A Fisher–Rao Metric for Curves Using the Information in Edges

Abstract

Two curves which are close together in an image are indistinguishable given a measurement, in that there is no compelling reason to associate the measurement with one curve rather than the other. This observation is made quantitative using the parametric version of the Fisher–Rao metric. A probability density function for a measurement conditional on a curve is constructed. The distance between two curves is then defined to be the Fisher–Rao distance between the two conditional pdfs. A tractable approximation to the Fisher–Rao metric is obtained for the case in which the measurements are compound in that they consist of a point \({\mathbf {x}}\) and an angle \(\alpha \) which specifies the direction of an edge at \({\mathbf {x}}\). If the curves are circles or straight lines, then the approximating metric is generalized to take account of inlying and outlying measurements. An estimate is made of the number of measurements required for the accurate location of a circle in the presence of outliers. A Bayesian algorithm for circle detection is defined. The prior density for the algorithm is obtained from the Fisher–Rao metric. The algorithm is tested on images from the CASIA iris interval database.

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Notes

  1. 1.

    http://uk.mathworks.com/help/images/ref/imfindcircles.html.

  2. 2.

    http://biometrics.idealtest.org/dbDetailForUser.do?id=4.

  3. 3.

    http://uk.mathworks.com/help/images/ref/imfindcircles.html.

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Correspondence to Stephen J. Maybank.

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Maybank, S.J. A Fisher–Rao Metric for Curves Using the Information in Edges. J Math Imaging Vis 54, 287–300 (2016). https://doi.org/10.1007/s10851-015-0603-y

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Keywords

  • Bayesian curve detection
  • CASIA iris database
  • Circle detection
  • Hough transform
  • Riemannian metric
  • Step edges