Combinatorial maps : efficient data structures for computer graphics and image processing


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Abstract: We address the problem of searching for a pattern in a plane graph, that is, a planar drawing of a planar graph. We define plane subgraph isomorphism and give a polynomial algorithm for this problem. We show that this algorithm may be used even when the pattern graph has holes. Abstract: In this paper, we address the problem of searching for a pattern in a plane graph, i. To do that, we propose to model plane graphs with 2-dimensional combinatorial maps, which provide nice data structures for modelling the topology of a subdivision of a plane into nodes, edges and faces.

We define submap isomorphism, we give a polynomial algorithm for this problem, and we show how this problem may be used to search for a pattern in a plane graph. First experimental results show the validity of this approach to efficiently search for patterns in images. Abstract: The extraction of a 3D topological map from an Oriented Boundary Graph can be needed to refine a 3D Split and Merge segmentation using topological information such as the Euler characteristic of regions.

A presegmentation could thus be efficiently obtained using a light structuring model, before proceeding to the extraction of more information on some regions of interest. In this paper, we present the topological map extraction algorithm which allows to reconstruct locally a set of regions from the Oriented Boundary Graph. A comparison of the two models is also presented. Abstract: Hierarchical data structures such as irregular pyramids are used by many applications related to image processing and segmentation.

The construction scheme of such pyramids is bottom-up. Such a scheme forbids the definition of a level according to more global information defined at upper levels in the hierarchy.


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Moreover, the base of the pyramid has to encode any single pixel of the initial image in order to allow the definition of regions of any shape at higher levels. This last constraint raises major issues of memory usage and processing costs when irregular pyramids are applied to large images. The objective of this paper is to define a top-down construction scheme for irregular pyramids. Each level of such a pyramid is encoded by a combinatorial map associated to an explicit encoding of the geometry and the inclusion relationships of the corresponding partition.

The resulting structure is a stack of finer and finer partitions obtained by successive splitting operations and is called a top-down pyramid. Abstract: In this paper, we present a generic topological and geometrical framework for the digital reconstruction of complex contours from labeled images.

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The proposed technique is based on combinatorial map simplifications guided by digital straight segments. We illustrate the genericity of the framework with a parallel contour reconstruction algorithm. Abstract: The comparison of two topological surfaces is a frequent question that may be solved by using accurate topological invariants.

Such invariants already exist and the main difficulty is to achieve their computation efficiently. In [VegYap90], authors propose an algorithm which allows to compute the canonical polygonal schema of a given surface by using particular transformation rules. This paper shows that these transformations can be defined on a 2-dimensional generalized map which encodes a surface. The algorithm proposed in [VegYap90] to compute canonical polygonal schema can now be transfered onto generalized maps.

Moreover, all transformations can be achieved in O 1 with generalized maps, which can help optimizing existing algorithms. Abstract: This paper presents incremental algorithms used to compute Betti numbers with topological maps. Their implementation, as topological criterion for an existing bottom-up segmentation, is explained and results on artificial images are shown to illustrate the process. Abstract: This 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 in order to handle topological maps. We study the complexity of the operation. Lastly, we present some experimental results mainly for 3D medical images.

These results show the process duration of our method and validate the interest to use 3D topological map in the context of image processing. Abstract: Hierarchical representations, as irregular pyramids, are the bases of several applications in the field of discrete imagery. So, n-dimensional "bottom-up" irregular pyramids can be defined as stacks of successively reduced n-dimensional generalized maps n-G-maps [13], each n-G-map being defined from the previous level by using removal and contraction operations defined in [18].

Our goal is to build a theoretical framework for the definition and handling of n-dimensional "top-down" irregular pyramids. To do that, we propose in this paper to study the definition of insertion and expansion operations that allows to conceive this kind of pyramids. Abstract: In 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. Abstract: Topological maps are a model which represents 2D or 3D image subdivisions. The aim of this structure is to provide an image representation which allows topological and geometrical features to be used by image processing operations.

Algorithm of region merging is an useful tool in image segmentation and image processing. 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. Abstract: In this paper, we present a topological algorithm which allows to fill contours images. The filling problem has been widely treated and it recently appeared that it can always be split into two different process : a generic topological process and a dedicated geometrical post-processing which depends on the application.

Our algorithm, based on a 2D topological map description of the image, addresses the first step of processing. It is fast, generic and robust. Moreover, the complete topological description allows to easily integrate geometrical constrains and makes our approach an interesting basis for every filling process. Abstract: This paper presents a new method for reconstructing geometry and topology of 3D buildings from 2D architectural plans. A complete topological model expresses incidence and adjacency relations between all the elements.

It is necessary for both recovering accurately 2D information and constructing a coherent 3D building. Based on an existing topological kernel, several high-level operations have been developped in 2D for creating walls, portals, stairs, etc. Semantic information is associated with all volumes for specifying openings, walls, rooms, stairs, facade, etc. The resulting 2D model is extruded for generating a 3D environment, taking the semantic information into account since doors are not processed as walls for instance.

Floors are superimposed using volumes corresponding to upper and lower ceilings linked according to stairways. Abstract: In this paper, we present an algorithm which allows to compute efficiently generators of the first homology group of a closed surface, orientable or not. Starting with an initial subdivision of a surface, we simplify it to its minimal form minimal refers to the number of cells , while preserving its homology.

Homology generators can thus be directly deduced from the minimal representation of the initial surface. Finally, we show how this algorithm can be used in a 3D labelled image in order to compute homology of each region described by its boundary. Abstract: Graph pyramids are often used to represent an image with various levels of details. Generalized pyramids have been recently defined in order to deal with images in any dimension. In this work, we show how to use generalized pyramids to represent 3D multi-level segmented images.

We show how to construct such a pyramid, by alternating segmentation and simplification steps.

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We present how cells to be removed are marked: by using an homogeneous criterion to mark faces and the cell degree to mark other cells. When the pyramid is constructed, the main problem consists in retrieving information on regions. In this work, we show how to retrieve two types of information. The first one is the low level cells that are merged into a unique high level cell. The second one is the inter-voxel cells that compose a given region.

Abstract: In this paper, we show how to use the three dimensional topological map in order to compute efficiently topological features on objects contained in a 3D image. These features are useful for example in image processing to control operations or in computer vision to characterize objects. Topological map is a combinatorial model which represents both topological and geometrical information of a three dimensional labeled image. This model can be computed incrementally by using only two basic operations: the removal and the fictive edge shifting. In this work, we show that Euler characteristic can be computed incrementally during the topological map construction.

This involves an efficient algorithm and open interesting perspectives for other features. Abstract: A pyramid of n-dimensional generalized maps is a hierarchical data structure. It can be used, for instance, in order to represent an irregular pyramid of n-dimensional images. In this paper, we define generalized orbits, which extend the classical notion of receptive fields. Generalized orbits allow to establish the correspondence between a cell of a pyramid level and the set of cells of previous levels, the removal or contraction of which have led to the creation of this cell. In order to define generalized orbits, we extend, for generalized map pyramids, the notion of connecting walk defined by Brun and Kropatsch.

Abstract: Graph 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.

Abstract: In this paper we compare two topological models of 3D segmented images representation. These models are based on a collaboration between a topological representation and a geometrical representation of the regions of the segmented image. In both models the description of the topology lays on topological maps with a combinatorial representation. A geometrical embedding of maps is used to describe the geometry of regions. Both models differ in the way of defining this embedding. The aim of this paper is to compare these two models from the point of view of their interest for image segmentation, and to explore the possibility and the interest of making both these models converge.

Such methods are often based upon graph-like representations which do not maintain all topological information, even for 2-dimensional images.

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Ebook Combinatorial Maps Efficient Data Structures For Computer Graphics And Image Processing

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.

Abstract: Topological 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 segmentation 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, 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. Abstract: Image segmentation methods based on Markovian assumption consist in optimizing a Gibbs energy function which depends on the observation field and the segmented field.

This energy function can be represented as a sum of potentials defined on cliques which are subsets of the grid of sites. The Potts model is the most commonly used to represent the segmented field. However, this model only expresses a potential on classes for nearest neighbor pixels. In this paper, we propose the integration of global informations, like the size of a region, in the local potentials of the Gibbs energy. To extract these informations, we use a representation model well known in geometric modeling: the topological map.

Results on synthetic and natural images are provided, showing improvements in the obtained segmented fields. Abstract: One of the most commonly used approach to segment a 2D image is the split and merge approach. In this paper, we are defining these two operations in 3D within the topological maps framework. This mathematic model of regions segmented image representation allows us to define these algorithms in a local and generic way.

Moreover, we are defining a new operation, the corefining, which allows to treat big images. They are cut into small units, treated separately, then the result of each of them are combined to reconstruct the final representation. These three operations let us view efficient 3D segmentation algorithms, which is a difficult problem due to the size of data to treat.

Graph Data Structure 1. Terminology and Representation (algorithms)

Abstract: In this paper we define formally and completely the 3d topological map introduced in BertrandAl a model which encodes totally and minimally all the topological information of a 3d image. In particular, we focus on the problem of disconnections induced by the constructive definition based on several levels of maps.

This multilevel representation is more or less a graph pyramid in sense that each level can be compute from the previous one in term of merge operations. Furthermore, algorithms extracting these maps from a segmented image have been given in BertrandAl00 and have been implemented and tested in practical applications. Abstract: In this paper we define the 3d topological map and give an optimal algorithm which computes it from a segmented image.

This data structure encodes totally all the information given by the segmentation. More, it allows to continue segmentation either algorithmically or interactively. We propose an original approach which uses several levels of maps. This allows us to propose a reasonable and implementable solution where other approaches don't allow suitable solutions. Moreover our solution has been implemented and the theoretical results translate very well in practical applications.

Abstract: The Trashcam project developed a rapid, automated method used to evaluate the seed coat fragment SCF potential of fibers from new cotton plant lines. A camera is used to acquire an image of the fiber web and this is then analyzed by image processing. The latest version of the product, has improved image quality by using a scanner for image acquisition. We compare the automated counts obtained with counts made by an 'expert' in cotton web images. Thanks to the new image acquisition technique and an algorithm whose parameters may be adjusted, the results obtained show that the square root of the automated counts are proportional to the square root of the visual counts.

Abstract: Different image-processing methods were used to count seed coat fragments SCF in cotton fiber webs. The results obtained showed that, similar to SCF detection on yarn, it is possible to use this technique to prdict the 'SCF potential' of lines during varietal improvement programs. The last method was tested on images derived from carded cottons of various geographic origins. Abstract: This chapter shows how combinatorial maps can be used for 2D or 3D image segmentation. We start by introducing combinatorial maps and we show how they can be used to describe image partitions.


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Then, we present a generic segmentation algorithm that uses and modifies the image partition represented by a combinatorial map. One advantage of this algorithm is that one can mix different criteria and use different image features which can be associated with the cells of the partition. In particular, it is interesting that the topological properties of the image partition can be controlled through this approach. This property is illustrated by the computation of classical topological invariants, known as Betti numbers, which are then used to control the number of cavities or the number of tunnels of regions in the image partition.

Finally, we present some experimental results of 2D and 3D image segmentation using different criteria detailed in this chapter. Dans ce chapitre nous montrons comment utiliser ces cartes combinatoires en analyse d'images. Abstract: This package implements Generalized Maps in d dimensions. A generalized map is a data structure enabling to represent an orientable or non orientable subdivided object by describing all the cells of the subdivision for example in 3D vertices, edges, faces, volumes and all the incidence and adjacency relationships between these cells.

Information can be associated to cells thanks to attributes. The package provides basic creation, modification operations, and several iterators enabling to run through some specific part of the object. Abstract: This package implements linear cell complexes, objects in d-dimension with linear geometry. The combinatorial part of objects is described either by a combinatorial or a generalized map, representing all the cells of the object plus the incidence and adjacency relations between cells.

GA and n -gmaps complement and reinforce each other.

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For instance; it improves the robustness when computing the structure from the geometrical information. It is capable of computing lengths, areas and volumes of any polyhedral complex with or without holes using the orbits of the n -gmap some examples are given. Finally the paper gives hints about other potentialities. We would like to thank: The help of Guillaume Damiand for his help in solving doubts about handling the [ 46 ] software and for sharing it freely. Randolph Franklin by his kind advice about the computation of lengths, areas and volumes.

The reviewers specialized in both disciplines; n -gmaps and GA, for their constructive and helpful comments. Skip to main content. Advertisement Hide. Article First Online: 08 January This is a preview of subscription content, log in to check access. Acknowledgements We would like to thank: The help of Guillaume Damiand for his help in solving doubts about handling the [ 46 ] software and for sharing it freely. Alayrangues, S. Accessed 07 May Imaging Vis. Baig, S.

Bellet, T. Braulio-Gonzalo, M. Energy Build. Brisson, E. Discrete Comput. Chard, J. New citations to this author. New articles related to this author's research. Email address for updates.

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My profile My library Metrics Alerts. Sign in. Get my own profile Cited by View all All Since Citations h-index 21 12 iindex 40 Colin de la Higuera university of Nantes Verified email at univ-nantes. Daniel Meneveaux University of Poitiers Verified email at univ-poitiers. View all. Verified email at liris. Computer graphics.

Combinatorial maps : efficient data structures for computer graphics and image processing Combinatorial maps : efficient data structures for computer graphics and image processing
Combinatorial maps : efficient data structures for computer graphics and image processing Combinatorial maps : efficient data structures for computer graphics and image processing
Combinatorial maps : efficient data structures for computer graphics and image processing Combinatorial maps : efficient data structures for computer graphics and image processing
Combinatorial maps : efficient data structures for computer graphics and image processing Combinatorial maps : efficient data structures for computer graphics and image processing
Combinatorial maps : efficient data structures for computer graphics and image processing Combinatorial maps : efficient data structures for computer graphics and image processing
Combinatorial maps : efficient data structures for computer graphics and image processing Combinatorial maps : efficient data structures for computer graphics and image processing
Combinatorial maps : efficient data structures for computer graphics and image processing Combinatorial maps : efficient data structures for computer graphics and image processing
Combinatorial maps : efficient data structures for computer graphics and image processing Combinatorial maps : efficient data structures for computer graphics and image processing

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