899 research outputs found
Time Discrete Geodesic Paths in the Space of Images
In this paper the space of images is considered as a Riemannian manifold
using the metamorphosis approach, where the underlying Riemannian metric
simultaneously measures the cost of image transport and intensity variation. A
robust and effective variational time discretization of geodesics paths is
proposed. This requires to minimize a discrete path energy consisting of a sum
of consecutive image matching functionals over a set of image intensity maps
and pairwise matching deformations. For square-integrable input images the
existence of discrete, connecting geodesic paths defined as minimizers of this
variational problem is shown. Furthermore, -convergence of the
underlying discrete path energy to the continuous path energy is proved. This
includes a diffeomorphism property for the induced transport and the existence
of a square-integrable weak material derivative in space and time. A spatial
discretization via finite elements combined with an alternating descent scheme
in the set of image intensity maps and the set of matching deformations is
presented to approximate discrete geodesic paths numerically. Computational
results underline the efficiency of the proposed approach and demonstrate
important qualitative properties.Comment: 27 pages, 7 figure
Geodesic Warps by Conformal Mappings
In recent years there has been considerable interest in methods for
diffeomorphic warping of images, with applications e.g.\ in medical imaging and
evolutionary biology. The original work generally cited is that of the
evolutionary biologist D'Arcy Wentworth Thompson, who demonstrated warps to
deform images of one species into another. However, unlike the deformations in
modern methods, which are drawn from the full set of diffeomorphism, he
deliberately chose lower-dimensional sets of transformations, such as planar
conformal mappings.
In this paper we study warps of such conformal mappings. The approach is to
equip the infinite dimensional manifold of conformal embeddings with a
Riemannian metric, and then use the corresponding geodesic equation in order to
obtain diffeomorphic warps. After deriving the geodesic equation, a numerical
discretisation method is developed. Several examples of geodesic warps are then
given. We also show that the equation admits totally geodesic solutions
corresponding to scaling and translation, but not to affine transformations
Constructing reparametrization invariant metrics on spaces of plane curves
Metrics on shape space are used to describe deformations that take one shape
to another, and to determine a distance between them. We study a family of
metrics on the space of curves, that includes several recently proposed
metrics, for which the metrics are characterised by mappings into vector spaces
where geodesics can be easily computed. This family consists of Sobolev-type
Riemannian metrics of order one on the space of
parametrized plane curves and the quotient space of unparametrized curves. For the space of open
parametrized curves we find an explicit formula for the geodesic distance and
show that the sectional curvatures vanish on the space of parametrized and are
non-negative on the space of unparametrized open curves. For the metric, which
is induced by the "R-transform", we provide a numerical algorithm that computes
geodesics between unparameterised, closed curves, making use of a constrained
formulation that is implemented numerically using the RATTLE algorithm. We
illustrate the algorithm with some numerical tests that demonstrate it's
efficiency and robustness.Comment: 27 pages, 4 figures. Extended versio
Dynamical Optimal Transport on Discrete Surfaces
We propose a technique for interpolating between probability distributions on
discrete surfaces, based on the theory of optimal transport. Unlike previous
attempts that use linear programming, our method is based on a dynamical
formulation of quadratic optimal transport proposed for flat domains by Benamou
and Brenier [2000], adapted to discrete surfaces. Our structure-preserving
construction yields a Riemannian metric on the (finite-dimensional) space of
probability distributions on a discrete surface, which translates the so-called
Otto calculus to discrete language. From a practical perspective, our technique
provides a smooth interpolation between distributions on discrete surfaces with
less diffusion than state-of-the-art algorithms involving entropic
regularization. Beyond interpolation, we show how our discrete notion of
optimal transport extends to other tasks, such as distribution-valued Dirichlet
problems and time integration of gradient flows
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