Super-fast Elastic Registration of Histologic Images of a Whole Rat Brain for Three-dimensional Reconstruction

We present a super-fast and parameter-free algorithm for non-rigid elastic registration of images of a serially sectioned whole rat brain. The purpose is to produce a three-dimensional high-resolution reconstruction. The registration is modelled as a minimization problem of a functional consisting of a distance measure and a regularizer based on the elastic potential of the displacement field. The minimization of the functional leads to a system of non-linear partial differential equations, the so-called Navier-Lame equations (NLE). Discretization of the NLE and a fixed point type iteration method lead to a linear system of equations, which has to be solved at each iteration step. We not only present a super-fast solution technique for this system, but also come up with sound strategies for accelerating the outer iteration. This does include a multi-scale approach based on a Gaussian pyramid as well as a clever estimation of the material constants for the elastic potential. The results of the registration process were controlled by an expert who was able to recognize histological details like laminations which was not possible before. Therefore, it is essential to apply elastic registration to this kind of imaging problem. Finally, the visually pleasing results were quantified by a distance measure leading to an improvement of about 79% after just 35 iteration steps.