A Piecewise Monotone Subgradient Algorithm for Accurate ${\rm L}^{1}$-TV Based Registration of Physical Slices With Discontinuities in Microscopy

Image registration tasks are often formulated in terms of minimization of a functional consisting of a data fidelity term penalizing the mismatch between the reference and the target image, and a term enforcing smoothness of shift between neighboring pairs of pixels (a min-sum problem). Most methods for deformable image registration use some form of interpolation between matching control points. The interpolation makes it impossible to account for isolated discontinuities in the deformation field that may appear, e.g., when a physical slice of a microscopy specimen is ruptured by the cutting tool. For registration of neighboring physical slices of microscopy specimens with discontinuities, Janácek proposed an L
¹
-distance data fidelity term and a total variation (TV) smoothness term, and used a graph-cut (GC) based iterative steepest descent algorithm for minimization. The L
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-TV functional is nonconvex; hence a steepest descent algorithm is not guaranteed to converge to the global minimum. Schlesinger presented transformation of max-sum problems to minimization of a dual quantity called problem power, which is - contrary to the original max-sum functional - convex. Based on Schlesinger's solution to max-sum problems we developed an algorithm for L
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-TV minimization by iterative multi-label steepest descent minimization of the convex dual problem. For Schlesinger's subgradient algorithm we proposed a novel step control heuristics that considerably enhances both speed and accuracy compared with standard step size strategies for subgradient methods. It is shown experimentally that our subgradient scheme achieves consistently better image registration than GC in terms of lower values both of the composite L
¹
-TV functional, and of its components, i.e., the L
¹
distance of the images and the transformation smoothness TV, and yields visually acceptable results even in cases where the GC based algorithm fails. The new algorithm allows easy parallelization and can thus be sped up by running on multi-core graphic processing units.