817 research outputs found
Quadrature Points via Heat Kernel Repulsion
We discuss the classical problem of how to pick weighted points on a
dimensional manifold so as to obtain a reasonable quadrature rule
This problem, naturally, has a long history; the purpose of our paper is to
propose selecting points and weights so as to minimize the energy functional
\sum_{i,j =1}^{N}{ a_i a_j \exp\left(-\frac{d(x_i,x_j)^2}{4t}\right) }
\rightarrow \min, \quad \mbox{where}~t \sim N^{-2/d}, is the
geodesic distance and is the dimension of the manifold. This yields point
sets that are theoretically guaranteed, via spectral theoretic properties of
the Laplacian , to have good properties. One nice aspect is that the
energy functional is universal and independent of the underlying manifold; we
show several numerical examples
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
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