Thinning grayscale well-composed images

Abstract

Usual approaches for constructing topological maps on discrete structures are based on cellular complexes topology. This paper aims to construct a coherent topological map defined on a square grid from a watershed transformation. The main idea behind the proposed approach is to impose some constraints on the original image in order to obtain good properties of the resulting watershed. We propose a definition of well-composed grayscale images based on the well-composed set theory and the cross-section topology. Properties of two different thinning algorithms are then studied and we show how to obtain a thin crest network. We derive an efficient algorithm that permits the construction of a meaningful topological map, resulting in a topological segmentation, i.e. a segmentation that describes in a coherent framework faces and contours. Finally, we demonstrate the usefulness of this algorithm for multilevel image segmentation.

Introduction

A digital image may be seen as the digitization of a piecewise continuous function. The discontinuities of this function are of primary importance in many shape recognition processes, as they usually describe the shapes of objects appearing on an image.

The notion of discontinuity is lost once a function is digitized. The structure describing domains where the underlying piecewise function is continuous is called a segmentation. Two main dual approaches to segmentation may be distinguished. The first approach consists of approximating the discrete function by a piecewise continuous function, and is usually referred to a region oriented segmentation. The second approach tries to directly catch the discontinuities of an underlying continuous function and is referred to a contour oriented segmentation. A topological segmentation may be viewed in this context as a process capturing both domains where an underlying piecewise function is continuous, and the set of points describing the discontinuities of the function.

A piecewise continuous function may be represented by a topological map, which can be viewed as a partition of the plane into three sets of points: a finite set S of points, a finite set A of disconnected Jordan arcs having elements of S as extremities, and a set of connected domains, the faces, whose boundaries are unions of elements of S and A. In this paper we aim to construct a topological segmentation that has a topological map structure consistent with a digital topological framework and with a topological map representing a continuous piecewise function.

Recent works (Braquelaire and Brun, 1998; Fiorio, 1996) aim to develop a coherent topological structure describing a digital image from the information provided by regions. The different structures described use a discrete topology based on the decomposition of the support domain of an image into three kinds of elements of different dimensions, i.e. surface elements, associated with the discrete points of the support, edge elements, which are the edges separating two surface elements, and vertices of the so defined grid (Fiorio, 1995; Kovalevsky, 1989). On one hand, such a partition has nice topological properties, but on the other hand, it suffers from many practical drawbacks, such as the amount of memory needed to store the entire partition, and difficulties faced when trying to construct the partition from the contour information.

Alternatively, a topological partition of an image may be directly defined by a digital topology involving only points on the square grid. In order to face the connectivity paradox, several neighborhood systems are usually used together. This is done either by considering different adjacency relations for points belonging to a set and its complement (Kong and Rosenfeld, 1989), or by assigning different neighborhoods to each point of $\text{Z}{}^2$ in a data independent manner, which can be formally stated using the framework proposed in (Khalimsky et al., 1990) (this framework can also be used in the cellular complex approach (Kovalevsky, 1989)).

Watersheds or more generally graytone skeletons can be used in this context to retrieve a vertex/arc network (crest lines) and faces (catchment basins) from a discrete topographic surface such as the modulus of the gradient viewed as a relief, which is exactly the sought for partition of the image.

However, many consistency problems are encountered on a square grid. Approaches that work by suppressing points from a potential crest network (grayscale thinning) or by adding points to connected sets of points (Arcelli, 1981; Bertrand et al., 1997) do not usually guarantee that the extracted crest network is thin (Fig. 1). Thick configurations of crests pose obvious problems when one is trying to link points from the resulting crests network in order to obtain digital curves and vertices of the topological partition. Approaches that work by linking potential crest points (Meyer, 1989; Pierrot Deseilligny et al., 1998), constructing a raster graph, do not usually guarantee that the faces defined by the cycles of the graph are composed of a unique connected component (Fig. 2).

Latecki proposed to face the problem of thickness of skeletons on digital binary images by forbidding some configurations of points (Latecki et al., 1995). He proposed a thinning operator that preserves the properties of the so called well-composed binary images, resulting in a thin skeleton. He also demonstrated a Jordan theorem that is verified on well-composed sets of points. He extended the property of well-composedness to multicolor images.

In this contribution, we first recall some classical notions of digital topology (Section 2) and some properties of well-composed sets (Section 3). We redefine grayscale well-composed images using the cross section topology formalism (Bertrand et al., 1996, Bertrand et al., 1997; Meyer, 1989), adapt a grayscale thinning algorithm to well-composed graylevel images, and prove some of its properties (Section 4). We derive an algorithm which constructs a topological partition from an irreductible thin well-composed image, and finally present an application (Section 5).