Geodesic Image Normalization in the Space of Diffeomorphisms

Abstract : Quantitative, deformable mappings between images are increasingly important as measurement tools in biology and medicine. The theory of diffeomorphisms (smooth differentiable bijections with differentiable inverse) provides a mathematical foundation for the computation and interpretation of these maps. Miller, in particular, has used this theory to develop image normalization techniques that rely on a distance measurement as a regularizer. Here, we use this metric within a general transformation framework that explicitly parameterizes the image to image mapping as a symmetric geodesic path in the space of diffeomorphisms. The key difference between our approach and Miller's is that ours does not choose a single end-point (or template image) from which to measure the map. Rather, we search for the shortest diffeomorphism (smallest deformation) between images by optimizing the transformation with respect to both of its end-points. This algorithm, geodesic normalization, provides solutions that are invariant to which of the input coordinate systems (images) is chosen as a reference. This allows the method to compute metric distances, have truly symmetric performance and to give full space-time solutions that are invertible and diffeomorphic in the discrete domain. Finally, our algorithm guarantees that our solutions and their inverses are consistent to a sub-pixel level.
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Xavier Pennec and Sarang Joshi. 1st MICCAI Workshop on Mathematical Foundations of Computational Anatomy: Geometrical, Statistical and Registration Methods for Modeling Biological Shape Variability, Oct 2006, Copenhagen, Denmark. pp.125-135, 2006
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Brian Avants, C.L. Epstein, James C. Gee. Geodesic Image Normalization in the Space of Diffeomorphisms. Xavier Pennec and Sarang Joshi. 1st MICCAI Workshop on Mathematical Foundations of Computational Anatomy: Geometrical, Statistical and Registration Methods for Modeling Biological Shape Variability, Oct 2006, Copenhagen, Denmark. pp.125-135, 2006. 〈inria-00636082〉

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