In the context of large deformations by diffeomorphisms, we propose a new diffeomorphic registration algorithm for 3D images that performs the optimization directly on the set of geodesic flows. The key contribution of this work is to provide an accurate estimation of the socalled initial momentum, which is a scalar function encoding the optimal deformation between two images through the Hamiltonian equations of geodesics. Since the initial momentum has proven to be a key tool for statistics on shape spaces, our algorithm enables more reliable statistical comparisons for 3D images.Our proposed algorithm is a gradient descent on the initial momentum, where the gradient is calculated using standard methods from optimal control theory. To improve the numerical efficiency of the gradient computation, we have developed an integral formulation of the adjoint equations associated with the geodesic equations.We then apply it successfully to the registration of 2D phantom images and 3D cerebral images. By comparing F.-X. Vialard ( ) our algorithm to the standard approach of Beg et al. (Int. J. Comput. Vis. 61:139-157, 2005), we show that it provides a more reliable estimation of the initial momentum for the optimal path. In addition to promising statistical applications, we finally discuss different perspectives opened by this work, in particular in the new field of longitudinal analysis of biomedical images.

We studied normal and tumorous three-dimensional (3D) microvascular networks in primate and rat brain. Tissues were prepared following a new preparation technique intended for high-resolution synchrotron tomography of microvascular networks. The resulting 3D images with a spatial resolution of less than the minimum capillary diameter permit a complete description of the entire vascular network for volumes as large as tens of cubic millimeters. The structural properties of the vascular networks were investigated by several multiscale methods such as fractal and powerspectrum analysis. These investigations gave a new coherent picture of normal and pathological complex vascular structures. They showed that normal cortical vascular networks have scaleinvariant fractal properties on a small scale from 1.4 lm up to 40 to 65 lm. Above this threshold, vascular networks can be considered as homogeneous. Tumor vascular networks show similar characteristics, but the validity range of the fractal regime extend to much larger spatial dimensions. These 3D results shed new light on previous two dimensional analyses giving for the first time a direct measurement of vascular modules associated with vessel-tissue surface exchange.