143 research outputs found
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
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
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
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
Geometric Registration of High-genus Surfaces
This paper presents a method to obtain geometric registrations between
high-genus () surfaces. Surface registration between simple surfaces,
such as simply-connected open surfaces, has been well studied. However, very
few works have been carried out for the registration of high-genus surfaces.
The high-genus topology of the surface poses great challenge for surface
registration. A possible approach is to partition surfaces into
simply-connected patches and registration is done patch by patch. Consistent
cuts are required, which are usually difficult to obtain and prone to error. In
this work, we propose an effective way to obtain geometric registration between
high-genus surfaces without introducing consistent cuts. The key idea is to
conformally parameterize the surface into its universal covering space, which
is either the Euclidean plane or the hyperbolic disk embedded in
. Registration can then be done on the universal covering space
by minimizing a shape mismatching energy measuring the geometric dissimilarity
between the two surfaces. Our proposed algorithm effectively computes a smooth
registration between high-genus surfaces that matches geometric information as
much as possible. The algorithm can also be applied to find a smooth and
bijective registration minimizing any general energy functionals. Numerical
experiments on high-genus surface data show that our proposed method is
effective for registering high-genus surfaces with geometric matching. We also
applied the method to register anatomical structures for medical imaging, which
demonstrates the usefulness of the 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
Parallelizable global conformal parameterization of simply-connected surfaces via partial welding
Conformal surface parameterization is useful in graphics, imaging and
visualization, with applications to texture mapping, atlas construction,
registration, remeshing and so on. With the increasing capability in scanning
and storing data, dense 3D surface meshes are common nowadays. While meshes
with higher resolution better resemble smooth surfaces, they pose computational
difficulties for the existing parameterization algorithms. In this work, we
propose a novel parallelizable algorithm for computing the global conformal
parameterization of simply-connected surfaces via partial welding maps. A given
simply-connected surface is first partitioned into smaller subdomains. The
local conformal parameterizations of all subdomains are then computed in
parallel. The boundaries of the parameterized subdomains are subsequently
integrated consistently using a novel technique called partial welding, which
is developed based on conformal welding theory. Finally, by solving the Laplace
equation for each subdomain using the updated boundary conditions, we obtain a
global conformal parameterization of the given surface, with bijectivity
guaranteed by quasi-conformal theory. By including additional shape
constraints, our method can be easily extended to achieve disk conformal
parameterization for simply-connected open surfaces and spherical conformal
parameterization for genus-0 closed surfaces. Experimental results are
presented to demonstrate the effectiveness of our proposed algorithm. When
compared to the state-of-the-art conformal parameterization methods, our method
achieves a significant improvement in both computational time and accuracy
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
Surface fluid registration of conformal representation: Application to detect disease burden and genetic influence on hippocampus
abstract: In this paper, we develop a new automated surface registration system based on surface conformal parameterization by holomorphic 1-forms, inverse consistent surface fluid registration, and multivariate tensor-based morphometty (mTBM). First, we conformally map a surface onto a planar rectangle space with holomorphic 1-forms. Second, we compute surface conformal representation by combining its local conformal factor and mean curvature and linearly scale the dynamic range of the conformal representation to form the feature image of the surface. Third, we align the feature image with a chosen template image via the fluid image registration algorithm, which has been extended into the curvilinear coordinates to adjust for the distortion introduced by surface parameterization. The inverse consistent image registration algorithm is also incorporated in the system to jointly estimate the forward and inverse transformations between the study and template images. This alignment induces a corresponding deformation on the surface. We tested the system on Alzheimer's Disease Neuroimaging Initiative (ADNI) baseline dataset to study AD symptoms on hippocampus. In our system, by modeling a hippocampus as a 3D parametric surface, we nonlinearly registered each surface with a selected template surface. Then we used mTBM to analyze the morphometry difference between diagnostic groups. Experimental results show that the new system has better performance than two publicly available subcortical surface registration tools: FIRST and SPHARM. We also analyzed the genetic influence of the Apolipoprotein E(is an element of)4 allele (ApoE4), which is considered as the most prevalent risk factor for AD. Our work successfully detected statistically significant difference between ApoE4 carriers and non-carriers in both patients of mild cognitive impairment (MCI) and healthy control subjects. The results show evidence that the ApoE genotype may be associated with accelerated brain atrophy so that our work provides a new MRI analysis tool that may help presymptomatic AD research.NOTICE: this is the author’s version of a work that was accepted for publication in NEUROIMAGE. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Neuroimage, 78, 111-134 [2013] http://dx.doi.org/10.1016/j.neuroimage.2013.04.01
Conformal Mapping and Brain Flattening
In this dissertation we study some of the main results concerning conformal mappings in the complex plane and between Riemann surfaces and we apply those results to the so-called brain flattening problem. In the first part of this thesis we prove the Riemann Mapping Theorem and we provide an introduction to the Uniformization Theorem for simply connected Riemann surfaces. The second part of the thesis is focused on the brain flattening problem, which deals with how to construct a conformal mapping from the brain's cortical surface to the unitary sphere. This procedure leads to a possible definition of the discrete mean curvature on a triangulated closed surface of genus zero. This flattening method has several applications in neuroscience
- …