13 research outputs found
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
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
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
Shape Analysis Using Spectral Geometry
Shape analysis is a fundamental research topic in computer graphics and computer vision. To date, more and more 3D data is produced by those advanced acquisition capture devices, e.g., laser scanners, depth cameras, and CT/MRI scanners. The increasing data demands advanced analysis tools including shape matching, retrieval, deformation, etc. Nevertheless, 3D Shapes are represented with Euclidean transformations such as translation, scaling, and rotation and digital mesh representations are irregularly sampled. The shape can also deform non-linearly and the sampling may vary. In order to address these challenging problems, we investigate Laplace-Beltrami shape spectra from the differential geometry perspective, focusing more on the intrinsic properties. In this dissertation, the shapes are represented with 2 manifolds, which are differentiable.
First, we discuss in detail about the salient geometric feature points in the Laplace-Beltrami spectral domain instead of traditional spatial domains. Simultaneously, the local shape descriptor of a feature point is the Laplace-Beltrami spectrum of the spatial region associated to the point, which are stable and distinctive. The salient spectral geometric features are invariant to spatial Euclidean transforms, isometric deformations and mesh triangulations. Both global and partial matching can be achieved with these salient feature points. Next, we introduce a novel method to analyze a set of poses, i.e., near-isometric deformations, of 3D models that are unregistered. Different shapes of poses are transformed from the 3D spatial domain to a geometry spectral one where all near isometric deformations, mesh triangulations and Euclidean transformations are filtered away. Semantic parts of that model are then determined based on the computed geometric properties of all the mapped vertices in the geometry spectral domain while semantic skeleton can be automatically built with joints detected. Finally we prove the shape spectrum is a continuous function to a scale function on the conformal factor of the manifold. The derivatives of the eigenvalues are analytically expressed with those of the scale function. The property applies to both continuous domain and discrete triangle meshes. On the triangle meshes, a spectrum alignment algorithm is developed. Given two closed triangle meshes, the eigenvalues can be aligned from one to the other and the eigenfunction distributions are aligned as well. This extends the shape spectra across non-isometric deformations, supporting a registration-free analysis of general motion data
Physics based supervised and unsupervised learning of graph structure
Graphs are central tools to aid our understanding of biological, physical, and social systems. Graphs also play a key role in representing and understanding the visual world around us, 3D-shapes and 2D-images alike. In this dissertation, I propose the use of physical or natural phenomenon to understand graph structure. I investigate four phenomenon or laws in nature: (1) Brownian motion, (2) Gauss\u27s law, (3) feedback loops, and (3) neural synapses, to discover patterns in graphs
Signal Processing on Textured Meshes
In this thesis we extend signal processing techniques originally formulated in the context of image processing to techniques that can be applied to signals on arbitrary triangles meshes.
We develop methods for the two most common representations of signals on triangle meshes: signals sampled at the vertices of a finely tessellated mesh, and signals mapped to a coarsely tessellated mesh through texture maps.
Our first contribution is the combination of Lagrangian Integration and the Finite Elements Method in the formulation of two signal processing tasks: Shock Filters for texture and geometry sharpening, and Optical Flow for texture registration.
Our second contribution is the formulation of Gradient-Domain processing within the texture atlas. We define a function space that handles chart discontinuities, and linear operators that capture the metric distortion introduced by the parameterization.
Our third contribution is the construction of a spatiotemporal atlas parameterization for evolving meshes. Our method introduces localized remeshing operations and a compact parameterization that improves geometry and texture video compression. We show temporally coherent signal processing using partial correspondences