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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Arsigny, V., Commowick, O., Ayache, N. et al. A Fast and Log-Euclidean Polyaffine Framework for Locally Linear Registration. J Math Imaging Vis 33, 222–238 (2009). https://doi.org/10.1007/s10851-008-0135-9
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DOI: https://doi.org/10.1007/s10851-008-0135-9