757 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
A discrete framework to find the optimal matching between manifold-valued curves
The aim of this paper is to find an optimal matching between manifold-valued
curves, and thereby adequately compare their shapes, seen as equivalent classes
with respect to the action of reparameterization. Using a canonical
decomposition of a path in a principal bundle, we introduce a simple algorithm
that finds an optimal matching between two curves by computing the geodesic of
the infinite-dimensional manifold of curves that is at all time horizontal to
the fibers of the shape bundle. We focus on the elastic metric studied in the
so-called square root velocity framework. The quotient structure of the shape
bundle is examined, and in particular horizontality with respect to the fibers.
These results are more generally given for any elastic metric. We then
introduce a comprehensive discrete framework which correctly approximates the
smooth setting when the base manifold has constant sectional curvature. It is
itself a Riemannian structure on the product manifold of "discrete curves"
given by a finite number of points, and we show its convergence to the
continuous model as the size of the discretization goes to infinity.
Illustrations of optimal matching between discrete curves are given in the
hyperbolic plane, the plane and the sphere, for synthetic and real data, and
comparison with dynamic programming is established
Why Use Sobolev Metrics on the Space of Curves
We study reparametrization invariant Sobolev metrics on spaces of regular curves. We discuss their completeness properties and the resulting usability for applications in shape analysis. In particular, we will argue, that the development of efficient numerical methods for higher order Sobolev type metrics is an extremely desirable goal
Elastic shape analysis of surfaces with second-order Sobolev metrics: a comprehensive numerical framework
This paper introduces a set of numerical methods for Riemannian shape
analysis of 3D surfaces within the setting of invariant (elastic) second-order
Sobolev metrics. More specifically, we address the computation of geodesics and
geodesic distances between parametrized or unparametrized immersed surfaces
represented as 3D meshes. Building on this, we develop tools for the
statistical shape analysis of sets of surfaces, including methods for
estimating Karcher means and performing tangent PCA on shape populations, and
for computing parallel transport along paths of surfaces. Our proposed approach
fundamentally relies on a relaxed variational formulation for the geodesic
matching problem via the use of varifold fidelity terms, which enable us to
enforce reparametrization independence when computing geodesics between
unparametrized surfaces, while also yielding versatile algorithms that allow us
to compare surfaces with varying sampling or mesh structures. Importantly, we
demonstrate how our relaxed variational framework can be extended to tackle
partially observed data. The different benefits of our numerical pipeline are
illustrated over various examples, synthetic and real.Comment: 25 pages, 16 figures, 1 tabl
Elastic shape matching of parameterized surfaces using square root normal fields.
In this paper we define a new methodology for shape analysis of parameterized surfaces, where the main issues are: (1) choice of metric for shape comparisons and (2) invariance to reparameterization. We begin by defining a general elastic metric on the space of parameterized surfaces. The main advantages of this metric are twofold. First, it provides a natural interpretation of elastic shape deformations that are being quantified. Second, this metric is invariant under the action of the reparameterization group. We also introduce a novel representation of surfaces termed square root normal fields or SRNFs. This representation is convenient for shape analysis because, under this representation, a reduced version of the general elastic metric becomes the simple \ensuremathL2\ensuremathL2 metric. Thus, this transformation greatly simplifies the implementation of our framework. We validate our approach using multiple shape analysis examples for quadrilateral and spherical surfaces. We also compare the current results with those of Kurtek et al. [1]. We show that the proposed method results in more natural shape matchings, and furthermore, has some theoretical advantages over previous methods
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
An inexact matching approach for the comparison of plane curves with general elastic metrics
This paper introduces a new mathematical formulation and numerical approach
for the computation of distances and geodesics between immersed planar curves.
Our approach combines the general simplifying transform for first-order elastic
metrics that was recently introduced by Kurtek and Needham, together with a
relaxation of the matching constraint using parametrization-invariant fidelity
metrics. The main advantages of this formulation are that it leads to a simple
optimization problem for discretized curves, and that it provides a flexible
approach to deal with noisy, inconsistent or corrupted data. These benefits are
illustrated via a few preliminary numerical results.Comment: 5 pages, 5 figure
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