90 research outputs found
Efficient conformal parameterization of multiply-connected surfaces using quasi-conformal theory
Conformal mapping, a classical topic in complex analysis and differential
geometry, has become a subject of great interest in the area of surface
parameterization in recent decades with various applications in science and
engineering. However, most of the existing conformal parameterization
algorithms only focus on simply-connected surfaces and cannot be directly
applied to surfaces with holes. In this work, we propose two novel algorithms
for computing the conformal parameterization of multiply-connected surfaces. We
first develop an efficient method for conformally parameterizing an open
surface with one hole to an annulus on the plane. Based on this method, we then
develop an efficient method for conformally parameterizing an open surface with
holes onto a unit disk with circular holes. The conformality and
bijectivity of the mappings are ensured by quasi-conformal theory. Numerical
experiments and applications are presented to demonstrate the effectiveness of
the proposed methods
Density-equalizing maps for simply-connected open surfaces
In this paper, we are concerned with the problem of creating flattening maps
of simply-connected open surfaces in . Using a natural principle
of density diffusion in physics, we propose an effective algorithm for
computing density-equalizing flattening maps with any prescribed density
distribution. By varying the initial density distribution, a large variety of
mappings with different properties can be achieved. For instance,
area-preserving parameterizations of simply-connected open surfaces can be
easily computed. Experimental results are presented to demonstrate the
effectiveness of our proposed method. Applications to data visualization and
surface remeshing are explored
Bijective Density-Equalizing Quasiconformal Map for Multiply-Connected Open Surfaces
This paper proposes a novel method for computing bijective density-equalizing
quasiconformal (DEQ) flattening maps for multiply-connected open surfaces. In
conventional density-equalizing maps, shape deformations are solely driven by
prescribed constraints on the density distribution, defined as the population
per unit area, while the bijectivity and local geometric distortions of the
mappings are uncontrolled. Also, prior methods have primarily focused on
simply-connected open surfaces but not surfaces with more complicated
topologies. Our proposed method overcomes these issues by formulating the
density diffusion process as a quasiconformal flow, which allows us to
effectively control the local geometric distortion and guarantee the
bijectivity of the mapping by solving an energy minimization problem involving
the Beltrami coefficient of the mapping. To achieve an optimal parameterization
of multiply-connected surfaces, we develop an iterative scheme that optimizes
both the shape of the target planar circular domain and the density-equalizing
quasiconformal map onto it. In addition, landmark constraints can be
incorporated into our proposed method for consistent feature alignment. The
method can also be naturally applied to simply-connected open surfaces. By
changing the prescribed population, a large variety of surface flattening maps
with different desired properties can be achieved. The method is tested on both
synthetic and real examples, demonstrating its efficacy in various applications
in computer graphics and medical imaging
Preconditioned Nonlinear Conjugate Gradient Method of Stretch Energy Minimization for Area-Preserving Parameterizations
Stretch energy minimization (SEM) is widely recognized as one of the most
effective approaches for the computation of area-preserving mappings. In this
paper, we propose a novel preconditioned nonlinear conjugate gradient method
for SEM with guaranteed theoretical convergence. Numerical experiments indicate
that our new approach has significantly improved area-preserving accuracy and
computational efficiency compared to another state-of-the-art algorithm.
Furthermore, we present an application of surface registration to illustrate
the practical utility of area-preserving mappings as parameterizations of
surfaces.Comment: 18 pages, 8 figure
Free-boundary conformal parameterization of point clouds
With the advancement in 3D scanning technology, there has been a surge of
interest in the use of point clouds in science and engineering. To facilitate
the computations and analyses of point clouds, prior works have considered
parameterizing them onto some simple planar domains with a fixed boundary shape
such as a unit circle or a rectangle. However, the geometry of the fixed shape
may lead to some undesirable distortion in the parameterization. It is
therefore more natural to consider free-boundary conformal parameterizations of
point clouds, which minimize the local geometric distortion of the mapping
without constraining the overall shape. In this work, we develop a
free-boundary conformal parameterization method for disk-type point clouds,
which involves a novel approximation scheme of the point cloud Laplacian with
accumulated cotangent weights together with a special treatment at the boundary
points. With the aid of the free-boundary conformal parameterization,
high-quality point cloud meshing can be easily achieved. Furthermore, we show
that using the idea of conformal welding in complex analysis, the point cloud
conformal parameterization can be computed in a divide-and-conquer manner.
Experimental results are presented to demonstrate the effectiveness of the
proposed method
A diffusion-driven Characteristic Mapping method for particle management
We present a novel particle management method using the Characteristic
Mapping framework. In the context of explicit evolution of parametrized curves
and surfaces, the surface distribution of marker points created from sampling
the parametric space is controlled by the area element of the parametrization
function. As the surface evolves, the area element becomes uneven and the
sampling, suboptimal. In this method we maintain the quality of the sampling by
pre-composition of the parametrization with a deformation map of the parametric
space. This deformation is generated by the velocity field associated to the
diffusion process on the space of probability distributions and induces a
uniform redistribution of the marker points. We also exploit the semigroup
property of the heat equation to generate a submap decomposition of the
deformation map which provides an efficient way of maintaining evenly
distributed marker points on curves and surfaces undergoing extensive
deformations
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