594 research outputs found
Diffeomorphic random sampling using optimal information transport
In this article we explore an algorithm for diffeomorphic random sampling of
nonuniform probability distributions on Riemannian manifolds. The algorithm is
based on optimal information transport (OIT)---an analogue of optimal mass
transport (OMT). Our framework uses the deep geometric connections between the
Fisher-Rao metric on the space of probability densities and the right-invariant
information metric on the group of diffeomorphisms. The resulting sampling
algorithm is a promising alternative to OMT, in particular as our formulation
is semi-explicit, free of the nonlinear Monge--Ampere equation. Compared to
Markov Chain Monte Carlo methods, we expect our algorithm to stand up well when
a large number of samples from a low dimensional nonuniform distribution is
needed.Comment: 8 pages, 3 figure
Diffeomorphic density registration
In this book chapter we study the Riemannian Geometry of the density
registration problem: Given two densities (not necessarily probability
densities) defined on a smooth finite dimensional manifold find a
diffeomorphism which transforms one to the other. This problem is motivated by
the medical imaging application of tracking organ motion due to respiration in
Thoracic CT imaging where the fundamental physical property of conservation of
mass naturally leads to modeling CT attenuation as a density. We will study the
intimate link between the Riemannian metrics on the space of diffeomorphisms
and those on the space of densities. We finally develop novel computationally
efficient algorithms and demonstrate there applicability for registering RCCT
thoracic imaging.Comment: 23 pages, 6 Figures, Chapter for a Book on Medical Image Analysi
Diffeomorphic density matching by optimal information transport
We address the following problem: given two smooth densities on a manifold,
find an optimal diffeomorphism that transforms one density into the other. Our
framework builds on connections between the Fisher-Rao information metric on
the space of probability densities and right-invariant metrics on the
infinite-dimensional manifold of diffeomorphisms. This optimal information
transport, and modifications thereof, allows us to construct numerical
algorithms for density matching. The algorithms are inherently more efficient
than those based on optimal mass transport or diffeomorphic registration. Our
methods have applications in medical image registration, texture mapping, image
morphing, non-uniform random sampling, and mesh adaptivity. Some of these
applications are illustrated in examples.Comment: 35 page
Higher-Order Momentum Distributions and Locally Affine LDDMM Registration
To achieve sparse parametrizations that allows intuitive analysis, we aim to
represent deformation with a basis containing interpretable elements, and we
wish to use elements that have the description capacity to represent the
deformation compactly. To accomplish this, we introduce in this paper
higher-order momentum distributions in the LDDMM registration framework. While
the zeroth order moments previously used in LDDMM only describe local
displacement, the first-order momenta that are proposed here represent a basis
that allows local description of affine transformations and subsequent compact
description of non-translational movement in a globally non-rigid deformation.
The resulting representation contains directly interpretable information from
both mathematical and modeling perspectives. We develop the mathematical
construction of the registration framework with higher-order momenta, we show
the implications for sparse image registration and deformation description, and
we provide examples of how the parametrization enables registration with a very
low number of parameters. The capacity and interpretability of the
parametrization using higher-order momenta lead to natural modeling of
articulated movement, and the method promises to be useful for quantifying
ventricle expansion and progressing atrophy during Alzheimer's disease
A Geometric Variational Approach to Bayesian Inference
We propose a novel Riemannian geometric framework for variational inference
in Bayesian models based on the nonparametric Fisher-Rao metric on the manifold
of probability density functions. Under the square-root density representation,
the manifold can be identified with the positive orthant of the unit
hypersphere in L2, and the Fisher-Rao metric reduces to the standard L2 metric.
Exploiting such a Riemannian structure, we formulate the task of approximating
the posterior distribution as a variational problem on the hypersphere based on
the alpha-divergence. This provides a tighter lower bound on the marginal
distribution when compared to, and a corresponding upper bound unavailable
with, approaches based on the Kullback-Leibler divergence. We propose a novel
gradient-based algorithm for the variational problem based on Frechet
derivative operators motivated by the geometry of the Hilbert sphere, and
examine its properties. Through simulations and real-data applications, we
demonstrate the utility of the proposed geometric framework and algorithm on
several Bayesian models
Diffeomorphic Deformation via Sliced Wasserstein Distance Optimization for Cortical Surface Reconstruction
Mesh deformation is a core task for 3D mesh reconstruction, but defining an
efficient discrepancy between predicted and target meshes remains an open
problem. A prevalent approach in current deep learning is the set-based
approach which measures the discrepancy between two surfaces by comparing two
randomly sampled point-clouds from the two meshes with Chamfer pseudo-distance.
Nevertheless, the set-based approach still has limitations such as lacking a
theoretical guarantee for choosing the number of points in sampled
point-clouds, and the pseudo-metricity and the quadratic complexity of the
Chamfer divergence. To address these issues, we propose a novel metric for
learning mesh deformation. The metric is defined by sliced Wasserstein distance
on meshes represented as probability measures that generalize the set-based
approach. By leveraging probability measure space, we gain flexibility in
encoding meshes using diverse forms of probability measures, such as
continuous, empirical, and discrete measures via \textit{varifold}
representation. After having encoded probability measures, we can compare
meshes by using the sliced Wasserstein distance which is an effective optimal
transport distance with linear computational complexity and can provide a fast
statistical rate for approximating the surface of meshes. Furthermore, we
employ a neural ordinary differential equation (ODE) to deform the input
surface into the target shape by modeling the trajectories of the points on the
surface. Our experiments on cortical surface reconstruction demonstrate that
our approach surpasses other competing methods in multiple datasets and
metrics
Spectral Convergence of the connection Laplacian from random samples
Spectral methods that are based on eigenvectors and eigenvalues of discrete
graph Laplacians, such as Diffusion Maps and Laplacian Eigenmaps are often used
for manifold learning and non-linear dimensionality reduction. It was
previously shown by Belkin and Niyogi \cite{belkin_niyogi:2007} that the
eigenvectors and eigenvalues of the graph Laplacian converge to the
eigenfunctions and eigenvalues of the Laplace-Beltrami operator of the manifold
in the limit of infinitely many data points sampled independently from the
uniform distribution over the manifold. Recently, we introduced Vector
Diffusion Maps and showed that the connection Laplacian of the tangent bundle
of the manifold can be approximated from random samples. In this paper, we
present a unified framework for approximating other connection Laplacians over
the manifold by considering its principle bundle structure. We prove that the
eigenvectors and eigenvalues of these Laplacians converge in the limit of
infinitely many independent random samples. We generalize the spectral
convergence results to the case where the data points are sampled from a
non-uniform distribution, and for manifolds with and without boundary
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