Efficient Encoding of n-D Combinatorial Pyramids

International audienceCombinatorial maps define a general framework which allows to encode any subdivision of an n-D orientable quasi-manifold with or without boundaries. Combinatorial pyramids are defined as stacks of successively reduced combinatorial maps. Such pyramids provide a rich framework which allows to encode fine properties of objects (either shapes or partitions). Combinatorial pyramids have first been defined in 2D, then extended using n-D generalized combinatorial maps. We motivate and present here an implicit and efficient way to encode pyramids of n-D combinatorial maps

Contains and Inside relationships within combinatorial Pyramids

Irregular pyramids are made of a stack of successively reduced graphs embedded in the plane. Such pyramids are used within the segmentation framework to encode a hierarchy of partitions. The different graph models used within the irregular pyramid framework encode different types of relationships between regions. This paper compares different graph models used within the irregular pyramid framework according to a set of relationships between regions. We also define a new algorithm based on a pyramid of combinatorial maps which allows to determine if one region contains the other using only local calculus.Comment: 35 page

Pyramids of n-Dimensional Generalized Maps

International audienceGraph pyramids are often used for representing irregular pyramids. Combinatorial pyramids have been recently defined for this purpose. We define here pyramids of n-dimensional generalized maps. This is the main contribution of this work: a generic definition in any dimension which extend and generalize the previous works. Moreover, such pyramids explicitly represent more topological information than graph pyramids. A pyramid can be implemented in several ways, and three representations are discussed in this paper

Hierarchical watersheds within the Combinatorial Pyramid framework

International audienceWatershed is the latest tool used in mathematical morphology. The algorithms which implement the watershed transform generally produce an over segmentation which includes the right image's boundaries. Based on this last assumption, the segmentation problem turns out to be equivalent to a proper valuation of the saliency of each contour. Using such a measure, hierarchical watershed algorithms use the edge's saliency conjointly with statistical tests to Decemberimate the initial partition. On the other hand, Irregular Pyramids encode a stack of successively reduced partitions. Combinatorial Pyramids consitute the latest model of this family. Within this framework, each partition is encoded by a combinatorial map which encodes all topological relationships between regions such as multInformation Processing Letterse boundaries and inclusion relationships. Moreover, the combinatorial pyramid framework provides a direct access to the embedding of the image's boundaries. We present in this paper a hierarchical watershed algorithm based on combinatorial pyramids. Our method overcomes the problems connected to the presence of noise both within the basins and along the watershed contours

Merge-and-simplify operation for compact combinatorial pyramid definition

International audienceImage pyramids are employed for years in digital image processing. They permit to store and use different scales/levels of details of an image. To represent all the topological information of the different levels, combinatorial pyramids have proved having many interests. But, when using an explicit representation, one drawback of this structure is the memory space required to store such a pyramid. In this paper, this drawback is solved by defining a compact version of combinatorial pyramids. This definition is based on the definition of a new operation, called "merge-and-simplify", which simultaneously merges regions and simplifies their boundaries. Our experiments show that the memory space of our solution is much smaller than the one of the original version. Moreover, the computation time of our solution is faster, because there are less levels in our pyramid than in the original one

Antiprismless, or: Reducing Combinatorial Equivalence to Projective Equivalence in Realizability Problems for Polytopes

This article exhibits a 4-dimensional combinatorial polytope that has no antiprism, answering a question posed by Bernt Lindst\"om. As a consequence, any realization of this combinatorial polytope has a face that it cannot rest upon without toppling over. To this end, we provide a general method for solving a broad class of realizability problems. Specifically, we show that for any semialgebraic property that faces inherit, the given property holds for some realization of every combinatorial polytope if and only if the property holds from some projective copy of every polytope. The proof uses the following result by Below. Given any polytope with vertices having algebraic coordinates, there is a combinatorial "stamp" polytope with a specified face that is projectively equivalent to the given polytope in all realizations. Here we construct a new stamp polytope that is closely related to Richter-Gebert's proof of universality for 4-dimensional polytopes, and we generalize several tools from that proof