125 research outputs found
Structural Surface Mapping for Shape Analysis
Natural surfaces are usually associated with feature graphs, such as the cortical surface with anatomical atlas structure. Such a feature graph subdivides the whole surface into meaningful sub-regions. Existing brain mapping and registration methods did not integrate anatomical atlas structures. As a result, with existing brain mappings, it is difficult to visualize and compare the atlas structures. And also existing brain registration methods can not guarantee the best possible alignment of the cortical regions which can help computing more accurate shape similarity metrics for neurodegenerative disease analysis, e.g., Alzheimer’s disease (AD) classification. Also, not much attention has been paid to tackle surface parameterization and registration with graph constraints in a rigorous way which have many applications in graphics, e.g., surface and image morphing.
This dissertation explores structural mappings for shape analysis of surfaces using the feature graphs as constraints. (1) First, we propose structural brain mapping which maps the brain cortical surface onto a planar convex domain using Tutte embedding of a novel atlas graph and harmonic map with atlas graph constraints to facilitate visualization and comparison between the atlas structures. (2) Next, we propose a novel brain registration technique based on an intrinsic atlas-constrained harmonic map which provides the best possible alignment of the cortical regions. (3) After that, the proposed brain registration technique has been applied to compute shape similarity metrics for AD classification. (4) Finally, we propose techniques to compute intrinsic graph-constrained parameterization and registration for general genus-0 surfaces which have been used in surface and image morphing applications
Computing quasiconformal folds
We propose a novel way of computing surface folding maps via solving a linear
PDE. This framework is a generalization to the existing quasiconformal methods
and allows manipulation of the geometry of folding. Moreover, the crucial
quantity that characterizes the geometry occurs as the coefficient of the
equation, namely the Beltrami coefficient. This allows us to solve an inverse
problem of parametrizing the folded surface given only partial data but with
known folding topology. Various interesting applications such as fold sculpting
on 3D models and self-occlusion reasoning are demonstrated to show the
effectiveness of our method
Maximization of Laplace-Beltrami eigenvalues on closed Riemannian surfaces
Let be a connected, closed, orientable Riemannian surface and denote
by the -th eigenvalue of the Laplace-Beltrami operator on
. In this paper, we consider the mapping .
We propose a computational method for finding the conformal spectrum
, which is defined by the eigenvalue optimization problem
of maximizing for fixed as varies within a conformal
class of fixed volume . We also propose a
computational method for the problem where is additionally allowed to vary
over surfaces with fixed genus, . This is known as the topological
spectrum for genus and denoted by . Our
computations support a conjecture of N. Nadirashvili (2002) that
, attained by a sequence of surfaces degenerating to
a union of identical round spheres. Furthermore, based on our computations,
we conjecture that ,
attained by a sequence of surfaces degenerating into a union of an equilateral
flat torus and identical round spheres. The values are compared to
several surfaces where the Laplace-Beltrami eigenvalues are well-known,
including spheres, flat tori, and embedded tori. In particular, we show that
among flat tori of volume one, the -th Laplace-Beltrami eigenvalue has a
local maximum with value . Several properties are also studied
computationally, including uniqueness, symmetry, and eigenvalue multiplicity.Comment: 43 pages, 18 figure
Spherical parameterization for genus zero surfaces using Laplace-Beltrami eigenfunctions
International audienceIn this work, we propose a fast and simple approach to obtain a spherical parameterization of a certain class of closed surfaces without holes. Our approach relies on empirical findings that can be mathematically investigated, to a certain extent, by using Laplace-Beltrami Operator and associated geometrical tools. The mapping proposed here is defined by considering only the three first non-trivial eigenfunctions of the Laplace-Beltrami Operator. Our approach requires a topological condition on those eigenfunctions, whose nodal domains must be 2. We show the efficiency of the approach through numerical experiments performed on cortical surface meshes
Mathematical Imaging and Surface Processing
Within the last decade image and geometry processing have become increasingly rigorous with solid foundations in mathematics. Both areas are research fields at the intersection of different mathematical disciplines, ranging from geometry and calculus of variations to PDE analysis and numerical analysis. The workshop brought together scientists from all these areas and a fruitful interplay took place. There was a lively exchange of ideas between geometry and image processing applications areas, characterized in a number of ways in this workshop. For example, optimal transport, first applied in computer vision is now used to define a distance measure between 3d shapes, spectral analysis as a tool in image processing can be applied in surface classification and matching, and so on. We have also seen the use of Riemannian geometry as a powerful tool to improve the analysis of multivalued images.
This volume collects the abstracts for all the presentations covering this wide spectrum of tools and application domains
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