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

Removal and Contraction for n-Dimensional Generalized Maps

International audienceRemoval and contraction are basic operations for several methods conceived in order to handle irregular image pyramids, for multi-level image analysis for instance. Such methods are often based upon graph-like representations which do not maintain all topological information, even for 2-dimensional images. We study the definitions of removal and contraction operations in the generalized maps framework. These combinatorial structures enable us to unambiguously represent the topology of a well-known class of subdivisions of n-dimensional (discrete) spaces. The results of this study make a basis for a further work about irregular pyramids of n-dimensional images

Open Issues and Chances for Topological Pyramids

High resolution image data require a huge amount of computational resources. Image pyramids have shown high performance and flexibility to reduce the amount of data while preserving the most relevant pieces of information, and still allowing fast access to those data that have been considered less important before. They are able to preserve an existing topological structure (Euler number, homology generators) when the spatial partitioning of the data is known at the time of construction. In order to focus on the topological aspects let us call this class of pyramids “topological pyramids”. We consider here four open problems, under the topological pyramids context: The minimality problem of volumes representation, the “contact”-relation representation, the orientation of gravity and time dimensions and the integration of different modalities as different topologies.Austrian Science Fund P20134-N13Junta de Andalucía FQM–296Junta de Andalucía PO6-TIC-0226

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

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

Tiled top-down pyramids and segmentation of large histological images

International audienceRecent microscopic imaging systems such as whole slide scanners provide very large (up to 18GB) high resolution images. Such amounts of memory raise major issues that prevent usual image representation models from being used. Moreover, using such high resolution images, global image features, such as tissues, do not clearly appear at full resolution. Such images contain thus different hierarchical information at different resolutions. This paper presents the model of tiled top-down pyramids which provides a framework to handle such images. This model encodes a hierarchy of partitions of large images defined at different resolutions. We also propose a generic construction scheme of such pyramids whose validity is evaluated on an histological image application

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

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

Algorithm to Compute a Minimal Length Basis of Representative Cocycles of Cohomology Generators

An algorithm to compute a minimal length basis of representative cocycles of cohomology generators for 2D images is proposed. We based the computations on combinatorial pyramids foreseeing its future extension to 3D objects. In our research we are looking for a more refined topological description of deformable 2D and 3D shapes, than they are the often used Betti numbers. We define contractions on the object edges toward the inner of the object until the boundaries touch each other, building an irregular pyramid with this purpose. We show the possible use of the algorithm seeking the minimal cocycles that connect the convex deficiencies on a human silhouette. We used minimality in the number of cocycle edges in the basis, which is a robust description to rotations and noise