Incremental Algorithm for Hierarchical Minimum Spanning Forests and Saliency of Watershed Cuts

Abstract

Abstract. We study hierarchical segmentations that are optimal in the sense of minimal spanning forests of the original image. We introduce a region-merging operation called uprooting, and we prove that optimal hierarchical segmentations are equivalent to the ones given by uprooting a watershed-cut based segmentation. Based on those theoretical results, we propose an efficient algorithm to compute such hierarchies, as well as the first saliency map algorithm compatible with the morphological filtering framework.

Constructive Links between Some Morphological Hierarchies on Edge-Weighted Graphs

“…Some of these methods (see [6,1]) also satisfy a "scale consistency property" that assesses the robustness of the detected contours and regions over scales. Given three image seed points x, y, and z that mark three objects of interest, a segmentation S into three regions obtained from the three seeds x, y and z (i.e., each region contains one seed) "is consistent" with a segmentation S into two regions obtained from the two seeds x and y if when a pixel belongs to the region of a seed in S, then it necessarily belongs to the region of S that contains this seed.…”

Optima on Hierarchies of Partitions

“…We start from the energy proposed in [10], and change it by adding new terms. A hierarchy H of Uppsala ducks has been obtained by previous segmentations of the luminance l = (r + g + b)/3 based on [5]. We want to find the best cut for a compression rate of 20.…”

Climbing: A Unified Approach for Global Constraints on Hierarchical Segmentation

“…A hierarchy, or pyramid, of image segmentations is classically understood as a series of progressive simplified versions of an initial image, which result in increasing partitions of the space. In the following, we do not aim to focus on the methods for obtaining pyramids of segmentation, and consider rather the whole hierarchies as a starting point 1 . Indeed, a multi-scale image description can rarely be considered as an end in itself.…”

“…It often requires to be completed by some energy function that allows us to formalize optima, and to summarizes a hierarchy into some "optimal cut". Three questions arise then, namely: 1 The main techniques for hierarchical segmentation include functional minimizations of Mumford and Shah type, semi-groups of morphological filters, and progressive floodings on watersheds [1]. In addition, the learning strategies for segmentation, as developed by [2], or by [3], among others, lead to very significant hierarchies.…”

“…The partitions of a hierarchy may be represented by their classes, or by the saliency map of the edges [13], [1], as depicted in Figure 1, or again by a family tree where each node of bifurcation is a class S, as depicted in Figure 2. The classes of π i−1 at level i − 1 which are included in class S i are said to be the sons of S i .…”