A Fast and Log-Euclidean Polyaffine Framework for Locally Linear Registration

Vincent Arsigny 1, * Olivier Commowick 1, 2, 3, * Nicholas Ayache 1, * Xavier Pennec 1, *
* Corresponding author
1 ASCLEPIOS - Analysis and Simulation of Biomedical Images
CRISAM - Inria Sophia Antipolis - Méditerranée
3 VisAGeS - Vision, Action et Gestion d'informations en Santé
INSERM - Institut National de la Santé et de la Recherche Médicale : U746, Inria Rennes – Bretagne Atlantique , IRISA-D5 - SIGNAUX ET IMAGES NUMÉRIQUES, ROBOTIQUE
Abstract : In this article, we focus on the parameterization of non-rigid geometrical deformations with a small number of flexible degrees of freedom. In previous work, we proposed a general framework called polyaffine to parameterize deformations with a finite number of rigid or affine components, while guaranteeing the invertibility of global deformations. However, this framework lacks some important properties: the inverse of a polyaffine transformation is not polyaffine in general, and the polyaffine fusion of affine components is not invariant with respect to a change of coordinate system. We present here a novel general framework, called Log-Euclidean polyaffine, which overcomes these defects. We also detail a simple algorithm, the Fast Polyaffine Transform, which allows to compute very efficiently Log-Euclidean polyaffine transformations and their inverses on regular grids. The results presented here on real 3D locally affine registration suggest that our novel framework provides a general and efficient way of fusing local rigid or affine deformations into a global invertible transformation without introducing artifacts, independently of the way local deformations are first estimated.
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Vincent Arsigny, Olivier Commowick, Nicholas Ayache, Xavier Pennec. A Fast and Log-Euclidean Polyaffine Framework for Locally Linear Registration. Journal of Mathematical Imaging and Vision, Springer Verlag, 2009, 33 (2), pp.222-238. ⟨10.1007/s10851-008-0135-9⟩. ⟨inria-00616084⟩

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