89 research outputs found
Multiple Shape Registration using Constrained Optimal Control
Lagrangian particle formulations of the large deformation diffeomorphic
metric mapping algorithm (LDDMM) only allow for the study of a single shape. In
this paper, we introduce and discuss both a theoretical and practical setting
for the simultaneous study of multiple shapes that are either stitched to one
another or slide along a submanifold. The method is described within the
optimal control formalism, and optimality conditions are given, together with
the equations that are needed to implement augmented Lagrangian methods.
Experimental results are provided for stitched and sliding surfaces
A relaxed approach for curve matching with elastic metrics
In this paper we study a class of Riemannian metrics on the space of
unparametrized curves and develop a method to compute geodesics with given
boundary conditions. It extends previous works on this topic in several
important ways. The model and resulting matching algorithm integrate within one
common setting both the family of -metrics with constant coefficients and
scale-invariant -metrics on both open and closed immersed curves. These
families include as particular cases the class of first-order elastic metrics.
An essential difference with prior approaches is the way that boundary
constraints are dealt with. By leveraging varifold-based similarity metrics we
propose a relaxed variational formulation for the matching problem that avoids
the necessity of optimizing over the reparametrization group. Furthermore, we
show that we can also quotient out finite-dimensional similarity groups such as
translation, rotation and scaling groups. The different properties and
advantages are illustrated through numerical examples in which we also provide
a comparison with related diffeomorphic methods used in shape registration.Comment: 27 page
Planar Curve Registration using Bayesian Inversion
We study parameterisation-independent closed planar curve matching as a
Bayesian inverse problem. The motion of the curve is modelled via a curve on
the diffeomorphism group acting on the ambient space, leading to a large
deformation diffeomorphic metric mapping (LDDMM) functional penalising the
kinetic energy of the deformation. We solve Hamilton's equations for the curve
matching problem using the Wu-Xu element [S. Wu, J. Xu, Nonconforming finite
element spaces for order partial differential equations on
simplicial grids when , Mathematics of Computation 88
(316) (2019) 531-551] which provides mesh-independent Lipschitz constants for
the forward motion of the curve, and solve the inverse problem for the momentum
using Bayesian inversion. Since this element is not affine-equivalent we
provide a pullback theory which expedites the implementation and efficiency of
the forward map. We adopt ensemble Kalman inversion using a negative Sobolev
norm mismatch penalty to measure the discrepancy between the target and the
ensemble mean shape. We provide several numerical examples to validate the
approach.Comment: 45 pages, 9 figure
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