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Integral Invariants and Shape Matching

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Statistics and Analysis of Shapes

Abstract

For shapes represented as closed planar contours, we introduce a class of functionals which are invariant with respect to the Euclidean group, and which are obtained by performing integral operations. While such integral invariants enjoy some of the desirable properties of their differential cousins, such as locality of computation (which allows matching under occlusions) and uniqueness of representation (asymptotically), they do not exhibit the noise sensitivity associated with differential quantities and therefore do not require pre-smoothing of the input shape. Our formulation allows the analysis of shapes at multiple scales. Based on integral invariants, we define a notion of distance between shapes. The proposed distance measure can be computed efficiently, it allows for shrinking and stretching of the boundary, and computes optimal correspondence. Numerical results on shape matching demonstrate that this framework can match shapes despite the deformation of subparts, missing parts, and noise. As a quantitative analysis, we report matching scores for shape retrieval from a database.

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Authors and Affiliations

  1. Lawrence Livermore National Laboratory, Livermore, CA, 94551, USA

    Siddharth Manay

  2. University of Bonn, 53113, Bonn, Germany

    Daniel Cremers

  3. University of California at Los Angeles, CA, 90095, USA

    Byung-Woo Hong & Stefano Soatto

  4. Georgia Institute of Technology, Atlanta, GA, 30332, USA

    Anthony Yezzi Jr.

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  1. Siddharth Manay
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  2. Daniel Cremers
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  3. Byung-Woo Hong
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  4. Anthony Yezzi Jr.
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  5. Stefano Soatto
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Editors and Affiliations

  1. Department of Electrical and Computer Engineering, North Carolina State University, Raleigh, NC, 27606-7914, USA

    Hamid Krim

  2. School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA, 30332-0250, USA

    Anthony Yezzi Jr.

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Manay, S., Cremers, D., Hong, BW., Yezzi, A., Soatto, S. (2006). Integral Invariants and Shape Matching. In: Krim, H., Yezzi, A. (eds) Statistics and Analysis of Shapes. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser Boston. https://doi.org/10.1007/0-8176-4481-4_6

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