234 research outputs found
Riemannian cubics on the group of diffeomorphisms and the Fisher-Rao metric
We study a second-order variational problem on the group of diffeomorphisms
of the interval [0, 1] endowed with a right-invariant Sobolev metric of order
2, which consists in the minimization of the acceleration. We compute the
relaxation of the problem which involves the so-called Fisher-Rao functional a
convex functional on the space of measures. This relaxation enables the
derivation of several optimality conditions and, in particular, a sufficient
condition which guarantees that a given path of the initial problem is also a
minimizer of the relaxed one. This sufficient condition is related to the
existence of a solution to a Riccati equation involving the path acceleration.Comment: 34 pages, comments welcom
On Completeness of Groups of Diffeomorphisms
We study completeness properties of the Sobolev diffeomorphism groups
endowed with strong right-invariant Riemannian metrics when
the underlying manifold is or compact without boundary. The
main result is that for , the group is
geodesically and metrically complete with a surjective exponential map. We then
present the connection between the Sobolev diffeomorphism group and the large
deformation matching framework in order to apply our results to diffeomorphic
image matching.Comment: 43 pages, revised versio
Variational Second-Order Interpolation on the Group of Diffeomorphisms with a Right-Invariant Metric
In this note, we propose a variational framework in which the minimization of
the acceleration on the group of diffeomorphisms endowed with a right-invariant
metric is well-posed. It relies on constraining the acceleration to belong to a
Sobolev space of higher-order than the order of the metric in order to gain
compactness. It provides the theoretical guarantee of existence of minimizers
which is compulsory for numerical simulations
Piecewise rigid curve deformation via a Finsler steepest descent
This paper introduces a novel steepest descent flow in Banach spaces. This
extends previous works on generalized gradient descent, notably the work of
Charpiat et al., to the setting of Finsler metrics. Such a generalized gradient
allows one to take into account a prior on deformations (e.g., piecewise rigid)
in order to favor some specific evolutions. We define a Finsler gradient
descent method to minimize a functional defined on a Banach space and we prove
a convergence theorem for such a method. In particular, we show that the use of
non-Hilbertian norms on Banach spaces is useful to study non-convex
optimization problems where the geometry of the space might play a crucial role
to avoid poor local minima. We show some applications to the curve matching
problem. In particular, we characterize piecewise rigid deformations on the
space of curves and we study several models to perform piecewise rigid
evolution of curves
Scaling Algorithms for Unbalanced Transport Problems
This article introduces a new class of fast algorithms to approximate
variational problems involving unbalanced optimal transport. While classical
optimal transport considers only normalized probability distributions, it is
important for many applications to be able to compute some sort of relaxed
transportation between arbitrary positive measures. A generic class of such
"unbalanced" optimal transport problems has been recently proposed by several
authors. In this paper, we show how to extend the, now classical, entropic
regularization scheme to these unbalanced problems. This gives rise to fast,
highly parallelizable algorithms that operate by performing only diagonal
scaling (i.e. pointwise multiplications) of the transportation couplings. They
are generalizations of the celebrated Sinkhorn algorithm. We show how these
methods can be used to solve unbalanced transport, unbalanced gradient flows,
and to compute unbalanced barycenters. We showcase applications to 2-D shape
modification, color transfer, and growth models
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