12,710 research outputs found
3D shape matching and Teichm\"uller spaces of pointed Riemann surfaces
Shape matching represents a challenging problem in both information
engineering and computer science, exhibiting not only a wide spectrum of
multimedia applications, but also a deep relation with conformal geometry.
After reviewing the theoretical foundations and the practical issues involved
in this fashinating subject, we focus on two state-of-the-art approaches
relying respectively on local features (landmark points) and on global
properties (conformal parameterizations). Finally, we introduce the
Teichm\"uller space of n-pointed Riemann surfaces of genus g into the realm of
multimedia, showing that its beautiful geometry provides a natural unified
framework for three-dimensional shape matching.Comment: Extended abstract submitted to MEGA 2011: Effective Methods in
Algebraic Geometr
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
Ricci Flow and Entropy Model for Avascular Tumor Growth and Decay Control
Prediction and control of cancer invasion is a vital problem in medical
science. This paper proposes a modern geometric Ricci-flow and entropy based
model for control of avascular multicellular tumor spheroid growth and decay.
As a tumor growth/decay control tool, a monoclonal antibody therapy is
proposed.
Keywords: avascular tumor growth and decay, multicellular tumor spheroid,
Ricci flow and entropy, nonlinear heat equation, monoclonal antibody cancer
therapyComment: 24 pages, 2 figures, Latex, revise
Level set flow in 3D steady gradient Ricci solitons
Let be a nontrivial 3-dimensional steady gradient Ricci
soliton. If the scalar curvature satisfies
for some , and , then the umbilical ratio of the
level sets of satisfies
Piecewise flat Ricci flow of compact without boundary three-manifolds
Using a recently developed piecewise flat method, numerical evolutions of the
Ricci flow are computed for a number of manifolds, using a number of different
mesh types, and shown to converge to the expected smooth behaviour as the mesh
resolution is increased. The manifolds were chosen to have varying degrees of
homogeneity, and include Nil and Gowdy manifolds, a three-torus initially
embedded in Euclidean four-space, and a perturbation of a flat three-torus. The
piecewise flat Ricci flow of the first two are shown to converge to analytic
and numerical partial differential equation solutions respectively, with the
remaining two flowing asymptotically to flat metrics
Ricci Flow and Nonlinear Reaction--Diffusion Systems in Biology, Chemistry, and Physics
This paper proposes the Ricci-flow equation from Riemannian geometry as a
general geometric framework for various nonlinear reaction-diffusion systems
(and related dissipative solitons) in mathematical biology. More precisely, we
propose a conjecture that any kind of reaction-diffusion processes in biology,
chemistry and physics can be modelled by the combined geometric-diffusion
system. In order to demonstrate the validity of this hypothesis, we review a
number of popular nonlinear reaction-diffusion systems and try to show that
they can all be subsumed by the presented geometric framework of the Ricci
flow.
Keywords: geometrical Ricci flow, nonlinear reaction-diffusion, dissipative
solitons and breathersComment: 30 pages, minor change
QCMC: Quasi-conformal Parameterizations for Multiply-connected domains
This paper presents a method to compute the {\it quasi-conformal
parameterization} (QCMC) for a multiply-connected 2D domain or surface. QCMC
computes a quasi-conformal map from a multiply-connected domain onto a
punctured disk associated with a given Beltrami differential. The
Beltrami differential, which measures the conformality distortion, is a
complex-valued function with supremum norm strictly less
than 1. Every Beltrami differential gives a conformal structure of . Hence,
the conformal module of , which are the radii and centers of the inner
circles, can be fully determined by , up to a M\"obius transformation. In
this paper, we propose an iterative algorithm to simultaneously search for the
conformal module and the optimal quasi-conformal parameterization. The key idea
is to minimize the Beltrami energy subject to the boundary constraints. The
optimal solution is our desired quasi-conformal parameterization onto a
punctured disk. The parameterization of the multiply-connected domain
simplifies numerical computations and has important applications in various
fields, such as in computer graphics and vision. Experiments have been carried
out on synthetic data together with real multiply-connected Riemann surfaces.
Results show that our proposed method can efficiently compute quasi-conformal
parameterizations of multiply-connected domains and outperforms other
state-of-the-art algorithms. Applications of the proposed parameterization
technique have also been explored.Comment: 26 pages, 23 figures, submitted. arXiv admin note: text overlap with
arXiv:1402.6908, arXiv:1307.2679 by other author
Area deficits and the Bel-Robinson tensor
The first law of causal diamonds relates the area deficit of a small ball
relative to flat space to the matter energy density it contains. At second
order in the Riemann normal coordinate expansion, this energy density should
receive contributions from the gravitational field itself. In this work, we
study the second-order area deficit of the ball in the absence of matter and
analyze its relation to possible notions of gravitational energy. In the small
ball limit, any proposed gravitational energy functional should evaluate to the
Bel-Robinson energy density in vacuum spacetimes. A direct calculation of
the area deficit reveals a result that is not simply proportional to . We
discuss how the deviation from is related to ambiguities in defining the
shape of the ball in curved space, and provide several proposals for fixing
these shape ambiguities.Comment: 30 page
A Linear Formulation for Disk Conformal Parameterization of Simply-Connected Open Surfaces
Surface parameterization is widely used in computer graphics and geometry
processing. It simplifies challenging tasks such as surface registrations,
morphing, remeshing and texture mapping. In this paper, we present an efficient
algorithm for computing the disk conformal parameterization of simply-connected
open surfaces. A double covering technique is used to turn a simply-connected
open surface into a genus-0 closed surface, and then a fast algorithm for
parameterization of genus-0 closed surfaces can be applied. The symmetry of the
double covered surface preserves the efficiency of the computation. A planar
parameterization can then be obtained with the aid of a M\"obius transformation
and the stereographic projection. After that, a normalization step is applied
to guarantee the circular boundary. Finally, we achieve a bijective disk
conformal parameterization by a composition of quasi-conformal mappings.
Experimental results demonstrate a significant improvement in the computational
time by over 60%. At the same time, our proposed method retains comparable
accuracy, bijectivity and robustness when compared with the state-of-the-art
approaches. Applications to texture mapping are presented for illustrating the
effectiveness of our proposed algorithm
TEMPO: Feature-Endowed Teichm\"uller Extremal Mappings of Point Clouds
In recent decades, the use of 3D point clouds has been widespread in computer
industry. The development of techniques in analyzing point clouds is
increasingly important. In particular, mapping of point clouds has been a
challenging problem. In this paper, we develop a discrete analogue of the
Teichm\"{u}ller extremal mappings, which guarantee uniform conformality
distortions, on point cloud surfaces. Based on the discrete analogue, we
propose a novel method called TEMPO for computing Teichm\"{u}ller extremal
mappings between feature-endowed point clouds. Using our proposed method, the
Teichm\"{u}ller metric is introduced for evaluating the dissimilarity of point
clouds. Consequently, our algorithm enables accurate recognition and
classification of point clouds. Experimental results demonstrate the
effectiveness of our proposed method
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