed method.
Sloven Dubois, Renaud Péteri, Michel Ménard
Texture Enhancing Based on Variational Image Decomposition
Abstract
In this paper we consider the Augmented Lagrangian Method for image decomposition. We propose a method which decomposes an image into texture, which is characterized to have finite l 1 curvelet coefficients, a cartoon part, which has finite total variation norm, and noise and oscillating patterns, which have finite G-norm. In the second part of the paper we utilize the equivalence of the Augmented Lagrangian Method and the iterative Bregman distance regularization to show that the dual variables can be used for enhancing of particular components. We concentrate on the enhancing feature for the texture and propose two different variants of the Augmented Lagrangian Method for decomposition and12.5pc]The first author has been considered as corresponding author. Please check. enhancing.
Florian Frühauf, Carsten Pontow, Otmar Scherzer
A Locally Anisotropic Model for Image Texture Extraction
Abstract
We present a variational model for image texture identification. We first use a second order model introduced in Bergounioux and Piffet (Set Valued Variational Anal. 18(3–4):277–306 (2010)) for image denoising. The model involves a L 2-data fitting term and a Tychonov-like regularization. We choose here the BV 2 norm, where BV 2 is the bounded hessian function space (see Bergounioux and Piffet, Set Valued Variational Anal. 18(3–4):277–306 (2010)). We observe that results are not satisfying since geometrical information appears in the oscillating component and should not. So we propose an anisotropic strategy, by setting components of the discrete hessian operator to 0 in order to focus on gradient directions. We precisely describe and illustrate the numerical methodology. Finally, we propose some numerical tests.
Loïc Piffet
A Neural Field Model for Motion Estimation
Abstract
We propose a bio-inspired approach to motion estimation based on recent neuroscience findings concerning the motion pathway. Our goal is to identify the key biological features in order to reach a good compromise between bio-inspiration and computational efficiency. Here we choose the neural field formalism which provides a sound mathematical framework to describe the model at a macroscopic scale. Within this framework we define the cortical activity as coupled integro-differential equations and we prove the well-posedness of the model. We show how our model performs on some classical computer vision videos, and we compare its behaviour against the visual system on a simple classical video used in psychophysics. As a whole, this article contributes to bring new ideas from computational neuroscience in the domain of computer vision, concerning modelling principles and mathematical formalism.
Émilien Tlapale, Pierre Kornprobst, Guillaume S. Masson, Olivier Faugeras
Non-Local Regularization and Registration of Multi-Valued Images By PDE’s and Variational Methods on Higher Dimensional Spaces
Abstract
We are interested in a simple transform that map any regular 2D multi-valued image into a high-dimensional Euler space of patches, so that each initial image patch is mapped as a single high-dimensional point. We propose a way to study the local geometry of this patch-projected image and we consider variational formulations and PDE’s in this particular space. Hence, we define a way to find natural patch-based counterparts of some classical image processing techniques, including Lucas–Kanade registration, Tikhonov regularization and tensor-driven anisotropic diffusion PDE’s. As a result, we end up with noteworthy variants of already known (non-variational) patch-based algorithms, namely the Non Local Means and Block Matching techniques. The interest of considering such variational or PDE approaches on high dimensional patch spaces is discussed and illustrated by comparison results with corresponding non-variational or non-patch methods.
David Tschumperlé, Luc Brun
Metadata
Title
Mathematical Image Processing
Editor
Maïtine Bergounioux
Copyright Year
2011
Publisher
Springer Berlin Heidelberg
Electronic ISBN
978-3-642-19604-1
Print ISBN
978-3-642-19603-4
DOI
https://doi.org/10.1007/978-3-642-19604-1

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