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Open AccessArticle

Inconsistency of Template Estimation by Minimizing of the Variance/Pre-Variance in the Quotient Space

1
Université Côte d’Azur, Inria, France
2
INSERM, UMRS 1138, CRC, team 22, Paris Descartes University, UPMC, Paris, France
3
CMLA, ENS Cachan, CNRS, Université Paris-Saclay, 94235 Cachan, France
*
Author to whom correspondence should be addressed.
This paper is an extended version of our paper published in Information Processing in Medical Imaging 2017, Boone, NC, USA, 25–30 June 2017.
Academic Editor: Geert Verdoolaege
Entropy 2017, 19(6), 288; https://doi.org/10.3390/e19060288
Received: 27 April 2017 / Revised: 7 June 2017 / Accepted: 17 June 2017 / Published: 20 June 2017
(This article belongs to the Special Issue Information Geometry II)
We tackle the problem of template estimation when data have been randomly deformed under a group action in the presence of noise. In order to estimate the template, one often minimizes the variance when the influence of the transformations have been removed (computation of the Fréchet mean in the quotient space). The consistency bias is defined as the distance (possibly zero) between the orbit of the template and the orbit of one element which minimizes the variance. In the first part, we restrict ourselves to isometric group action, in this case the Hilbertian distance is invariant under the group action. We establish an asymptotic behavior of the consistency bias which is linear with respect to the noise level. As a result the inconsistency is unavoidable as soon as the noise is enough. In practice, template estimation with a finite sample is often done with an algorithm called “max-max”. In the second part, also in the case of isometric group finite, we show the convergence of this algorithm to an empirical Karcher mean. Our numerical experiments show that the bias observed in practice can not be attributed to the small sample size or to a convergence problem but is indeed due to the previously studied inconsistency. In a third part, we also present some insights of the case of a non invariant distance with respect to the group action. We will see that the inconsistency still holds as soon as the noise level is large enough. Moreover we prove the inconsistency even when a regularization term is added. View Full-Text
Keywords: Fréchet mean; Hilbert space; deformable model; template estimation; quotient space; inconsistency; regularization Fréchet mean; Hilbert space; deformable model; template estimation; quotient space; inconsistency; regularization
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Devilliers, L.; Allassonnière, S.; Trouvé, A.; Pennec, X. Inconsistency of Template Estimation by Minimizing of the Variance/Pre-Variance in the Quotient Space. Entropy 2017, 19, 288.

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