The basic theory and applications of a set-theoretic approach to image analysis called mathematical morphology are reviewed. The goals are to show how the concepts of mathematical morphology geometrical structure in signals to illuminate the ways that morphological systems can enrich the theory and applications of multidimensional signal processing. The topics covered include: applications to nonlinear filtering (morphological and rank-order filters, multiscale smoothing, morphological sampling, and morphological correlation); applications to image analysis (feature extraction, shape representation and description, size distributions, and fractals); and representation theorems, which shows how a large class of nonlinear and linear signal operators can be realized as a combination of simple morphological operations. >

Morphological systems for multidimensional signal processing

Why mathematical morphology needs complete lattices

We develop a general theory of spatially-variant (SV) mathematical morphology for binary images in the euclidean space. The basic SV morphological operators (that is, SV erosion, SV dilation, SV opening, and SV closing) are defined. We demonstrate the ubiquity of SV morphological operators by providing an SV kernel representation of increasing operators. The latter representation is a generalization of Matheron's representation theorem of increasing and translation-invariant operators. The SV kernel representation is redundant, in the sense that a smaller subset of the SV kernel is sufficient for the representation of increasing operators. We provide sufficient conditions for the existence of the basis representation in terms of upper-semicontinuity in the hit-or-miss topology. The latter basis representation is a generalization of Maragos' basis representation for increasing and translation-invariant operators. Moreover, we investigate the upper-semicontinuity property of the basic SV morphological operators. Several examples are used to demonstrate that the theory of spatially-variant mathematical morphology provides a general framework for the unification of various morphological schemes based on spatially-variant geometrical structuring elements (for example, circular, affine, and motion morphology). Simulation results illustrate the theory of the proposed spatially-variant morphological framework and show its potential power in various image processing applications.

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Theoretical Foundations of Spatially-Variant Mathematical Morphology Part I: Binary Images

Locally adaptable mathematical morphology using distance transformations

This paper deals with the theory and applications of spatially-variant discrete mathematical morphology. We review and formalize the definition of spatially variant dilation/erosion and opening/closing for binary and gray-level images using exclusively the structuring function, without resorting to complement. This theoretical framework allows to build morphological operators whose structuring elements can locally adapt their shape and orientation across the dominant direction of the structures in the image. The shape and orientation of the structuring element at each pixel are extracted from the image under study: the orientation is given by means of a diffusion process of the average square gradient field, which regularizes and extends the orientation information from the edges of the objects to the homogeneous areas of the image; and the shape of the orientated structuring elements can be linear or it can be given by the distance to relevant edges of the objects. The proposed filters are used on binary and gray-level images for enhancement of anisotropic features such as coherent, flow-like structures. Results of spatially-variant erosions/dilations and openings/closings-based filters prove the validity of this theoretical sound and novel approach.

H. J. A. M. Heijmans

Papers

Mathematical morphology for structure without translation symmetry