Morphological Image Analysis

Mathematical morphology (MM) or simply morphology can be defined as a theory for the analysis of spatial structures. It is called morphology because it aims at analysing the shape and form of objects. It is mathematical in the sense that the analysis is based on set theory, integral geometry, and lattice algebra. MM is not only a theory, but also a powerful image analysis technique. The purpose of this book is to provide a detailed presentation of the principles and applications of morphological image analysis. The emphasis is therefore put on the technique rather than the theory underlying MM. Besides, any non-specialist faced with an image analysis problem rapidly realises that a unique image transformation usually fails to solve it. Indeed, most image analysis problems are very complex and can only possibly be solved by a combination of many elementary transformations. In this context, knowledge of the individual image processing operators is a necessary but not sufficient condition to find a solution: guidelines and expert knowledge on the way to combine the elementary transformations are also required. Hence, beyond the presentation of the morphological operators, we will describe many real applications to help the reader acquiring the expert knowledge necessary for building the chain of operators to resolve his/her own image analysis problem.

Morphological operators aim at extracting relevant structures of the image considered through its subgraph representation. This is achieved by probing the image with another set of known shape called structuring element (SE). The shape of the SE is usually chosen according to some a priori knowledge about the geometry of the relevant and irrelevant image structures. By irrelevant structures, we mean either noise or objects we would like to suppress.

The erosion of an image not only removes all structures that cannot contain the structuring element but it also shrinks all the other ones. The search for an operator recovering most structures lost by the erosion leads to the definition of the morphological opening operator. The principle consists in dilating the image previously eroded using the same structuring element. In general, not all structures are recovered. For example, objects completely destroyed by the erosion are not recovered at all. This behaviour is at the very basis of the filtering properties of the opening operator: image structures are selectively filtered out, the selection depending on the shape and size of the SE. The dual operator of the morphological opening is the morphological closing. Both operators are at the basis of the morphological approach to image filtering developed in Chap. 8.

All morphological transformations discussed so far involved combinations of one input image with specific structuring elements. The approach taken with geodesic transformations is to consider two input images. A morphological transformation is applied to the first image and it is then forced to remain either greater or lower to the second image. Authorised morphological transformations are restricted to elementary erosions and dilations. The choice of specific structuring elements is therefore eluded. In practice, geodesic transformations are iterated until stability making the choice of a size unnecessary. It is actually the combination of appropriate pairs of input images which produces new morphological primitives. These primitives are at the basis of formal definitions of many important image structures for both binary and grey scale images.

The distance between Sydney and Berlin is referred to as a geodesic distance because, contrary to the Euclidean distance, the shortest path linking two points on the earth is constrained to follow the surface of the geoid. In image analysis, geodesic distances are used wherever paths linking image pixels are constrained to remain within a subset of the image plane. The region thus defined is called geodesic mask. For example, when planning the path of a robot, the geodesic mask corresponds to the regions where it can move.

In signal processing, a filter is usually defined as a linear, shift-invariant operation.

The most commonly used abstract model for pattern recognition is the classification model. This model contains three parts: a transducer, a feature extractor, and a classifier (Duda & Hart, 1973). The transducer senses the input and converts it to a form suitable for computer processing. The feature extractor extracts presumably relevant information from the input data. The classifier uses this information to assign the input data to one of a finite number of known categories or classes.

Many examples of applications have been presented and referred to throughout the book. This last chapter provides a brief overview of other published papers reporting on applications that have been tackled with morphological algorithms. The papers are sorted by application fields (Secs. 11.1–11.7): geosciences and remote sensing, materials science, biomedical imaging, industrial applications, identification and security control, document processing, and image coding. Topics not fitting into these categories are discussed in Sec. 11.8. The chapter concludes with further links to morphology and the reference list.