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# Application of the Fisher-Rao Metric to Ellipse Detection

## Abstract

The parameter space for the ellipses in a two dimensional image is a five dimensional manifold, where each point of the manifold corresponds to an ellipse in the image. The parameter space becomes a Riemannian manifold under a Fisher-Rao metric, which is derived from a Gaussian model for the blurring of ellipses in the image. Two points in the parameter space are close together under the Fisher-Rao metric if the corresponding ellipses are close together in the image. The Fisher-Rao metric is accurately approximated by a simpler metric under the assumption that the blurring is small compared with the sizes of the ellipses under consideration. It is shown that the parameter space for the ellipses in the image has a finite volume under the approximation to the Fisher-Rao metric. As a consequence the parameter space can be replaced, for the purpose of ellipse detection, by a finite set of points sampled from it. An efficient algorithm for sampling the parameter space is described. The algorithm uses the fact that the approximating metric is flat, and therefore locally Euclidean, on each three dimensional family of ellipses with a fixed orientation and a fixed eccentricity. Once the sample points have been obtained, ellipses are detected in a given image by checking each sample point in turn to see if the corresponding ellipse is supported by the nearby image pixel values. The resulting algorithm for ellipse detection is implemented. A multiresolution version of the algorithm is also implemented. The experimental results suggest that ellipses can be reliably detected in a given low resolution image and that the number of false detections can be reduced using the multiresolution algorithm.

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## Author information

Correspondence to Stephen J. Maybank.

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Maybank, S.J. Application of the Fisher-Rao Metric to Ellipse Detection. Int J Comput Vision 72, 287–307 (2007). https://doi.org/10.1007/s11263-006-9033-z

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### Keywords

• ellipse detection
• Fisher-Rao metric
• flat metric
• geodesic
• Hough transform
• Kullback-Leibler distance
• lattice
• multiresolution
• Riemannian manifold
• volume of a Riemannian manifold
• Voronoi’s principal lattice