Workshop on Bioimaging II / PDEs, Wed, 21 Nov, 2007
Speaker: Stephen Keeling
Abstract
A generalization of rigid image registration and interpolation is achieved variationally by penalizing a departure from rigidity of an optical flow. The image similarity measure is based upon the difference between scaled intensities. This similarity measure achieves contrast invariance purely by composition with scaling functions, while other contrast invariant measures are based upon differential formulations. Such rescaling is additionally found to have a smoothing effect on noisy images. Results obtained by Tikhonov regularization of scaling functions are superior to those obtained by a restricted set of basis functions. It is found that the computational cost for the implementation of scaling functions is very small, and yet the resulting image similarity can provide a match between images which is closer than that obtained by a sum of squared differences alone. The optical flow approach is investigated theoretically and numerically as well as with respect to a geometric multigrid solution process. For computations, a lumped finite element Eulerian discretization is used to solve the optimality system for the optical flow under natural boundary conditions. Also, a Lagrangian integration of the intensity along optical flow trajectories has the advantage of prohibiting diffusion among trajectories which would otherwise blur interpolated images. For the solution of the optimality system determining the optical flow, it is shown that the Hackbusch convergence criteria are met. Specifically, the Galerkin formalism is used together with a multi-colored ordering of unknowns to permit vectorization of a symmetric successive over-relaxation on image processing systems. The procedure is shown to be independent of image order. Also, it is shown that non-autonomous optical flows are theoretically possible, but that autonomous flows may be used in practice to achieve a great computational savings. This work is motivated by applications in histological reconstruction and in dynamic medical imaging, and results are shown for such realistic examples.
URL: www.ricam.oeaw.ac.at/specsem/ssqbm/participants/abstracts/index.php
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