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A Subdivision Scheme for Continuous-Scale B-Splines and Affine-Invariant Progressive Smoothing

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Abstract

Multiscale representations and progressive smoothing constitutean important topic in different fields as computer vision, CAGD,and image processing. In this work, a multiscale representationof planar shapes is first described. The approach is based oncomputing classical B-splines of increasing orders, andtherefore is automatically affine invariant. The resultingrepresentation satisfies basic scale-space properties at least ina qualitative form, and is simple to implement.

The representation obtained in this way is discrete in scale,since classical B-splines are functions in \(C^{k - 2}\), where k isan integer bigger or equal than two. We present a subdivisionscheme for the computation of B-splines of finite support atcontinuous scales. With this scheme, B-splines representationsin \(C^r\) are obtained for any real r in [0, ∞), andthe multiscale representation is extended to continuous scale.

The proposed progressive smoothing receives a discrete set ofpoints as initial shape, while the smoothed curves arerepresented by continuous (analytical) functions, allowing astraightforward computation of geometric characteristics of theshape.

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Authors and Affiliations

  1. Hewlett-Packard Labs, 1501 Page Mill Rd., Palo Alto, CA, 94304

    Guillermo Sapiro & Alfred M. Bruckstein

  2. Laboratoire d'Analyse Numerique, Université Pierre et Marie Curie, 4 Place Jussieu, Paris, 75005, France

    Albert Cohen

Authors
  1. Guillermo Sapiro
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  2. Albert Cohen
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  3. Alfred M. Bruckstein
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Sapiro, G., Cohen, A. & Bruckstein, A.M. A Subdivision Scheme for Continuous-Scale B-Splines and Affine-Invariant Progressive Smoothing. Journal of Mathematical Imaging and Vision 7, 23–40 (1997). https://doi.org/10.1023/A:1008261923192

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