294 research outputs found
Computing Conformal Structure of Surfaces
This paper solves the problem of computing conformal structures of general
2-manifolds represented as triangle meshes. We compute conformal structures in
the following way: first compute homology bases from simplicial complex
structures, then construct dual cohomology bases and diffuse them to harmonic
1-forms. Next, we construct bases of holomorphic differentials. We then obtain
period matrices by integrating holomorphic differentials along homology bases.
We also study the global conformal mapping between genus zero surfaces and
spheres, and between general meshes and planes. Our method of computing
conformal structures can be applied to tackle fundamental problems in computer
aid design and computer graphics, such as geometry classification and
identification, and surface global parametrization.Comment: 14 pages, 3 figures, simplified version, full version upon reques
LMap: Shape-Preserving Local Mappings for Biomedical Visualization
Visualization of medical organs and biological structures is a challenging
task because of their complex geometry and the resultant occlusions. Global
spherical and planar mapping techniques simplify the complex geometry and
resolve the occlusions to aid in visualization. However, while resolving the
occlusions these techniques do not preserve the geometric context, making them
less suitable for mission-critical biomedical visualization tasks. In this
paper, we present a shape-preserving local mapping technique for resolving
occlusions locally while preserving the overall geometric context. More
specifically, we present a novel visualization algorithm, LMap, for conformally
parameterizing and deforming a selected local region-of-interest (ROI) on an
arbitrary surface. The resultant shape-preserving local mappings help to
visualize complex surfaces while preserving the overall geometric context. The
algorithm is based on the robust and efficient extrinsic Ricci flow technique,
and uses the dynamic Ricci flow algorithm to guarantee the existence of a local
map for a selected ROI on an arbitrary surface. We show the effectiveness and
efficacy of our method in three challenging use cases: (1) multimodal brain
visualization, (2) optimal coverage of virtual colonoscopy centerline
flythrough, and (3) molecular surface visualization.Comment: IEEE Transactions on Visualization and Computer Graphics, 24(12):
3111-3122, 2018 (12 pages, 11 figures
Rigidity of Infinite Hexagonal Triangulation of the Plane
In the paper, we consider the rigidity problem of the infinite hexagonal
triangulation of the plane under the piecewise linear conformal changes
introduced by Luo in [5]. Our result shows that if a geometric hexagonal
triangulation of the plane is PL conformal to the regular hexagonal
triangulation and all inner angles are in for any
constant , then it is the regular hexagonal triangulation. This
partially solves a conjecture of Luo [4]. The proof uses the concept of
\emph{quasi-harmonic} functions to unfold the properties of the mesh.Comment: 17 pages, 8 figure
A Conformal Approach for Surface Inpainting
We address the problem of surface inpainting, which aims to fill in holes or
missing regions on a Riemann surface based on its surface geometry. In
practical situation, surfaces obtained from range scanners often have holes
where the 3D models are incomplete. In order to analyze the 3D shapes
effectively, restoring the incomplete shape by filling in the surface holes is
necessary. In this paper, we propose a novel conformal approach to inpaint
surface holes on a Riemann surface based on its surface geometry. The basic
idea is to represent the Riemann surface using its conformal factor and mean
curvature. According to Riemann surface theory, a Riemann surface can be
uniquely determined by its conformal factor and mean curvature up to a rigid
motion. Given a Riemann surface , its mean curvature and conformal
factor can be computed easily through its conformal parameterization.
Conversely, given and , a Riemann surface can be uniquely
reconstructed by solving the Gauss-Codazzi equation on the conformal parameter
domain. Hence, the conformal factor and the mean curvature are two geometric
quantities fully describing the surface. With this - representation
of the surface, the problem of surface inpainting can be reduced to the problem
of image inpainting of and on the conformal parameter domain.
Once and are inpainted, a Riemann surface can be reconstructed
which effectively restores the 3D surface with missing holes. Since the
inpainting model is based on the geometric quantities and , the
restored surface follows the surface geometric pattern. We test the proposed
algorithm on synthetic data as well as real surface data. Experimental results
show that our proposed method is an effective surface inpainting algorithm to
fill in surface holes on an incomplete 3D models based their surface geometry.Comment: 19 pages, 12 figure
Convergence of an iterative algorithm for Teichm\"uller maps via generalized harmonic maps
Finding surface mappings with least distortion arises from many applications
in various fields. Extremal Teichm\"uller maps are surface mappings with least
conformality distortion. The existence and uniqueness of the extremal
Teichm\"uller map between Riemann surfaces of finite type are theoretically
guaranteed [1]. Recently, a simple iterative algorithm for computing the
Teichm\"uller maps between connected Riemann surfaces with given boundary
value was proposed in [11]. Numerical results was reported in the paper to show
the effectiveness of the algorithm. The method was successfully applied to
landmark-matching registration. The purpose of this paper is to prove the
iterative algorithm proposed in [11] indeed converges.Comment: 18 pages, 11 figures. arXiv admin note: text overlap with
arXiv:1005.3292 by other author
Corresponding Supine and Prone Colon Visualization Using Eigenfunction Analysis and Fold Modeling
We present a method for registration and visualization of corresponding
supine and prone virtual colonoscopy scans based on eigenfunction analysis and
fold modeling. In virtual colonoscopy, CT scans are acquired with the patient
in two positions, and their registration is desirable so that physicians can
corroborate findings between scans. Our algorithm performs this registration
efficiently through the use of Fiedler vector representation (the second
eigenfunction of the Laplace-Beltrami operator). This representation is
employed to first perform global registration of the two colon positions. The
registration is then locally refined using the haustral folds, which are
automatically segmented using the 3D level sets of the Fiedler vector. The use
of Fiedler vectors and the segmented folds presents a precise way of
visualizing corresponding regions across datasets and visual modalities. We
present multiple methods of visualizing the results, including 2D flattened
rendering and the corresponding 3D endoluminal views. The precise fold modeling
is used to automatically find a suitable cut for the 2D flattening, which
provides a less distorted visualization. Our approach is robust, and we
demonstrate its efficiency and efficacy by showing matched views on both the 2D
flattened colons and in the 3D endoluminal view. We analytically evaluate the
results by measuring the distance between features on the registered colons,
and we also assess our fold segmentation against 20 manually labeled datasets.
We have compared our results analytically to previous methods, and have found
our method to achieve superior results. We also prove the hot spots conjecture
for modeling cylindrical topology using Fiedler vector representation, which
allows our approach to be used for general cylindrical geometry modeling and
feature extraction.Comment: IEEE Transactions on Visualization and Computer Graphics,
23(1):751-760, 2017 (11 pages, 13 figures
A discrete uniformization theorem for polyhedral surfaces
A discrete conformality for polyhedral metrics on surfaces is introduced in
this paper which generalizes earlier work on the subject. It is shown that each
polyhedral metric on a surface is discrete conformal to a constant curvature
polyhedral metric which is unique up to scaling. Furthermore, the constant
curvature metric can be found using a discrete Yamabe flow with surgery.Comment: 17 pages, 4 figure
Kernel Estimation from Salient Structure for Robust Motion Deblurring
Blind image deblurring algorithms have been improving steadily in the past
years. Most state-of-the-art algorithms, however, still cannot perform
perfectly in challenging cases, especially in large blur setting. In this
paper, we focus on how to estimate a good kernel estimate from a single blurred
image based on the image structure. We found that image details caused by
blurring could adversely affect the kernel estimation, especially when the blur
kernel is large. One effective way to eliminate these details is to apply image
denoising model based on the Total Variation (TV). First, we developed a novel
method for computing image structures based on TV model, such that the
structures undermining the kernel estimation will be removed. Second, to
mitigate the possible adverse effect of salient edges and improve the
robustness of kernel estimation, we applied a gradient selection method. Third,
we proposed a novel kernel estimation method, which is capable of preserving
the continuity and sparsity of the kernel and reducing the noises. Finally, we
developed an adaptive weighted spatial prior, for the purpose of preserving
sharp edges in latent image restoration. The effectiveness of our method is
demonstrated by experiments on various kinds of challenging examples.Comment: This work has been accepted by Signal Processing: Image
Communication, 201
Variational Principles for Minkowski Type Problems, Discrete Optimal Transport, and Discrete Monge-Ampere Equations
In this paper, we develop several related finite dimensional variational
principles for discrete optimal transport (DOT), Minkowski type problems for
convex polytopes and discrete Monge-Ampere equation (DMAE). A link between the
discrete optimal transport, discrete Monge-Ampere equation and the power
diagram in computational geometry is established.Comment: 13 pages, 5 figure
Discrete Laplace-Beltrami Operator Determines Discrete Riemannian Metric
The Laplace-Beltrami operator of a smooth Riemannian manifold is determined
by the Riemannian metric. Conversely, the heat kernel constructed from its
eigenvalues and eigenfunctions determines the Riemannian metric. This work
proves the analogy on Euclidean polyhedral surfaces (triangle meshes), that the
discrete Laplace-Beltrami operator and the discrete Riemannian metric (unique
up to a scaling) are mutually determined by each other. Given an Euclidean
polyhedral surface, its Riemannian metric is represented as edge lengths,
satisfying triangle inequalities on all faces. The Laplace-Beltrami operator is
formulated using the cotangent formula, where the edge weight is defined as the
sum of the cotangent of angles against the edge. We prove that the edge lengths
can be determined by the edge weights unique up to a scaling using the
variational approach. First, we show that the space of all possible metrics of
a polyhedral surface is convex. Then, we construct a special energy defined on
the metric space, such that the gradient of the energy equals to the edge
weights. Third, we show the Hessian matrix of the energy is positive definite,
restricted on the tangent space of the metric space, therefore the energy is
convex. Finally, by the fact that the parameter on a convex domain and the
gradient of a convex function defined on the domain have one-to-one
correspondence, we show the edge weights determines the polyhedral metric
unique up to a scaling. The constructive proof leads to a computational
algorithm that finds the unique metric on a topological triangle mesh from a
discrete Laplace-Beltrami operator matrix
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