Removal and Contraction Operations in nnD Generalized Maps for Efficient Homology Computation

In this paper, we show that contraction operations preserve the homology of $n$D generalized maps, under some conditions. Removal and contraction operations are used to propose an efficient algorithm that compute homology generators of $n$D generalized maps. Its principle consists in simplifying a generalized map as much as possible by using removal and contraction operations. We obtain a generalized map having the same homology than the initial one, while the number of cells decreased significantly. Keywords: $n$D Generalized Maps; Cellular Homology; Homology Generators; Contraction and Removal Operations.Comment: Research repor

First Results for 3D Image Segmentation with Topological Map

International audienceThis paper presents the first segmentation operation defined within the 3D topological map framework. Firstly we show how a traditional segmentation algorithm, found in the literature, can be transposed on a 3D image represented by a topological map. We show the consistency of the results despite of the modifications made to the segmentation algorithm and we study the complexity of the operation. Lastly, we present some experimental results made on 3D medical images. These results show the process duration of this method and validate the interest to use 3D topological map in the context of image processing

Comparison of Local and Global Region Merging in the Topological Map

International audienceThe topological map is a model that represents 2D and 3D images subdivision. It aims to allow the use of topological and geometrical features of the subdivision in image processing operations. When handling regions in an image, one of the main operation is the region merging, for example in segmentation process. This paper presents two algorithms of region merging in 3D topological maps: one local which modifies locally the map around merged regions, and another one global which runs through all the elements of the map. We study their complexities and present experimental results to compare both approaches

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

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

Computing Canonical Polygonal Schemata with Generalized Maps

International audienceThis paper shows that a well-known algorithm proposed to compute the canonical polygonal schema of a surface can be transferred onto a 2-dimensional generalized map. We show that transformation rules on polygonal schemata can be achieved in O(1) with generalized maps, which can help optimizing existing algorithms

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

Computing Homology Generators for Volumes Using Minimal Generalized Maps

International audienceIn this paper, we present an algorithm for computing efficiently homology generators of 3D subdivided orientable objects which can contain tunnels and cavities. Starting with an initial subdivision, represented with a generalized map where every cell is a topological ball, the number of cells is reduced using simplification operations (removal of cells), while preserving homology. We obtain a minimal representation which is homologous to the initial object. A set of homology generators is then directly deduced on the simplified 3D object

Incremental Updating of 3D Topological Maps to Describe Videos

International audienceA topological map is an efficient mathematical model for representing an image subdivision where all cells and adjacency relations between elements are represented. It has been proved to be a very good tool for video processing when video is seen as a 3D image. However the construction of a topological map for representing a video needs the availability of the complete image sequence. In this paper we propose a procedure for online updating a topological map in order to build it as the video is produced, allowing to use it in real time

Using 2D Topological Map Information in a Markovian Image Segmentation

International audienceTopological map is a mathematical model of labeled image representation which contains both topological and geometrical information. In this work, we use this model to improve a Markovian seg-mentation algorithm. Image segmentation methods based on Markovian assumption consist in optimizing a Gibbs energy function. This energy function can be given by a sum of potentials which could be based on the shape or the size of a region, the number of adjacencies,.. . and can be computed by using topological map. In this work we propose the integration of a new potential: the global linearity of the boundaries, and show how this potential can be extracted from the topological map. Moreover, to decrease the complexity of our algorithm, we propose a local modification of the topological map in order to avoid the reconstruction of the entire structure