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 $\mathbb{R}^3$. 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

Area-preserving mapping of 3D ultrasound carotid artery images using density-equalizing reference map

Carotid atherosclerosis is a focal disease at the bifurcations of the carotid artery. To quantitatively monitor the local changes in the vessel-wall-plus-plaque thickness (VWT) and compare the VWT distributions for different patients or for the same patients at different ultrasound scanning sessions, a mapping technique is required to adjust for the geometric variability of different carotid artery models. In this work, we propose a novel method called density-equalizing reference map (DERM) for mapping 3D carotid surfaces to a standardized 2D carotid template, with an emphasis on preserving the local geometry of the carotid surface by minimizing the local area distortion. The initial map was generated by a previously described arc-length scaling (ALS) mapping method, which projects a 3D carotid surface onto a 2D non-convex L-shaped domain. A smooth and area-preserving flattened map was subsequently constructed by deforming the ALS map using the proposed algorithm that combines the density-equalizing map and the reference map techniques. This combination allows, for the first time, one-to-one mapping from a 3D surface to a standardized non-convex planar domain in an area-preserving manner. Evaluations using 20 carotid surface models show that the proposed method reduced the area distortion of the flattening maps by over 80% as compared to the ALS mapping method

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

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 $k$ holes onto a unit disk with $k$ 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

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