2,274 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
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
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
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
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
Optimization of Surface Registrations using Beltrami Holomorphic Flow
In shape analysis, finding an optimal 1-1 correspondence between surfaces
within a large class of admissible bijective mappings is of great importance.
Such process is called surface registration. The difficulty lies in the fact
that the space of all surface diffeomorphisms is a complicated functional
space, making exhaustive search for the best mapping challenging. To tackle
this problem, we propose a simple representation of bijective surface maps
using Beltrami coefficients (BCs), which are complex-valued functions defined
on surfaces with supreme norm less than 1. Fixing any 3 points on a pair of
surfaces, there is a 1-1 correspondence between the set of surface
diffeomorphisms between them and the set of BCs. Hence, every bijective surface
map can be represented by a unique BC. Conversely, given a BC, we can
reconstruct the unique surface map associated to it using the Beltrami
Holomorphic flow (BHF) method. Using BCs to represent surface maps is
advantageous because it is a much simpler functional space, which captures many
essential features of a surface map. By adjusting BCs, we equivalently adjust
surface diffeomorphisms to obtain the optimal map with desired properties. More
specifically, BHF gives us the variation of the associated map under the
variation of BC. Using this, a variational problem over the space of surface
diffeomorphisms can be easily reformulated into a variational problem over the
space of BCs. This makes the minimization procedure much easier. More
importantly, the diffeomorphic property is always preserved. We test our method
on synthetic examples and real medical applications. Experimental results
demonstrate the effectiveness of our proposed algorithm for surface
registration
Densely tracking sequences of 3D face scans
3D face dense tracking aims to find dense inter-frame correspondences in a
sequence of 3D face scans and constitutes a powerful tool for many face
analysis tasks, e.g., 3D dynamic facial expression analysis. The majority of
the existing methods just fit a 3D face surface or model to a 3D target surface
without considering temporal information between frames. In this paper, we
propose a novel method for densely tracking sequences of 3D face scans, which
ex- tends the non-rigid ICP algorithm by adding a novel specific criterion for
temporal information. A novel fitting framework is presented for automatically
tracking a full sequence of 3D face scans. The results of experiments carried
out on the BU4D-FE database are promising, showing that the proposed algorithm
outperforms state-of-the-art algorithms for 3D face dense tracking.Comment: 8 page
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
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
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