Random Projection Depth for Multivariate Mathematical Morphology

Abstract

Abstract-The 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.Index Terms-Hyperspectral images, multivariate morphology, statistical depth function.

Complete lattice learning for multivariate mathematical morphology

“…The first approach relying on such a mapping was the bit-mixing approach [15] that employs a transformation exploiting the binary representation of each component. This approach cannot be easily extended to high-dimensional spaces and recent works towards this have considered either distance-based [7,10] or projection-based [8,6,9] h-orderings. Given a mapping h, one can then define h-erosion and h-dilation [9, 7, 10].…”

“…In [6,9] Velasco-Forero and Angulo have proposed a P-ordering to produce an ordering by using statistical depth functions. We can call such h-orderings unsupervised h-orderings.…”

“…We refer the reader to [2,3,1] for a comprehensive review of vector morphology. Several recent approaches have been proposed in literature for e.g., color and hyperspectral images [4,5,6,7,8,9,10]. This paper introduces a systematic approach towards the construction of complete lattices for any kind of multivariate data.…”

“…This paper introduces a systematic approach towards the construction of complete lattices for any kind of multivariate data. Following recent approaches [8,9], we propose to learn, in an unsupervised manner, the construction of a complete lattice from the values of an image. To do so, we rely on the theoretical framework of h-orderings [11], suitable for the definition of complete lattices.…”

Quantization of Hyperspectral Image Manifold Using Probabilistic Distances

“…The second one is inspired from the work [17,19,21], where thanks to random projections they succeed to have a good estimator of the Mahalanobis distance. We will denote respectively the corresponding distance D Mahal1 and D Mahal2 .…”