Image Registration Based on Fast Algorithms for the Solution of Elliptic Equations

Abstract

Image registration arises whenever we need to extract information from different images. Here we present elastic registration method focused on medical usage. The finite difference approximation with periodic boundary conditions is usually used, which leads to equation system with highly structured matrix. Then Fast Fourier transform can be utilize to diagonalize and invert the matrix, thus the solution has complexity O(Nlog N), where N is number of pixels. But this approach doesn't work with the constraints, which destroy the structure of the matrix. Here we propose to solve the related elasticity problems by a variant of the finite element method - domain decomposition method for massively parallel numerical solution of elliptic partial differential equations with optimal complexity. Besides parallelization, this approach enables to improve the precision of the registration by controlling the ``stiffness'' of the parts of image such as bones. We also don't need to use structured mesh and solution can be enhanced by adaptive mesh generation. We use the libMesh library to generate mesh and input data to PermonFLLOP library, the PERMON package for constrained quadratic programming and domain decomposition. Both libraries allow us to exploit high performance computing. Several results of the proposed algorithm based on synthetic and real medical data were computed on SALOMON (IT4Innovations National Supercomputing Center).