Advances in Constrained Connectivity
The concept of constrained connectivity [Soille 2008, PAMI] is summarised. We then introduce a variety of measurements for characterising connected components generated by constrained connectivity relations. We also propose a weighted mean for estimating
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Abstract. The concept of constrained connectivity [Soille 2008, PAMI] is summarised. We then introduce a variety of measurements for characterising connected components generated by constrained connectivity relations. We also propose a weighted mean for estimating a representative value of each connected component. Finally, we define the notion of spurious connected components and investigate a variety of methods for suppressing them.
1
Introduction
A segmentation of the definition domain X of an image is usually defined as a partition of X into disjoint connected subsets Xi , . . . , Xn (called segments) such that there exists a logical predicate P returning true on each segment but false on any union of adjacent segments [1]. That is, a series of subsets Xi of the definition domain X of an image forms a segmentation of this image if and only if the following four conditions are met (i) ∪i (Xi ) = X, (ii) Xi ∩ Xj = ∅ for all i = j, (iii) P (Xi ) = true for all i, and (iv) P (Xi ∪ Xj ) = false if Xi and Xj are adjacent. With this classical definition of image segmentation, given an arbitrary logical predicate, there may exist more than one valid segmentation. For example, the logical predicate returning true on segments containing one and only one regional minimum and false otherwise lead to many possible segmentations. The watershed transformation definition considers the additional constraint that there should exist a steepest slope path linking each pixel of the segment to its corresponding minimum for the logical predicate to return true. Still, this does not guarantee that there is a unique solution because the steepest slope path of a pixel is not necessarily unique (problem of ties). If uniqueness of the result is required, logical predicates based on equivalence relations should be considered1 . Indeed, it has been known for a long time that there exists a one-to-one correspondence between the partitions of a set and the equivalence relations on it, e.g., [2, p. 48]. Since connectivity relations are equivalence relations, logical predicates based on connectivity relations naturally lead 1
A binary relation which is reflexive, symmetric, and transitive is called an equivalence relation.
D. Coeurjolly et al. (Eds.): DGCI 2008, LNCS 4992, pp. 423–433, 2008. c Springer-Verlag Berlin Heidelberg 2008
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P. Soille and J. Grazzini
to unique segmentations. For example, the trivial connectivity relation stating that two pixels are connected if and only if they can be joined by an iso-intensity path breaks digital images into segments of uniform grey scale [3]. They are called plateaus in fuzzy digital topology [4] and flat zones in mathematical morphology [5]. In most cases, the equality of grey scale is a too strong homogeneity criterion so that it produces too many segments. Consequently, the resulting partition is too fine. A weaker connectivity relation consists in stating that two pixels of a grey tone image are connected if there exists a path of pixels linking these pixels and such that the grey level diffe
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