Computer Vision, IEEE International Conference on (2005)

Beijing, China

Oct. 17, 2005 to Oct. 20, 2005

ISSN: 1550-5499

ISBN: 0-7695-2334-X

pp: 1132-1139

DOI Bookmark: http://doi.ieeecomputersociety.org/10.1109/ICCV.2005.100

Davi Geiger , New York University

Richard Cole , New York University

Hiroshi Ishikawa , Nagoya City University

ABSTRACT

The representation of objects in images as tree structures is of great interest to vision, as they can represent articulated objects such as people as well as other structured objects like arteries in human bodies, roads, circuit board patterns, etc. Tree structures are often related to the symmetry axis representation of shapes, which captures their local symmetries. Algorithms have been introduced to detect (i) open contours in images in quadratic time (ii) closed contours in images in cubic time, and (iii) tree structures from contours in quadratic time. The algorithms are based on dynamic programming and Single Source Shortest Path algorithms. However, in this paper, we show that the problem of finding tree structures in images in a principled manner is a much harder problem. We argue that the optimization problem of finding tree structures in images is essentially equivalent to a variant of the Steiner Tree problem, which is NP-hard. Nevertheless, an approximate polynomial-time algorithm for this problem exists: we apply a fast implementation of the Goemans-Williamson approximate algorithm to the problem of finding a tree representation after an image is transformed by a local symmetry mapping. Examples of extracting tree structures from images illustrate the idea and applicability of the approximate method.

INDEX TERMS

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CITATION

Davi Geiger,
Richard Cole,
Hiroshi Ishikawa,
"Finding Tree Structures by Grouping Symmetries",

*Computer Vision, IEEE International Conference on*, vol. 02, no. , pp. 1132-1139, 2005, doi:10.1109/ICCV.2005.100