Dougherty E.R. (ed.) Mathematical Morphology in Image Processing

Marcel Dekker, 1993. — 545 p.As a nonlinear branch of image and signal processing, mathematical morphology represents a dramatic break with classical linear processing and facilitates the application of various mathematical disciplines to the processing and analysis of images. These areas include nonlinear statistics, logic, geometry, geometrical probabil ity, topology, and various algebraic systems such as the theories of lattices and groups. It is not that these disciplines have not played imaging roles outside mathematical morphology, but rather that they appear naturally within the context of mathematical morphology and are central to its development in both theory and application.
The present volume is testimony to the ferti lity of the original Matheron-Serra conception and to the great expansion of interest in mathematical morphology that has taken place over the past few years. Many researchers-mathematicians, statistician s, engineers, computer scienti sts, and natural scientistshave taken interest in the field. As a result, a number of productive research veins are currently being explored. These directions are well represented in this volume. Although it is not possible to characterize the contributions crisply, they can be roughly placed into five active areas within morphological image processing. The first three chapters are statistical, dealing with approaches to finding well-petfonning structuring elements and the statistical analysis of morphological operations as noise filters. Chapters 4, 5, and 11 concern morphological feature generation for classification. Chapters 6, 7, and 13 concern extension of the morphological paradigm, and they illustrate the degree to wmch mathematical morphology provides promising ground for algebraists. Chapters 8, 9, and 10 treat topics having to do with efficient morphological algorithms, the latter containing theoretical results whose importance extends well beyond algorithmic efficiency. Finally, Chapter 12 describes a key paradigm for morphological image segmentation. Each chapter is self-contained and can be read in any order, depending on preference. Perhaps brief descriptions of the chapters' places within mathematical morphology will prove useful.
Successful application of mathematical morphology often depends on the selection of one or more appropriate structuring elements. Except in simple problems, human structuring-element selection can prove daunting. This is especially true when one wishes to penorm image restoration or shape recognition in the presence of noise. Currently, much interest is centered on automatic des ign of structuring elements. In the first chapter, Stephen Wilson introduces a train ing method based on Hebbian learning to find appropriate hit-and-miss structuring elements for character recognition in noise. Generally, overly dense structuring elements lower detection rates, whereas too sparse structuring elements raise false-positive rates. The network approach attempts to limit both types of errors by judiciously placing structuring-element pixels.
Image restoration by morphological fi ltering can be placed into the classical mean-square-error paradigm; however, here too human structuring-element design is not practical. Using the Matheron erosion representation as the general restoration operator, one needs to find a basis of structuring elements providing an optimal filter. Because of the nonlinear and combinatoric nature of the problem, even when computers are used design-time efficiency is problematic. In the second chapter, Robert P. Loce and I discuss design strategies based on search-space constraint and efficient estimation of image statistics. The chapter provides a practical multiple-level approach to circumventing computation limitations in the automatic design ofrestoration structuring elements.
If a filter is to be treated as a statistical estimator, the input image must be treated as a random process From a statistical perspective, the input and output processes possess distributions and one would like to express the output distribution in terms of the input distribution, or, with less completeness, some output moments in terms of input moments. In the third chapter, Jaakko Astola, Lasse Koskinen, and Yrjo Neuvo derive output statistics resulting from application of certain basic flat morphological filters to independent random variables. They utilize the equivalence between stack and flat-morphological filters in their derivations. Their treatment includes analysis of flat openings and flat closings.Training Structuring Elements in Morphological Networks.
Efficient Design Strategies for the Optimal Binary Digital Morphological Filter: Probabilities, Constraints, and Structuring-Element Libraries.
Statistical Properties of Discrete Morphologica1 Filters.
Morphological Ana1ysis of Pavement Surface Condition.
On Two Inverse Problems in Mathematical Morphology.
Graph Morphology in [mage Analysis.
Mathematical Morphology with Noncommutative Symmetry Groups.
Morphological Algorithms.
Discrete Half-Plane Morphology for Restricted Domains.
On a Distance Function Approach for Gray-Level Mathematical Morphology.
Invariant Characterizations and Pseudocharacterizations of FiniteMultidimensional Sets Based on Mathemati cal Morphology.
The Morphological Approach to Segmentation: The Watershed Transformation.
Anamorphoses and Function Lattices (Multivalued Morphology).