Quasi-Linear Algorithms for the Topological Watershed

Abstract

The watershed transformation is an efficient tool for segmenting grayscale images. An original approach to the watershed [1,9] consists in modifying the original image by lowering some points while preserving some topological properties, namely, the connectivity of each lower cross-section. Such a transformation (and its result) is called a W-thinning, a topological watershed being an "ultimate" W-thinning. In this paper, we study algorithms to compute topological watersheds. We propose and prove a characterization of the points that can be lowered during a W-thinning, which may be checked locally and efficiently implemented thanks to a data structure called component tree. We introduce the notion of M-watershed of an image F , which is a W-thinning of F in which the minima cannot be extended anymore without changing the connectivity of the lower cross-sections. The set of points in an M-watershed of F which do not belong to any regional minimum corresponds to a binary watershed of F . We propose quasi-linear algorithms for computing Mwatersheds and topological watersheds. These algorithms are proved to give correct results with respect to the definitions, and their time complexity is analyzed.

Watershed Cuts: Thinnings, Shortest Path Forests, and Topological Watersheds

“…As far as we know, the watershed algorithms available in the literature (e.g., [4], [8], [9], [13], [14], [18]) all require either a sorting step, a hierarchical queue, or a data structure to maintain a collection of disjoint sets under the operation of union. On one hand, the global complexities of a sorting step and of a (monotone) hierarchical queue (i.e., a structure from which the elements can be removed in the order of their altitude) are equivalent [34]: They both run in linear time only if the range of the weights is sufficiently small.…”

“…8e). A topological watershed and its divide constitute an interesting segmentation, which satisfies important properties (see [18], [23], [25]) not guaranteed by most popular watershed algorithms. In particular, in [23], [25], the equivalence between a class of transformations which preserves the connection value and the W-thinnings is proved.…”

“…We stress that Algorithm M-kernel and the one introduced in [16] are the first watershed algorithms that satisfy such a property. Indeed, as far as we know, the watershed algorithms available in the literature (e.g., [4], [8], [9], [13], [14], [18]) all require either a sorting step, a hierarchical queue, or a data structure to maintain a collection of disjoint sets under the operation of union, and none of these operations can be performed in linear time whatever the range of the weight function. Moreover, in practice, the algorithm proposed in this paper is more flexible than the one proposed in [16].…”

Ultrametric Watersheds

“…Future papers will propose novel algorithms (based on the topological watershed algorithm [31]) to compute ultrametric watersheds, with proof of correctness.…”

“…For example, scale-sets theory considers a rather general formulation of the partitioning problem which involves minimizing a two-term-based energy, of the form λC + D, where D is a goodness-of-fit term and C is a regularization term, and proposes an algorithm to compute the hierarchical segmentation we obtain by varying the λ parameter. We can hope that the topological watershed algorithm [31] can be used on a specific energy function to directly obtain the hierarchy. -Subdominant theory (mentionned at the end of section 4) links hierachical classification and optimisation.…”