Multivariate risks and depth-trimmed regions

We describe a general framework for measuring risks, where the risk measure takes values in an abstract cone. It is shown that this approach naturally includes the classical risk measures and set-valued risk measures and yields a natural definition of vector-valued risk measures. Several main constructions of risk measures are described in this abstract axiomatic framework. It is shown that the concept of depth-trimmed (or central) regions from the multivariate statistics is closely related to the definition of risk measures. In particular, the halfspace trimming corresponds to the Value-at-Risk, while the zonoid trimming yields the expected shortfall. In the abstract framework, it is shown how to establish a both-ways correspondence between risk measures and depth-trimmed regions. It is also demonstrated how the lattice structure of the space of risk values influences this relationship.Comment: 26 pages. Substantially revised version with a number of new results adde

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

Mathematical morphology for tensor data induced by the Loewner orderingin higher dimensions

Positive semidefinite matrix fields are becoming increasingly important in digital imaging. One reason for this tendency consists of the introduction of diffusion tensor magnetic resonance imaging (DTMRI). In order to perform shape analysis, enhancement or segmentation of such tensor fields, appropriate image processing tools must be developed. This paper extends fundamental morphological operations to the matrix-valued setting. We start by presenting novel definitions for the maximum and minimum of a set of matrices since these notions lie at the heart of the morphological operations. In contrast to naive approaches like the component-wise maximum or minimum of the matrix channels, our approach is based on the Loewner ordering for symmetric matrices. The notions of maximum and minimum deduced from this partial ordering satisfy desirable properties such as rotation invariance, preservation of positive semidefiniteness, and continuous dependence on the input data. We introduce erosion, dilation, opening, closing, top hats, morphological derivatives, shock filters, and mid-range filters for positive semidefinite matrix-valued images. These morphological operations incorporate information simultaneously from all matrix channels rather than treating them independently. Experiments on DT-MRI images with ball- and rod-shaped structuring elements illustrate the properties and performance of our morphological operators for matrix-valued data

Remote Sensing Image Classification Using Attribute Filters Defined over the Tree of Shapes

International audience—Remotely sensed images with very high spatial resolution provide a detailed representation of the surveyed scene with a geometrical resolution that at the present can be up to 30 cm (WorldView-3). A set of powerful image processing operators have been defined in the mathematical morphology framework. Among those, connected operators (e.g., attribute filters) have proven their effectiveness in processing very high resolution images. Attribute filters are based on attributes which can be efficiently implemented on tree-based image representations. In this work, we considered the definition of min, max, direct and subtractive filter rules for the computation of attribute filters over the tree of shapes representation. We study their performance on the classification of remotely sensed images. We compare the classification results over the tree of shapes with the results obtained when the same rules are applied on the component trees. The random forest is used as a baseline classifier and the experiments are conducted using multispectral data sets acquired by QuickBird and IKONOS sensors over urban areas

Aerospace medicine and biology: A continuing bibliography with indexes (supplement 346)

This bibliography lists 134 reports, articles and other documents introduced into the NASA Scientific and Technical Information System during Jan. 1991. Subject coverage includes: aerospace medicine and psychology, life support systems and controlled environments, safety equipment, exobiology and extraterrestrial life, and flight crew behavior and performance

MULTIVARIATE RISKS AND DEPTH-TRIMMED REGIONS

We describe a general framework for measuring risks, where the risk measure takes values in an abstract cone. It is shown that this approach naturally includes the classical risk measures and set-valued risk measures and yields a natural definition of vector-valued risk measures. Several main constructions of risk measures are described in this axiomatic framework. It is shown that the concept of depth-trimmed (or central) regions from the multivariate statistics is closely related to the definition of risk measures. In particular, the halfspace trimming corresponds to the Value-at-Risk, while the zonoid trimming yields the expected shortfall. In the abstract framework, it is shown how to establish a both-ways correspondence between risk measures and depth-trimmed regions. It is also demonstrated how the lattice structure of the space of risk values influences this relationship.

Vector attribute profiles for hyperspectral image classification

International audienceMorphological attribute profiles are among the most prominent spectral-spatial pixel description methods. They are efficient, effective and highly customizable multi-scale tools based on hierarchical representations of a scalar input image. Their application to multivariate images in general, and hyperspectral images in particular, has been so far conducted using the marginal strategy, i.e. by processing each image band (eventually obtained through a dimension reduction technique) independently. In this paper, we investigate the alternative vector strategy, which consists in processing the available image bands simultaneously. The vector strategy is based on a vector ordering relation that leads to the computation of a single max-and min-tree per hyperspectral dataset, from which attribute profiles can then be computed as usual. We explore known vector ordering relations for constructing such max-trees and subsequently vector attribute profiles, and introduce a combination of marginal and vector strategies. We provide an experimental comparison of these approaches in the context of hyperspectral classification with common datasets, where the proposed approach outperforms the widely used marginal strategy

Low Complexity Regularization of Linear Inverse Problems

Inverse problems and regularization theory is a central theme in contemporary signal processing, where the goal is to reconstruct an unknown signal from partial indirect, and possibly noisy, measurements of it. A now standard method for recovering the unknown signal is to solve a convex optimization problem that enforces some prior knowledge about its structure. This has proved efficient in many problems routinely encountered in imaging sciences, statistics and machine learning. This chapter delivers a review of recent advances in the field where the regularization prior promotes solutions conforming to some notion of simplicity/low-complexity. These priors encompass as popular examples sparsity and group sparsity (to capture the compressibility of natural signals and images), total variation and analysis sparsity (to promote piecewise regularity), and low-rank (as natural extension of sparsity to matrix-valued data). Our aim is to provide a unified treatment of all these regularizations under a single umbrella, namely the theory of partial smoothness. This framework is very general and accommodates all low-complexity regularizers just mentioned, as well as many others. Partial smoothness turns out to be the canonical way to encode low-dimensional models that can be linear spaces or more general smooth manifolds. This review is intended to serve as a one stop shop toward the understanding of the theoretical properties of the so-regularized solutions. It covers a large spectrum including: (i) recovery guarantees and stability to noise, both in terms of $\ell^2$-stability and model (manifold) identification; (ii) sensitivity analysis to perturbations of the parameters involved (in particular the observations), with applications to unbiased risk estimation ; (iii) convergence properties of the forward-backward proximal splitting scheme, that is particularly well suited to solve the corresponding large-scale regularized optimization problem

On morphological hierarchical representations for image processing and spatial data clustering

Hierarchical data representations in the context of classi cation and data clustering were put forward during the fties. Recently, hierarchical image representations have gained renewed interest for segmentation purposes. In this paper, we briefly survey fundamental results on hierarchical clustering and then detail recent paradigms developed for the hierarchical representation of images in the framework of mathematical morphology: constrained connectivity and ultrametric watersheds. Constrained connectivity can be viewed as a way to constrain an initial hierarchy in such a way that a set of desired constraints are satis ed. The framework of ultrametric watersheds provides a generic scheme for computing any hierarchical connected clustering, in particular when such a hierarchy is constrained. The suitability of this framework for solving practical problems is illustrated with applications in remote sensing