Contributions to 3D diffeomorphic atlas estimation: application to brain images

This paper focuses on the estimation of statistical atlases of 3D images by means of diffeomorphic transformations. Within a Log-Euclidean framework, the exponential and logarithm maps of diffeomorphisms need to be computed. In this framework, the Inverse Scaling and Squaring (ISS) method has been recently extended for the computation of the logarithm map, which is one of the most time demanding stages. In this work we propose to apply the Baker-Campbell-Hausdorff (BCH) formula instead. In a 3D simulation study, BCH formula and ISS method obtained similar accuracy but BCH formula was more than 100 times faster. This approach allowed us to estimate a 3D statistical brain atlas in a reasonable time, including the average and the modes of variation. Details for the computation of the modes of variation in the Sobolev tangent space of diffeomorphisms are also provided.