4,841 research outputs found
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
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
The Square Root Velocity Framework for Curves in a Homogeneous Space
In this paper we study the shape space of curves with values in a homogeneous
space , where is a Lie group and is a compact Lie subgroup. We
generalize the square root velocity framework to obtain a reparametrization
invariant metric on the space of curves in . By identifying curves in
with their horizontal lifts in , geodesics then can be computed. We can also
mod out by reparametrizations and by rigid motions of . In each of these
quotient spaces, we can compute Karcher means, geodesics, and perform principal
component analysis. We present numerical examples including the analysis of a
set of hurricane paths.Comment: To appear in 3rd International Workshop on Diff-CVML Workshop, CVPR
201
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