273 research outputs found
Beltrami Representation and its applications to texture map and video compression
Surface parameterizations and registrations are important in computer
graphics and imaging, where 1-1 correspondences between meshes are computed. In
practice, surface maps are usually represented and stored as 3D coordinates
each vertex is mapped to, which often requires lots of storage memory. This
causes inconvenience in data transmission and data storage. To tackle this
problem, we propose an effective algorithm for compressing surface
homeomorphisms using Fourier approximation of the Beltrami representation. The
Beltrami representation is a complex-valued function defined on triangular
faces of the surface mesh with supreme norm strictly less than 1. Under
suitable normalization, there is a 1-1 correspondence between the set of
surface homeomorphisms and the set of Beltrami representations. Hence, every
bijective surface map is associated with a unique Beltrami representation.
Conversely, given a Beltrami representation, the corresponding bijective
surface map can be exactly reconstructed using the Linear Beltrami Solver
introduced in this paper. Using the Beltrami representation, the surface
homeomorphism can be easily compressed by Fourier approximation, without
distorting the bijectivity of the map. The storage memory can be effectively
reduced, which is useful for many practical problems in computer graphics and
imaging. In this paper, we proposed to apply the algorithm to texture map
compression and video compression. With our proposed algorithm, the storage
requirement for the texture properties of a textured surface can be
significantly reduced. Our algorithm can further be applied to compressing
motion vector fields for video compression, which effectively improve the
compression ratio.Comment: 30 pages, 23 figure
Teichm\"uller extremal mapping and its applications to landmark matching registration
Registration, which aims to find an optimal 1-1 correspondence between
shapes, is an important process in different research areas. Conformal mappings
have been widely used to obtain a diffeomorphism between shapes that minimizes
angular distortion. Conformal registrations are beneficial since it preserves
the local geometry well. However, when landmark constraints are enforced,
conformal mappings generally do not exist. This motivates us to look for a
unique landmark matching quasi-conformal registration, which minimizes the
conformality distortion. Under suitable condition on the landmark constraints,
a unique diffeomporphism, called the Teichm\"uller extremal mapping between two
surfaces can be obtained, which minimizes the maximal conformality distortion.
In this paper, we propose an efficient iterative algorithm, called the
Quasi-conformal (QC) iterations, to compute the Teichm\"uller mapping. The
basic idea is to represent the set of diffeomorphisms using Beltrami
coefficients (BCs), and look for an optimal BC associated to the desired
Teichm\"uller mapping. The associated diffeomorphism can be efficiently
reconstructed from the optimal BC using the Linear Beltrami Solver(LBS). Using
BCs to represent diffeomorphisms guarantees the diffeomorphic property of the
registration. Using our proposed method, the Teichm\"uller mapping can be
accurately and efficiently computed within 10 seconds. The obtained
registration is guaranteed to be bijective. The proposed algorithm can also be
extended to compute Teichm\"uller mapping with soft landmark constraints. We
applied the proposed algorithm to real applications, such as brain landmark
matching registration, constrained texture mapping and human face registration.
Experimental results shows that our method is both effective and efficient in
computing a non-overlap landmark matching registration with least amount of
conformality distortion.Comment: 26 pages, 21 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
Image retargeting via Beltrami representation
Image retargeting aims to resize an image to one with a prescribed aspect
ratio. Simple scaling inevitably introduces unnatural geometric distortions on
the important content of the image. In this paper, we propose a simple and yet
effective method to resize an image, which preserves the geometry of the
important content, using the Beltrami representation. Our algorithm allows
users to interactively label content regions as well as line structures. Image
resizing can then be achieved by warping the image by an orientation-preserving
bijective warping map with controlled distortion. The warping map is
represented by its Beltrami representation, which captures the local geometric
distortion of the map. By carefully prescribing the values of the Beltrami
representation, images with different complexity can be effectively resized.
Our method does not require solving any optimization problems and tuning
parameters throughout the process. This results in a simple and efficient
algorithm to solve the image retargeting problem. Extensive experiments have
been carried out, which demonstrate the efficacy of our proposed method.Comment: 13pages, 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
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
Efficient Feature-based Image Registration by Mapping Sparsified Surfaces
With the advancement in the digital camera technology, the use of high
resolution images and videos has been widespread in the modern society. In
particular, image and video frame registration is frequently applied in
computer graphics and film production. However, conventional registration
approaches usually require long computational time for high resolution images
and video frames. This hinders the application of the registration approaches
in the modern industries. In this work, we first propose a new image
representation method to accelerate the registration process by triangulating
the images effectively. For each high resolution image or video frame, we
compute an optimal coarse triangulation which captures the important features
of the image. Then, we apply a surface registration algorithm to obtain a
registration map which is used to compute the registration of the high
resolution image. Experimental results suggest that our overall algorithm is
efficient and capable to achieve a high compression rate while the accuracy of
the registration is well retained when compared with the conventional
grid-based approach. Also, the computational time of the registration is
significantly reduced using our triangulation-based approach
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
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
A Low-rank Spline Approximation of Planar Domains
Construction of spline surfaces from given boundary curves is one of the
classical problems in computer aided geometric design, which regains much
attention in isogeometric analysis in recent years and is called domain
parameterization. However, for most of the state-of-the-art parameterization
methods, the rank of the spline parameterization is usually large, which
results in higher computational cost in solving numerical PDEs. In this paper,
we propose a low-rank representation for the spline parameterization of planar
domains using low-rank tensor approximation technique, and apply
quasi-conformal map as the framework of the spline parameterization. Under
given correspondence of boundary curves, a quasi-conformal map with low rank
and low distortion between a unit square and the computational domain can be
modeled as a non-linear optimization problem. We propose an efficient algorithm
to compute the quasi-conformal map by solving two convex optimization problems
alternatively. Experimental results show that our approach can produce a
bijective and low-rank parametric spline representation of planar domains,
which results in better performance than previous approaches in solving
numerical PDEs
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