## Approximations of Shape Metrics and Application to Shape Warping and Empirical Shape Statistics (2004)

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Venue: | FOUNDATIONS OF COMPUTATIONAL MATHEMATICS |

Citations: | 90 - 20 self |

### BibTeX

@ARTICLE{Charpiat04approximationsof,

author = {Guillaume Charpiat and Olivier Faugeras and Renaud Keriven},

title = {Approximations of Shape Metrics and Application to Shape Warping and Empirical Shape Statistics},

journal = {FOUNDATIONS OF COMPUTATIONAL MATHEMATICS},

year = {2004},

volume = {5},

pages = {1--58}

}

### Years of Citing Articles

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### Abstract

This paper proposes a framework for dealing with several problems related to the analysis of shapes. Two related such problems are the definition of the relevant set of shapes and that of defining a metric on it. Following a recent research monograph by Delfour and Zolesio [11], we consider the characteristic functions of the subsets of R² and their distance functions. The L² norm of the difference of characteristic functions, the L # and the W norms of the difference of distance functions define interesting topologies, in particular the well-known Hausdorff distance. Because of practical considerations arising from the fact that we deal with

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- 1970
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- 1944
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- 2003
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- Federer
- 1951
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- 2000
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- 1969
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Citation Context ...d only if f (x) ≤ lim inf f (y), for all x ∈ E. y→x Therefore f is l.s.c. by construction. The existence of a minimum of an l.s.c. function defined on a compact metric space is well-known (see, e.g., =-=[7]-=-, [15]) and will be needed later to prove that some of our minimization problems are well-posed. 4. Deforming Shapes The problem of continuously deforming a shape so that it turns into another is cent... |

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