Morphology for matrix data: Ordering versus PDE-based approach

Abstract

Matrix fields are becoming increasingly important in digital imaging. In order to perform shape analysis, enhancement or segmentation of such matrix fields, appropriate image processing tools must be developed. This paper extends fundamental morphological operations to the setting of matrices, in the literature sometimes referred to as tensors despite the fact that matrices are only rank two tensors. The goal of this paper is to introduce and explore two approaches to mathematical morphology for matrix-valued data: One is based on a partial ordering, the other utilises nonlinear partial differential equations (PDEs). We start by presenting definitions for the maximum and minimum of a set of symmetric matrices since these notions are the cornerstones of the morphological operations. Our first approach is based on the Loewner ordering for symmetric matrices, and is in contrast to the unsatisfactory component-wise techniques. The notions of maximum and minimum deduced from the Loewner ordering satisfy desirable properties such as rotation invariance, preservation of positive semidefiniteness, and continuous dependence on the input data. These properties are also shared by the dilation and erosion processes governed by a novel nonlinear system of PDEs we are proposing for our second approach to morphology on matrix data. These PDEs are a suitable counterpart of the nonlinear equations known from scalar continuous-scale morphology. Both approaches incorporate information simultaneously from all matrix channels rather than treating them independently. In experiments on artificial and real medical positive semidefinite matrix-valued images we contrast the resulting notions of erosion, dilation, opening, closing, top hats, morphological derivatives, and shock filters stemming from these two alternatives. Using a ball shaped structuring element we illustrate the properties and performance of our ordering-or PDE-driven morphological operators for matrix-valued data.

Adaptive Continuous-Scale Morphology for Matrix Fields

“…Figure 1(b) shows the result of applying coherence-enhancing diffusion (CED) [19]. Figure 1(c) contains the result of isotropic (classical) dilation [16] using the Rouy-Tourin scheme, and Figure 1(d) the result of the proposed adaptive anisotropic dilation after applying the directional Rouy-Tourin scheme. The parameters used were chosen in a way that every method fills in the missing tensors of both stripes.…”

PDE-based Morphology for Matrix Fields: Numerical Solution Schemes

“…To extend the notions of dilation (2) and erosion (3) to matrix fields a different method to calculate the supremum/infimum of a larger number of matrices A i has to be employed. Such a technique has been developed and described in detail in [10,12]. However, in order to keep this presentation as self-contained as possible we sketch this approach in the following.…”

“…In [10] novel matrix-valued partial differential equations governing dilation and erosion for matrix fields have been proposed and numerically solved by a first-order scheme of Osher and Sethian generalised to the setting of matrices. However, in the case of scalar images the morphological differential equations of dilation and erosion can also be solved numerically by a flux corrected transport (FCT) scheme as introduced in [8].…”

Morphological Processing

“…If Mathematical Morphology is well defined for binary and gray scale images, there exist no general extension which permits to perform basic operations on multivariate data since there is no natural ordering on vectors. Several ordering have been reported in literature to consider that problem but they are reduced to considering one specific type of images (color [1] or tensor [5] images).…”