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Geodesic Regression and Its Application to Shape Analysis

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Innovations for Shape Analysis

Part of the book series: Mathematics and Visualization ((MATHVISUAL))

Abstract

In this chapter, I present a regression method for modeling the relationship between a manifold-valued random variable and a real-valued independent parameter. The principle is to fit a geodesic curve, parameterized by the independent parameter, that best fits the data. Error in the model is evaluated as the sum-of-squared geodesic distances from the model to the data, and this provides an intrinsic least squares criterion. Geodesic regression is, in some sense, the simplest parametric model that one could choose, and it provides a direct generalization of linear regression to the manifold setting. A generalization of the coefficient of determination and a resulting hypothesis test for determining the significance of the estimated trend is developed. Also, a diagnostic test for the quality of the fit of the estimated geodesic is demonstrated. While the method can be generally applied to data on any manifold, specific examples are given for a set of synthetically generated rotation data and an application to analyzing shape changes in the corpus callosum due to age.

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Authors and Affiliations

  1. University of Utah, Salt Lake City, 84112, UT, USA

    P. Thomas Fletcher

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  1. P. Thomas Fletcher
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Correspondence to P. Thomas Fletcher .

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Editors and Affiliations

  1. , Inst. for Appl. Math.&Scient.Comp., Brandenburg Technical University, Platz der Deutschen Einheit 1, Cottbus, 03046, Germany

    Michael Breuß

  2. Department of Computer Science, Technion-Israel Institute of Technology, Haifa, 32000, Israel

    Alfred Bruckstein

  3. School of Electrical &, Computer Engineering, National Technical University of Athens, Athens, 15773, Greece

    Petros Maragos

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Fletcher, P.T. (2013). Geodesic Regression and Its Application to Shape Analysis. In: Breuß, M., Bruckstein, A., Maragos, P. (eds) Innovations for Shape Analysis. Mathematics and Visualization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-34141-0_2

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