Elastic geometric shape matching for translations under the Manhattan norm (bibtex)

Abstract:

The term elastic geometric shape matching (EGSM) refers to geometric optimization problems that are a generalization of many classical and well-studied geometric shape matching problems. In a geometric shape matching problem, one seeks a single transformation that, if applied to a geometric object – the pattern – minimizes the distance of the transformed object to another geometric object – the model. In an EGSM problem, the pattern is partitioned into parts which are transformed by a collection of transformations, called a transformation ensemble, in order to minimize the distance of the individually transformed parts to the model under the constraint that specific pairs of transformations of the ensemble have to be similar. These constraints are defined by an abstract graph on the parts of the model, called the neighborhood graph. We present algorithms for an EGSM problem for point sets under translations where the neighborhood graph is a tree. We measure the similarity of the shapes by the L1-Hausdorff distance (and the Hausdorff distance induced by other polygonal metrics).

Reference:

Elastic geometric shape matching for translations under the Manhattan norm (Christian Knauer, Luise Sommer, Fabian Stehn), In Computational Geometry, volume 73, 2018. (Secial Issue on EuroCG2015)

Bibtex Entry:

@article{journals/KNAUER201857, title = "Elastic geometric shape matching for translations under the Manhattan norm", journal = "Computational Geometry", volume = "73", pages = "57 - 69", year = "2018", note = "Secial Issue on EuroCG2015", issn = "0925-7721", doi = "10.1016/j.comgeo.2018.01.002", url = "http://www.sciencedirect.com/science/article/pii/S0925772118300026", author = "Christian Knauer and Luise Sommer and Fabian Stehn", keywords = "Computational geometry, Exact elastic shape matching, Polygonal norms, Approximation", abstract = "The term elastic geometric shape matching (EGSM) refers to geometric optimization problems that are a generalization of many classical and well-studied geometric shape matching problems. In a geometric shape matching problem, one seeks a single transformation that, if applied to a geometric object – the pattern – minimizes the distance of the transformed object to another geometric object – the model. In an EGSM problem, the pattern is partitioned into parts which are transformed by a collection of transformations, called a transformation ensemble, in order to minimize the distance of the individually transformed parts to the model under the constraint that specific pairs of transformations of the ensemble have to be similar. These constraints are defined by an abstract graph on the parts of the model, called the neighborhood graph. We present algorithms for an EGSM problem for point sets under translations where the neighborhood graph is a tree. We measure the similarity of the shapes by the L1-Hausdorff distance (and the Hausdorff distance induced by other polygonal metrics)." }

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