6 research outputs found
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
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
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
Structural Surface Mapping for Shape Analysis
Natural surfaces are usually associated with feature graphs, such as the cortical surface with anatomical atlas structure. Such a feature graph subdivides the whole surface into meaningful sub-regions. Existing brain mapping and registration methods did not integrate anatomical atlas structures. As a result, with existing brain mappings, it is difficult to visualize and compare the atlas structures. And also existing brain registration methods can not guarantee the best possible alignment of the cortical regions which can help computing more accurate shape similarity metrics for neurodegenerative disease analysis, e.g., Alzheimer’s disease (AD) classification. Also, not much attention has been paid to tackle surface parameterization and registration with graph constraints in a rigorous way which have many applications in graphics, e.g., surface and image morphing.
This dissertation explores structural mappings for shape analysis of surfaces using the feature graphs as constraints. (1) First, we propose structural brain mapping which maps the brain cortical surface onto a planar convex domain using Tutte embedding of a novel atlas graph and harmonic map with atlas graph constraints to facilitate visualization and comparison between the atlas structures. (2) Next, we propose a novel brain registration technique based on an intrinsic atlas-constrained harmonic map which provides the best possible alignment of the cortical regions. (3) After that, the proposed brain registration technique has been applied to compute shape similarity metrics for AD classification. (4) Finally, we propose techniques to compute intrinsic graph-constrained parameterization and registration for general genus-0 surfaces which have been used in surface and image morphing applications