On the consistency of Fréchet means in deformable models for curve and image analysis

Abstract

A new class of statistical deformable models is introduced to study high-dimensional curves or images. In addition to the standard measurement error term, these deformable models include an extra error term modeling the individual variations in intensity around a mean pattern. It is shown that an appropriate tool for statistical inference in such models is the notion of sample Fréchet means, which leads to estimators of the deformation parameters and the mean pattern. The main contribution of this paper is to study how the behavior of these estimators depends on the number n of design points and the number J of observed curves (or images). Numerical experiments are given to illustrate the finite sample performances of the procedure.Acknowledgements -We would like to thank Dominique Bakry for fruitful discussions. Both authors would like to thank the Center for Mathematical Modeling and the CNRS for financial support and excellent hospitality while visiting Santiago where part of this work was carried out. We are very much indebted to the referees and the Associate Editor for their constructive comments and remarks that resulted in a major revision of the original manuscript.consistency of the proposed estimators in various asymptotic settings: either when both the number n of design points and the number J of curves (or images) tend to infinity, or when n (resp. J) remains fixed while J (resp. n) tends to infinity.In many situations, data sets of curves or images exhibit a source of geometric variations in time or shape. In such settings, the usual Euclidean meanȲ ℓ = 1 J J j=1 Y ℓ j in model (1.1) cannot be used to recover a meaningful mean pattern. Indeed, consider the following simple model of randomly shifted curves (with d = 1) which is commonly used in many applied areas such as neuroscience [TIR10] where f : Ω −→ R is the mean pattern of the observed curves, and the θ * j 's are i.i.d. random variables in R with density g and independent of the ε ℓ j 's. In model (1.2), the shifts θ * j represent a source of variability in time. However, in (1.2) the Euclidean mean is not a consistent estimator of the mean pattern f since by the law of large numbersThe randomly shifted curves model (1.2) is close to the perturbation model introduced by [Goo91] in shape analysis for the study of consistent estimation of a mean pattern from a set of random planar shapes. The mean pattern to estimate in [Goo91] is called a population mean, but to stress the fact that it comes from a perturbation model [Huc10] uses the term perturbation mean. To achieve consistency in such models, a Procrustean procedure is used in [Goo91], which leads to the statistical analysis of sample Fréchet means [Fré48] which are extensions of the usual Euclidean mean to non-linear spaces using non-Euclidean metrics. For random variables belonging to a nonlinear manifold, a well-known example is the computation of the mean of a set of planar shapes in the Kendall's shape space [Ken84] which leads to the Procrustean means studied in [Goo91]. Consistent ...

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