4,721 research outputs found
Conformal Surface Morphing with Applications on Facial Expressions
Morphing is the process of changing one figure into another. Some numerical
methods of 3D surface morphing by deformable modeling and conformal mapping are
shown in this study. It is well known that there exists a unique Riemann
conformal mapping from a simply connected surface into a unit disk by the
Riemann mapping theorem. The dilation and relative orientations of the 3D
surfaces can be linked through the M\"obius transformation due to the conformal
characteristic of the Riemann mapping. On the other hand, a 3D surface
deformable model can be built via various approaches such as mutual
parameterization from direct interpolation or surface matching using landmarks.
In this paper, we take the advantage of the unique representation of 3D
surfaces by the mean curvatures and the conformal factors associated with the
Riemann mapping. By registering the landmarks on the conformal parametric
domains, the correspondence of the mean curvatures and the conformal factors
for each surfaces can be obtained. As a result, we can construct the 3D
deformation field from the surface reconstruction algorithm proposed by Gu and
Yau. Furthermore, by composition of the M\"obius transformation and the 3D
deformation field, the morphing sequence can be generated from the mean
curvatures and the conformal factors on a unified mesh structure by using the
cubic spline homotopy. Several numerical experiments of the face morphing are
presented to demonstrate the robustness of our approach.Comment: 8 pages, 13 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
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
Computing Quasiconformal Maps on Riemann surfaces using Discrete Curvature Flow
Surface mapping plays an important role in geometric processing. They induce
both area and angular distortions. If the angular distortion is bounded, the
mapping is called a {\it quasi-conformal} map. Many surface maps in our
physical world are quasi-conformal. The angular distortion of a quasi-conformal
map can be represented by Beltrami differentials. According to quasi-conformal
Teichm\"uller theory, there is an 1-1 correspondence between the set of
Beltrami differentials and the set of quasi-conformal surface maps. Therefore,
every quasi-conformal surface map can be fully determined by the Beltrami
differential and can be reconstructed by solving the so-called Beltrami
equation.
In this work, we propose an effective method to solve the Beltrami equation
on general Riemann surfaces. The solution is a quasi-conformal map associated
with the prescribed Beltrami differential. We firstly formulate a discrete
analog of quasi-conformal maps on triangular meshes. Then, we propose an
algorithm to compute discrete quasi-conformal maps. The main strategy is to
define a discrete auxiliary metric of the source surface, such that the
original quasi-conformal map becomes conformal under the newly defined discrete
metric. The associated map can then be obtained by using the discrete Yamabe
flow method. Numerically, the discrete quasi-conformal map converges to the
continuous real solution as the mesh size approaches to 0. We tested our
algorithm on surfaces scanned from real life with different topologies.
Experimental results demonstrate the generality and accuracy of our auxiliary
metric method
The Theory of Computational Quasi-conformal Geometry on Point Clouds
Quasi-conformal (QC) theory is an important topic in complex analysis, which
studies geometric patterns of deformations between shapes. Recently,
computational QC geometry has been developed and has made significant
contributions to medical imaging, computer graphics and computer vision.
Existing computational QC theories and algorithms have been built on
triangulation structures. In practical situations, many 3D acquisition
techniques often produce 3D point cloud (PC) data of the object, which does not
contain connectivity information. It calls for a need to develop computational
QC theories on PCs. In this paper, we introduce the concept of computational QC
geometry on PCs. We define PC quasi-conformal (PCQC) maps and their associated
PC Beltrami coefficients (PCBCs). The PCBC is analogous to the Beltrami
differential in the continuous setting. Theoretically, we show that the PCBC
converges to its continuous counterpart as the density of the PC tends to zero.
We also theoretically and numerically validate the ability of PCBCs to measure
local geometric distortions of PC deformations. With these concepts, many
existing QC based algorithms for geometry processing and shape analysis can be
easily extended to PC data
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
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
Spherical Conformal Parameterization of Genus-0 Point Clouds for Meshing
Point cloud is the most fundamental representation of 3D geometric objects.
Analyzing and processing point cloud surfaces is important in computer graphics
and computer vision. However, most of the existing algorithms for surface
analysis require connectivity information. Therefore, it is desirable to
develop a mesh structure on point clouds. This task can be simplified with the
aid of a parameterization. In particular, conformal parameterizations are
advantageous in preserving the geometric information of the point cloud data.
In this paper, we extend a state-of-the-art spherical conformal
parameterization algorithm for genus-0 closed meshes to the case of point
clouds, using an improved approximation of the Laplace-Beltrami operator on
data points. Then, we propose an iterative scheme called the North-South
reiteration for achieving a spherical conformal parameterization. A balancing
scheme is introduced to enhance the distribution of the spherical
parameterization. High quality triangulations and quadrangulations can then be
built on the point clouds with the aid of the parameterizations. Also, the
meshes generated are guaranteed to be genus-0 closed meshes. Moreover, using
our proposed spherical conformal parameterization, multilevel representations
of point clouds can be easily constructed. Experimental results demonstrate the
effectiveness of our proposed framework
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
Fast Disk Conformal Parameterization of Simply-connected Open Surfaces
Surface parameterizations have been widely used in computer graphics and
geometry processing. In particular, as simply-connected open surfaces are
conformally equivalent to the unit disk, it is desirable to compute the disk
conformal parameterizations of the surfaces. In this paper, we propose a novel
algorithm for the conformal parameterization of a simply-connected open surface
onto the unit disk, which significantly speeds up the computation, enhances the
conformality and stability, and guarantees the bijectivity. The conformality
distortions at the inner region and on the boundary are corrected by two steps,
with the aid of an iterative scheme using quasi-conformal theories.
Experimental results demonstrate the effectiveness of our proposed method
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