Jesús Angulo

Probability Density Estimation on the Hyperbolic Space Applied to Radar Processing

The two main techniques of probability density estimation on symmetric spaces are reviewed in the hyperbolic case. For computational reasons we chose to focus on the kernel density estimation and we provide the expression of Pelletier estimator on hyperbolic space. The method is applied to density estimation of reflection coefficients derived from radar observations.

Morphological PDE and dilation/erosion semigroups on length spaces

This paper gives a survey of recent research on Hamilton-Jacobi partial differential equations (PDE) on length spaces. This theory provides the background to formulate morphological PDEs for processing data and images supported on a length space, without the need of a Riemmanian structure. We first introduce the most general pair of dilation/erosion semigroups on a length space, whose basic ingredients are the metric distance and a convex shape function. The second objective is to show under which conditions the solution of a morphological PDE in the length space framework is equal to the dilation/erosion semigroups.

Ordering on the probability simplex of endmembers for hyperspectral morphological image processing

A hyperspectral image can be represented as a set of materials called endmembers, where each pixel corresponds to a mixture of several of these materials. More precisely pixels are described by the quantity of each material, this quantity is often called abundance and is positive and of sum equal to one. This leads to the characterization of a hyperspectral image as a set of points in a probability simplex. The geometry of the simplex has been particularly studied in the theory of quantum information, giving rise to different notions of distances and interesting preorders. In this paper, we present total orders based on theory of the ordering on the simplex. Thanks to this theory, we can give a physical interpretation of our orders.

Bagging Stochastic Watershed on Natural Color Image Segmentation

The stochastic watershed is a probabilistic segmentation approach which estimates the probability density of contours of the image from a given gradient. In complex images, the stochastic watershed can enhance insignificant contours. To partially address this drawback, we introduce here a fully unsupervised multi-scale approach including bagging. Re-sampling and bagging is a classical stochastic approach to improve the estimation. We have assessed the performance, and compared to other version of stochastic watershed, using the Berkeley segmentation database.

Morphological Scale-Space Operators for Images Supported on Point Clouds

The aim of this paper is to develop the theory, and to propose an algorithm, for morphological processing of images painted on point clouds, viewed as a length metric measure space ((X,d,mu )). In order to extend morphological operators to process point cloud supported images, one needs to define dilation and erosion as semigroup operators on ((X,d)). That corresponds to a supremal convolution (and infimal convolution) using admissible structuring function on ((X,d)). From a more theoretical perspective, we introduce the notion of abstract structuring functions formulated on length metric Maslov idempotent measurable spaces, which is the appropriate setting for ((X,d)). In practice, computation of Maslov structuring function is approached by a random walks framework to estimate heat kernel on ((X,d,mu )), followed by the logarithmic trick.

Adaptive morphological filters based on a multiple orientation vector field dependent on image local features

This paper addresses the formulation of adaptive morphological filters based on spatially-variant structuring elements. The adaptivity of these filters is achieved by modifying the shape and orientation of the structuring elements according to a multiple orientation vector field. This vector field is provided by means of a bank of directional openings which can take into account the possible multiple orientations of the contours in the image. After reviewing and formalizing the definition of the spatially-variant dilation, erosion, opening and closing, the proposed structuring elements are described. These spatially-variant structuring elements are based on ellipses which vary over the image domain adapting locally their orientation according to the multiple orientation vector field and their shape (the eccentricity of the ellipses) according to the distance to relevant contours of the objects. The proposed adaptive morphological filters are used on gray-level images and are compared with spatially-invariant filters, with spatially-variant filters based on a single orientation vector field, and with adaptive morphological bilateral filters. Results show that the morphological filters based on a multiple orientation vector field are more adept at enhancing and preserving structures which contains more than one orientation.

Hyperspectral image classification with support vector machines on kernel distribution embeddings

We propose a novel approach for pixel classification in hyperspectral images, leveraging on both the spatial and spectral information in the data. The introduced method relies on a recently proposed framework for learning on distributions - by representing them with mean elements in reproducing kernel Hilbert spaces (RKHS) and formulating a classification algorithm therein. In particular, we associate each pixel to an empirical distribution of its neighbouring pixels, a judicious representation of which in an RKHS, in conjunction with the spectral information contained in the pixel itself, give a new explicit set of features that can be fed into a suite of standard classification techniques - we opt for a well established framework of support vector machines (SVM). Furthermore, the computational complexity is reduced via random Fourier features formalism. We study the consistency and the convergence rates of the proposed method and the experiments demonstrate strong performance on hyperspectral data with gains in comparison to the state-of-the-art results.

Quantization of Hyperspectral Image Manifold Using Probabilistic Distances

A technique of spatial-spectral quantization of hyperspectral images is introduced. Thus a quantized hyperspectral image is just summarized by K spectra which represent the spatial and spectral structures of the image. The proposed technique is based on (alpha )-connected components on a region adjacency graph. The main ingredient is a dissimilarity metric. In order to choose the metric that best fit the hyperspectral data manifold, a comparison of different probabilistic dissimilarity measures is achieved.

Enhanced EDX images by fusion of multimodal SEM images using pansharpening techniques

The goal of this paper is to explore the potential interest of image fusion in the context of multimodal scanning electron microscope (SEM) imaging. In particular, we aim at merging the backscattered electron images that usually have a high spatial resolution but do not provide enough discriminative information to physically classify the nature of the sample, with energy‐dispersive X‐ray spectroscopy (EDX) images that have discriminative information but a lower spatial resolution. The produced images are named enhanced EDX. To achieve this goal, we have compared the results obtained with classical pansharpening techniques for image fusion with an original approach tailored for multimodal SEM fusion of information. Quantitative assessment is obtained by means of two SEM images and a simulated dataset produced by a software based on PENELOPE.

Morphological Semigroups and Scale-Spaces on Ultrametric Spaces

Ultrametric spaces are the natural mathematical structure to deal with data embedded into a hierarchical representation. This kind of representations is ubiquitous in morphological image processing, from pyramids of nested partitions to more abstract dendrograms from minimum spanning trees. This paper is a formal study of morphological operators for functions defined on ultrametric spaces. First, the notion of ultrametric structuring function is introduced. Then, using as basic ingredient the convolution in (max,min)-algebra, the multi-scale ultrametric dilation and erosion are defined and their semigroup properties are stated. It is proved in particular that they are idempotent operators and consequently they are algebraically ultrametric closing and opening too. Some preliminary examples illustrate the behavior and practical interest of ultrametric dilations/erosions.