Template estimation in computational anatomy: Fréchet means in top and quotient spaces are not consistent

Loïc Devilliers (1) , Stéphanie Allassonnière (2) , Alain Trouvé (3) , Xavier Pennec (1)

1 ASCLEPIOS - Analysis and Simulation of Biomedical Images
2 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
3 CMLA - Centre de Mathématiques et de Leurs Applications

In this article we study the consistency of the template estimation with the Fréchet mean in the quotient space when the observations are shifted by a group action. We show that in most cases this estimator is actually inconsistent. We exhibit a sufficient condition for this inconsistency, which amounts to the folding of the distribution of the noisy template when it is projected to the quotient space. This condition appears to be fulfilled as soon as the support of the noise is large enough. To quantify this inconsistency we provide lower and upper bounds of the bias as a function of the variability (the noise level). This shows that the consistency bias cannot be neglected when the variability increases.

Mots clés: inner vector space; empirical Fréchet mean; inconsistency; group action; quotient space; Template; Fréchet mean; consistency bias; manifold

Domaines: Mathématiques [math]; Statistiques [math.ST]

Loïc Devilliers, Stéphanie Allassonnière, Alain Trouvé, Xavier Pennec. Template estimation in computational anatomy: Fréchet means in top and quotient spaces are not consistent. 2016. ⟨hal-01352707v2⟩