In the shape analysis approach to computer vision problems, one treats shapes as points in an infinite-dimensional Riemannian manifold, thereby facilitating algorithms for statistical calculations such as geodesic distance between shapes and averaging of a collection of shapes. The performance of these algorithms depends heavily on the choice of the Riemannian metric. In the setting of plane curve shapes, attention has largely been focused on a two-parameter family of first order Sobolev metrics, referred to as elastic metrics. They are particularly useful due to the existence of simplifying coordinate transformations for particular parameter values, such as the well-known square-root velocity transform. In this paper, we extend the transformations appearing in the existing literature to a family of isometries, which take any elastic metric to the flat $L^2$ metric. We also extend the transforms to treat piecewise linear curves and demonstrate the existence of optimal matchings over the diffeomorphism group in this setting. We conclude the paper with multiple examples of shape geodesics for open and closed curves. We also show the benefits of our approach in a simple classification experiment.


  1. elastic shape analysis
  2. statistical shape analysis
  3. infinite-dimensional geometry
  4. Sobolev metrics
  5. curve matching

MSC codes

  1. Primary
  2. 58B20
  3. 58E50; Secondary
  4. 68U05

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Information & Authors


Published In

cover image SIAM Journal on Imaging Sciences
SIAM Journal on Imaging Sciences
Pages: 445 - 473
ISSN (online): 1936-4954


Submitted: 30 May 2019
Accepted: 13 January 2020
Published online: 12 March 2020


  1. elastic shape analysis
  2. statistical shape analysis
  3. infinite-dimensional geometry
  4. Sobolev metrics
  5. curve matching

MSC codes

  1. Primary
  2. 58B20
  3. 58E50; Secondary
  4. 68U05



Funding Information

National Institutes of Health https://doi.org/10.13039/100000002
National Science Foundation https://doi.org/10.13039/100000001 : DMS-1613054, CCF-1740761, CCF-1839252

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