Scale-Sets Image Analysis

Abstract

This paper introduces a multi-scale theory of piecewise image modelling, called the scale-sets theory, and which can be regarded as a region-oriented scale-space theory. The first part of the paper studies the general structure of a geometrically unbiased region-oriented multi-scale image description and introduces the scale-sets representation, a representation which allows to handle such a description exactly. The second part of the paper deals with the way scale-sets image analyses can be built according to an energy minimization principle. We consider 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. We describe the way such energies arise from basic principles of approximate modelling and we relate them to operational rate/distorsion problems involved in lossy compression problems. We then show that an important subset of these energies constitutes a class of multi-scale energies in that the minimal cut of a hierarchy gets coarser and coarser as parameter λ increases. This allows us to devise a fast dynamic-programming procedure to find the complete scale-sets representation of this family of minimal cuts. Considering then the construction of the hierarchy from which the minimal cuts are extracted, we end up with an exact and parameterfree algorithm to build scale-sets image descriptions whose sections constitute a monotone sequence of upward global minima of a multi-scale energy, which is called the "scale climbing" algorithm. This algorithm can be viewed as a continuation method along the scale dimension or as a minimum pursuit along the operational rate/distorsion curve. Furthermore, the solution verifies a linear scale invariance property which allows to completely postpone the tuning of the scale parameter to a subsequent stage. For computational reasons, the scale climbing algorithm is approximated by a pair-wise region merging scheme: however the principal properties of the solutions are kept. Some results obtained with Mumford-Shah's piece-wise constant model and a variant are provided and different applications of the proposed multi-scale analyses are finally sketched.

Constructive Links between Some Morphological Hierarchies on Edge-Weighted Graphs

“…In fact, hierarchical segmentation methods are not limited to edge-weighted graphs (see e.g., [7][8][9][10]). In particular, in mathematical morphology, component trees [11], quasi-flat zones [12,13], binary partition trees [14] and watersheds [15][16][17] are hierarchies at the basis of efficient segmentation and filtering methods.…”

Optima on Hierarchies of Partitions

“…Computationally, the h-increasing condition (3) allows us to reach the optimal cut in one ascending pass, by the following Guigues'algorithm [6]:…”

“…under the name of separable energies [6], for partitions with connected classes. For the aim of scale increasingness, the energy ω λ (S) is written as a linear function of λ:…”

“…Some of them [10], [6] simplify the combinatorial complexity by assuming that the energy of a partition equals the sum of the energies of its classes, which permits the treat the above questions 1) and 3). However, one can wonder whether additivity is the very underlying cause, since alternative approaches [12] replace the sums by suprema and obtain similar properties.…”

A Stochastic Evaluation of the Contour Strength

“…There a two main classes of methods for extracting the contours of interest from the hierarchy, in order to get the final segmentation. In the first a functional is defined and the set of contours maximizing this functional is extracted from the hierarchy [1,2]. Another approach to segmentation uses seeds or markers for the regions of interest (including the background).…”