This paper demonstrates the impact of a phase field method on shape
registration to align shapes of possibly different topology. It yields new
insights into the building of discrepancy measures between shapes regardless of
topology, which would have applications in fields of image data analysis such
as computational anatomy. A soft end-point optimal control problem is
introduced whose minimum measures the minimal control norm required to align an
initial shape to a final shape, up to a small error term. The initial data is
spatially integrable, the paths in control spaces are integrable and the
evolution equation is a generalized convective Allen-Cahn. Binary images are
used to represent shapes for the initial data. Inspired by level-set methods
and large diffeomorphic deformation metric mapping, the controls spaces are
integrable scalar functions to serve as a normal velocity and smooth
reproducing kernel Hilbert spaces to serve as velocity vector fields. The
existence of mild solutions to the evolution equation is proved, the minimums
of the time discretized optimal control problem are characterized, Und
numerical simulations of minimums to the fully discretized optimal control
problem are displayed. The numerical implementation enforces the
maximum-bounded principle, although it is not proved for these mild solutions.
This research offers a novel discrepancy measure that provides valuable ways to
analyze diverse image data sets. Future work involves proving the existence of
minimums, existence and uniqueness of strong solutions and the maximum bounded
principle.

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