Inner-Cheeger Opening and Applications

International audienceThe aim of this paper is to study an optimal opening in the sense of minimize the relationship perimeter over area. We analyze theoretical properties of this opening by means of classical results from variational calculus. Firstly, we explore the optimal radius as attribute in morphological attribute filtering for grey scale images. Secondly, an application of this optimal opening that yields a decomposition into meaningful parts in the case of binary image is explored. We provide different examples of 2D, 3D images and mesh-points datasets

Parameters Selection Of Morphological Scale-Space Decomposition For Hyperspectral Images Using Tensor Modeling

International audienceDimensionality reduction (DR) using tensor structures in morphological scale-space decomposition (MSSD) for HSI has been investigated in order to incorporate spatial information in DR.We present results of a comprehensive investigation of two issues underlying DR in MSSD. Firstly, information contained in MSSD is reduced using HOSVD but its nonconvex formulation implicates that in some cases a large number of local solutions can be found. For all experiments, HOSVD always reach an unique global solution in the parameter region suitable to practical applications. Secondly, scale parameters in MSSD are presented in relation to connected components size and the influence of scale parameters in DR and subsequent classification is studied

Morphological scale-space for hyperspectral images and dimensionality exploration using tensor modeling

International audienceThis paper proposes a framework to integrate spatial information into unsupervised feature extraction for hyperspectral images. In this approach a nonlinear scale-space representation using morphological levelings is formulated. In order to apply feature extraction, tensor principal components are computed involving spatial and spectral information. The proposed method has shown significant gain over the conventional schemes used with real hyperspectral images. In addition, the proposed framework opens a wide field for future developments in which spatial information can be easily integrated into the feature extraction stage. Examples using real hyperspectral images with high spatial resolution showed excellent performance even with a low number of training samples

Riemannian mathematical morphology

This paper introduces mathematical morphology operators for real-valued images whose support space is a Riemannian manifold. The starting point consists in replacing the Euclidean distance in the canonic quadratic structuring function by the Riemannian distance used for the adjoint dilation/erosion. We then extend the canonic case to a most general framework of Riemannian operators based on the notion of admissible Riemannian structuring function. An alternative paradigm of morphological Riemannian operators involves an external structuring function which is parallel transported to each point on the manifold. Besides the definition of the various Riemannian dilation/erosion and Riemannian opening/closing, their main properties are studied. We show also how recent results on Lasry-Lions regularization can be used for non-smooth image filtering based on morphological Riemannian operators. Theoretical connections with previous works on adaptive morphology and manifold shape morphology are also considered. From a practical viewpoint, various useful image embedding into Riemannian manifolds are formalized, with some illustrative examples of morphological processing real-valued 3D surfaces

Vector ordering and multispectral morphological image processing

International audienceThis chapter illustrates the suitability of recent multivariate ordering approaches to morphological analysis of colour and multispectral images working on their vector representation. On the one hand, supervised ordering renders machine learning no-tions and image processing techniques, through a learning stage to provide a total ordering in the colour/multispectral vector space. On the other hand, anomaly-based ordering, automatically detects spectral diversity over a majority background, al-lowing an adaptive processing of salient parts of a colour/multispectral image. These two multivariate ordering paradigms allow the definition of morphological operators for multivariate images, from algebraic dilation and erosion to more advanced techniques as morphological simplification, decomposition and segmentation. A number of applications are reviewed and implementation issues are discussed in detail

Statistical Shape Modeling Using Morphological Representations

ISBN : 978-1-4244-7542-1International audienceThe aim of this paper is to propose tools for statistical analysis of shape families using morphological operators. Given a series of shape families (or shape categories), the approach consists in empirically computing shape statistics (i.e., mean shape and variance of shape) and then to use simple algorithms for random shape generation, for empirical shape confidence boundaries computation and for shape classification using Bayes rules. The main required ingredients for the present methods are well known in image processing, such as watershed on distance functions or log-polar transformation. Performance of classification is presented in a well-known shape database

Semi-Supervised Hyperspectral Image Segmentation Using Regionalized Stochastic Watershed

International audienceStochastic watershed is a robust method to estimate the probability density function (pdf) of contours of a multi-variate image using MonteCarlo simulations of watersheds from random markers. The aim of this paper is to propose a stochastic watershed-based algorithm for segmenting hyperspectral images using a semi-supervised approach. Starting from a training dataset consisting in a selection of representative pixel vectors of each spectral class of the image, the algorithm calculate for each class a membership probability map (MPM). Then, the MPM of class k is considered as a regionalized density function which is used to simulate the random markers for the MonteCarlo estimation of the pdf of contours of the corresponding class k. This pdf favours the spatial regions of the image spectrally close to the class k. After applying the same technique to each class, a series of pdf are obtained for a single image. Finally, the pdf's can be segmented hierarchically either separately for each class or after combination, as a single pdf function. In the results, besides the generic spatial-spectral segmentation of hyperspectral images, the interest of the approach is also illustrated for target segmentation

Random projection depth for multivariate mathematical morphology

International audienceThe open problem of the generalization of mathematical morphology to vector images is handled in this paper using the paradigm of depth functions. Statistical depth functions provide from the "deepest" point a "center-outward ordering" of a multidimensional data distribution and they can be therefore used to construct morphological operators. The fundamental assumption of this data-driven approach is the existence of "background/foreground" image representation. Examples in real color and hyperspectral images illustrate the results

Classification of hyperspectral images by tensor modeling and additive morphological decomposition

International audiencePixel-wise classification in high-dimensional multivariate images is investigated. The proposed method deals with the joint use of spectral and spatial information provided in hyperspectral images. Additive morphological decomposition (AMD) based on morphological operators is proposed. AMD defines a scale-space decomposition for multivariate images without any loss of information. AMD is modeled as a tensor structure and tensor principal components analysis is compared as dimensional reduction algorithm versus classic approach. Experimental comparison shows that the proposed algorithm can provide better performance for the pixel classification of hyperspectral image than many other well-known techniques

Contributions en morphologie mathématique pour l'analyse d'images multivariées

This thesis contributes to the field of mathematical morphology and illustrates how multivariate statistics and machine learning techniques can be exploited to design vector ordering and to include results of morphological operators in the pipeline of multivariate image analysis. In particular, we make use of supervised learning, random projections, tensor representations and conditional transformations to design new kinds of multivariate ordering, and morphological filters for color and multi/hyperspectral images. Our key contributions include the following points:• Exploration and analysis of supervised ordering based on kernel methods.• Proposition of an unsupervised ordering based on statistical depth function computed by random projections. We begin by exploring the properties that an image requires to ensure that the ordering and the associated morphological operators can be interpreted in a similar way than in the case of grey scale images. This will lead us to the notion of background/foreground decomposition. Additionally, invariance properties are analyzed and theoretical convergence is showed.• Analysis of supervised ordering in morphological template matching problems, which corresponds to the extension of hit-or-miss operator to multivariate image by using supervised ordering.• Discussion of various strategies for morphological image decomposition, specifically, the additive morphological decomposition is introduced as an alternative for the analysis of remote sensing multivariate images, in particular for the task of dimensionality reduction and supervised classification of hyperspectral remote sensing images.• Proposition of an unified framework based on morphological operators for contrast enhancement and salt- and-pepper denoising.• Introduces a new framework of multivariate Boolean models using a complete lattice formulation. This theoretical contribution is useful for characterizing and simulation of multivariate textures.Cette thèse contribue au domaine de la morphologie mathématique et illustre comment la statistique multivariée et les techniques d'apprentissage numérique peuvent être exploitées pour concevoir un ordre dans l'espace des vecteurs et pour inclure les résultats d'opérateurs morphologiques au processus d'analyse d'images multivariées. En particulier, nous utilisons l'apprentissage supervisé, les projections aléatoires, les représentations tensorielles et les transformations conditionnelles pour concevoir de nouveaux types d'ordres multivariés et de nouveaux filtres morphologiques pour les images multi/hyperspectrales. Nos contributions clés incluent les points suivants :• Exploration et analyse d'ordre supervisé, basé sur les méthodes à noyaux.• Proposition d'un ordre nonsupervisé, basé sur la fonction de profondeur statistique calculée par projections aléatoires. Nous commençons par explorer les propriétés nécessaires à une image pour assurer que l'ordre ainsi que les opérateurs morphologiques associés, puissent être interprétés de manière similaire au cas d'images en niveaux de gris. Cela nous amènera à la notion de décomposition en arrière plan. De plus, les propriétés d'invariance sont analysées et la convergence théorique est démontrée.• Analyse de l'ordre supervisé dans les problèmes de correspondance morphologique de patrons, qui correspond à l'extension de l'opérateur tout-ou-rien aux images multivariées grâce à l‘utilisation de l'ordre supervisé.• Discussion sur différentes stratégies pour la décomposition morphologique d'images. Notamment, la décomposition morphologique additive est introduite comme alternative pour l'analyse d'images de télédétection, en particulier pour les tâches de réduction de dimension et de classification supervisée d'images hyperspectrales de télédétection.• Proposition d'un cadre unifié basé sur des opérateurs morphologiques, pour l'amélioration de contraste et pour le filtrage du bruit poivre-et-sel.• Introduction d'un nouveau cadre de modèles Booléens multivariés en utilisant une formulation en treillis complets. Cette contribution théorique est utile pour la caractérisation et la simulation de textures multivariées