Image Segmentation with Topological Maps and Inter-pixel Representation

Braquelaire, Jean-Pierre; Brun, Luc

doi:10.1006/jvci.1998.0374

Cited by 54 publications

(51 citation statements)

References 33 publications

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“…In 2D, combinatorial and generalized maps may be used to model the topology of an embedding of a planar graph in a plane. In particular, these models are very well suited for scene modeling [4], for 2D and 3D image segmentation [5], and there exist efficient algorithms to extract maps from images [6].…”

Section: Motivationsmentioning

confidence: 99%

Map Edit Distance vs. Graph Edit Distance for Matching Images

Combier

Damiand

Solnon

2013

Graph-Based Representations in Pattern Recognition

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Abstract. Generalized maps are widely used to model the topology of nD objects (such as 2D or 3D images) by means of incidence and adjacency relationships between cells (0D vertices, 1D edges, 2D faces, 3D volumes, ...). We have introduced in [1] a map edit distance. This distance compares maps by means of a minimum cost sequence of edit operations that should be performed to transform a map into another map. In this paper, we introduce labelled maps and we show how the map edit distance may be extended to compare labeled maps. We experimentally compare our map edit distance to the graph edit distance for matching regions of different segmentations of a same image. MotivationsIn many computer vision applications we have to match interest points or regions extracted from different images in order to, e.g., recognize objects or reconstitute 3D models from 2D images. When looking for such matchings, graph-based approaches offer a good compromise between local approaches, which match each point with the most similar point of the other image independently from its relationships with other points, and global approaches such as RANSAC, which consider rigid transformations. Indeed, graph-based approaches are able to exploit local relationships while being more tolerant to deformations than global approaches such as RANSAC.There exist different kinds of graph matchings [2], ranging from subgraph isomorphism to more error-tolerant matchings such as the graph edit distance. The graph edit distance is a generic measure, which is parametrized by edition costs, and it is widely used to match graphs. It defines the distance between two graphs G 1 and G 2 as the minimum cost sequence of edit operations for transforming G 1 into G 2 . Edit operations are vertex and edge deletion, insertion and substitution. A vertex matching may be derived from the sequence of edit operations in a straightforward way: Any vertex v 1 of G 1 which is substituted to a vertex v 2 of G 2 is actually matched with v 2 .Graphs are well suited to model binary relationships such as point proximity or region adjacency. However, graphs are less well suited to model the topology of the subdivision of a plane in faces, edges, and vertices. Combinatorial maps

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Section: Motivationsmentioning

confidence: 99%

Map Edit Distance vs. Graph Edit Distance for Matching Images

Combier

Damiand

Solnon

2013

Graph-Based Representations in Pattern Recognition

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Section: Motivationsmentioning

confidence: 99%

“…In 2D, maps may be used to model the topology of an embedding of a planar graph in a plane. In particular, these models are very well suited for scene modeling [2], and for 2D and 3D image segmentation [3].In [4], we have defined a basic tool for comparing 2D maps, i.e., submap isomorphism (which involves deciding if a copy of a pattern map may be found in a target map), and we have proposed an efficient polynomial-time algorithm for solving this problem when the pattern map is connected. This work has been generalized to nD maps in [5].…”

mentioning

confidence: 99%

On the Complexity of Submap Isomorphism

Solnon

Damiand

Higuera

et al. 2013

Graph-Based Representations in Pattern Recognition

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Abstract. Generalized maps describe the subdivision of objects in cells, and incidence and adjacency relations between cells, and they are widely used to model 2D and 3D images. Recently, we have defined submap isomorphism, which involves deciding if a copy of a pattern map may be found in a target map, and we have described a polynomial time algorithm for solving this problem when the pattern map is connected. In this paper, we show that submap isomorphism becomes NP-complete when the pattern map is not connected, by reducing the NP-complete problem Planar-4 3-SAT to it. MotivationsCombinatorial maps and generalized maps [1] are very nice data structures to model the topology of nD objects subdivided in cells (e.g., 0D vertices, 1D edges, 2D faces, 3D volumes, . . . ) by means of incidence and adjacency relationships between these cells. In 2D, maps may be used to model the topology of an embedding of a planar graph in a plane. In particular, these models are very well suited for scene modeling [2], and for 2D and 3D image segmentation [3].In [4], we have defined a basic tool for comparing 2D maps, i.e., submap isomorphism (which involves deciding if a copy of a pattern map may be found in a target map), and we have proposed an efficient polynomial-time algorithm for solving this problem when the pattern map is connected. This work has been generalized to nD maps in [5]. The subisomorphism defined in [5] is based on induced submap relations, such that submaps are obtained by removing some darts and all their seams, just like induced subgraphs are obtained by removing some vertices and all their incident edges. In [6], we have introduced a new kind of submap relation, called partial submap: partial submaps are obtained by removing not only some darts (and all their seams), but also some other seams, just like partial subgraphs are obtained by removing not only some vertices (and their incident edges), but also some other edges. The polynomial time algorithm described in [5] for solving the induced submap isomorphism problem may be extended to the partial case in a very straightforward way. However, this algorithm still assumes that the pattern map is connected. In this paper, we show that the submap isomorphism problem becomes NP-complete when the pattern map is not connected, both for partial and induced submaps.

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“…In particular, this makes algorithms independent of the embedding model and allows to use either inter-pixel boundaries [18], 8-connected pixel boundaries [16], or sub-pixel precise polygonal boundaries [17,8].…”

Section: Contraction Kernelsmentioning

confidence: 99%

Annotated Contraction Kernels for Interactive Image Segmentation

Meine

2009

Graph-Based Representations in Pattern Recognition

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Abstract. This article shows how the interactive segmentation tool termed "Active Paintbrush" and a fully automatic region merging can both be based on the theoretical framework of contraction kernels within irregular pyramids instead of their own, specialized data structures. We introduce "continous pyramids" in which we purposely drop the common requirement of a fixed reduction factor between successive levels, and we show how contraction kernels can be annotated for a fast navigation of such pyramids. Finally, we use these concepts for improving the integration of the automatic region merging and the interactive tool.

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Image Segmentation with Topological Maps and Inter-pixel Representation

Cited by 54 publications

References 33 publications

Map Edit Distance vs. Graph Edit Distance for Matching Images

Map Edit Distance vs. Graph Edit Distance for Matching Images

On the Complexity of Submap Isomorphism

Annotated Contraction Kernels for Interactive Image Segmentation

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