371 research outputs found
Functional Maps Representation on Product Manifolds
We consider the tasks of representing, analyzing and manipulating maps
between shapes. We model maps as densities over the product manifold of the
input shapes; these densities can be treated as scalar functions and therefore
are manipulable using the language of signal processing on manifolds. Being a
manifold itself, the product space endows the set of maps with a geometry of
its own, which we exploit to define map operations in the spectral domain; we
also derive relationships with other existing representations (soft maps and
functional maps). To apply these ideas in practice, we discretize product
manifolds and their Laplace--Beltrami operators, and we introduce localized
spectral analysis of the product manifold as a novel tool for map processing.
Our framework applies to maps defined between and across 2D and 3D shapes
without requiring special adjustment, and it can be implemented efficiently
with simple operations on sparse matrices.Comment: Accepted to Computer Graphics Foru
Functional maps representation on product manifolds
We consider the tasks of representing, analysing and manipulating maps between shapes. We model maps as densities over the product manifold of the input shapes; these densities can be treated as scalar functions and therefore are manipulable using the language of signal processing on manifolds. Being a manifold itself, the product space endows the set of maps with a geometry of its own, which we exploit to define map operations in the spectral domain; we also derive relationships with other existing representations (soft maps and functional maps). To apply these ideas in practice, we discretize product manifolds and their Laplace–Beltrami operators, and we introduce localized spectral analysis of the product manifold as a novel tool for map processing. Our framework applies to maps defined between and across 2D and 3D shapes without requiring special adjustment, and it can be implemented efficiently with simple operations on sparse matrices
Unified Heat Kernel Regression for Diffusion, Kernel Smoothing and Wavelets on Manifolds and Its Application to Mandible Growth Modeling in CT Images
We present a novel kernel regression framework for smoothing scalar surface
data using the Laplace-Beltrami eigenfunctions. Starting with the heat kernel
constructed from the eigenfunctions, we formulate a new bivariate kernel
regression framework as a weighted eigenfunction expansion with the heat kernel
as the weights. The new kernel regression is mathematically equivalent to
isotropic heat diffusion, kernel smoothing and recently popular diffusion
wavelets. Unlike many previous partial differential equation based approaches
involving diffusion, our approach represents the solution of diffusion
analytically, reducing numerical inaccuracy and slow convergence. The numerical
implementation is validated on a unit sphere using spherical harmonics. As an
illustration, we have applied the method in characterizing the localized growth
pattern of mandible surfaces obtained in CT images from subjects between ages 0
and 20 years by regressing the length of displacement vectors with respect to
the template surface.Comment: Accepted in Medical Image Analysi
Metric Optimization for Surface Analysis in the Laplace-Beltrami Embedding Space
In this paper we present a novel approach for the intrinsic mapping of anatomical surfaces and its application in brain mapping research. Using the Laplace-Beltrami eigen-system, we represent each surface with an isometry invariant embedding in a high dimensional space. The key idea in our system is that we realize surface deformation in the embedding space via the iterative optimization of a conformal metric without explicitly perturbing the surface or its embedding. By minimizing a distance measure in the embedding space with metric optimization, our method generates a conformal map directly between surfaces with highly uniform metric distortion and the ability of aligning salient geometric features. Besides pairwise surface maps, we also extend the metric optimization approach for group-wise atlas construction and multi-atlas cortical label fusion. In experimental results, we demonstrate the robustness and generality of our method by applying it to map both cortical and hippocampal surfaces in population studies. For cortical labeling, our method achieves excellent performance in a cross-validation experiment with 40 manually labeled surfaces, and successfully models localized brain development in a pediatric study of 80 subjects. For hippocampal mapping, our method produces much more significant results than two popular tools on a multiple sclerosis study of 109 subjects
Spectrum of the Laplace-Beltrami Operator and the Phase Structure of Causal Dynamical Triangulation
We propose a new method to characterize the different phases observed in the
non-perturbative numerical approach to quantum gravity known as Causal
Dynamical Triangulation. The method is based on the analysis of the eigenvalues
and the eigenvectors of the Laplace-Beltrami operator computed on the
triangulations: it generalizes previous works based on the analysis of
diffusive processes and proves capable of providing more detailed information
on the geometric properties of the triangulations. In particular, we apply the
method to the analysis of spatial slices, showing that the different phases can
be characterized by a new order parameter related to the presence or absence of
a gap in the spectrum of the Laplace-Beltrami operator, and deriving an
effective dimensionality of the slices at the different scales. We also propose
quantities derived from the spectrum that could be used to monitor the running
to the continuum limit around a suitable critical point in the phase diagram,
if any is found.Comment: 21 pages, 26 figures, 2 table
Steklov Spectral Geometry for Extrinsic Shape Analysis
We propose using the Dirichlet-to-Neumann operator as an extrinsic
alternative to the Laplacian for spectral geometry processing and shape
analysis. Intrinsic approaches, usually based on the Laplace-Beltrami operator,
cannot capture the spatial embedding of a shape up to rigid motion, and many
previous extrinsic methods lack theoretical justification. Instead, we consider
the Steklov eigenvalue problem, computing the spectrum of the
Dirichlet-to-Neumann operator of a surface bounding a volume. A remarkable
property of this operator is that it completely encodes volumetric geometry. We
use the boundary element method (BEM) to discretize the operator, accelerated
by hierarchical numerical schemes and preconditioning; this pipeline allows us
to solve eigenvalue and linear problems on large-scale meshes despite the
density of the Dirichlet-to-Neumann discretization. We further demonstrate that
our operators naturally fit into existing frameworks for geometry processing,
making a shift from intrinsic to extrinsic geometry as simple as substituting
the Laplace-Beltrami operator with the Dirichlet-to-Neumann operator.Comment: Additional experiments adde
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