Abstract. This paper proposes a novel method for computing bijective density-equalizing quasiconformal 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.

Abstract. Density-equalizing maps are a class of mapping methods in which the shape deformation is driven by prescribed density information. In recent years, they have been widely used for data visualization on planar domains and planar parameterization of open surfaces. However, the theory and computation of density-equalizing maps for closed surfaces are much less explored. In this work, we develop a novel method for computing spherical density-equalizing maps for genus-0 closed surfaces. Specifically, we first compute a conformal parameterization of the given genus-0 closed surface onto the unit sphere. Then we perform density equalization on the spherical domain based on the given density information to achieve a spherical density-equalizing map. The bijectivity of the mapping is guaranteed by introducing an overlap correction scheme based on quasi-conformal theory throughout the density-equalizing iterative process. We further propose a method for incorporating the harmonic energy and landmark constraints into our formulation to achieve landmark-aligned spherical density-equalizing maps balancing different distortion measures. Using the proposed methods, a large variety of spherical parameterizations can be achieved. Applications to surface registration, remeshing, and data visualization are presented to demonstrate the effectiveness of our methods.

With the advancement of computer technology, there is a surge of interest in effective mapping methods for objects in higher-dimensional spaces. To establish a one-to-one correspondence between objects, higher-dimensional quasi-conformal theory can be utilized for ensuring the bijectivity of the mappings. In addition, it is often desirable for the mappings to satisfy certain prescribed geometric constraints and possess low distortion in conformality or volume. In this work, we develop a unifying framework for computing $n$-dimensional quasi-conformal mappings. More specifically, we propose a variational model that integrates quasi-conformal distortion, volumetric distortion, landmark correspondence, intensity mismatch, and volume prior information to handle a large variety of deformation problems. We further prove the existence of a minimizer for the proposed model and devise efficient numerical methods to solve the optimization problem. We demonstrate the effectiveness of the proposed framework using various experiments in two and three dimensions, with applications to medical image registration, adaptive remeshing, and shape modeling.

Conformal and quasi-conformal mappings have widespread applications in imaging science, computer vision, and computer graphics and can be used in surface registration, segmentation, remeshing, and texture map compression. While various conformal and quasi-conformal parameterization methods for simply connected surfaces have been proposed, efficient parameterization algorithms for multiply connected surfaces have been less explored. In this paper, we propose a novel parallelizable algorithm for computing the global conformal and quasi-conformal parameterizations of multiply connected surfaces onto a 2D circular domain using variants of the partial welding method and the Koebe's iteration. The main idea is to first partition a multiply connected surface into several subdomains and compute the free-boundary conformal and quasi-conformal parameterizations of them, respectively, and then apply a variant of the partial welding algorithm to reconstruct the global mapping. We apply the Koebe's iteration, together with the geodesic algorithm, to the boundary points and welding paths before and after the global welding to transform all the boundaries into circles conformally. After getting all the updated boundary conditions, we obtain the global parameterization of the multiply connected surface by solving the Laplace equation for each subdomain. Using this divide-and-conquer approach, the global conformal and quasi-conformal parameterizations of surfaces can be efficiently computed. Experimental results are presented to demonstrate the effectiveness of our proposed algorithm. More broadly, the proposed shift in perspective from solving a global quasi-conformal mapping problem to solving multiple local mapping problems paves a new way for computational quasi-conformal geometry.

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.

Recent advances in 3D human shape estimation build upon parametric representations that model very well the shape of the naked body, but are not appropriate to represent the clothing geometry. In this paper, we present an approach to model dressed humans and predict their geometry from single images. We contribute in three fundamental aspects of the problem, namely, a new dataset, a novel shape parameterization algorithm and an end-to-end deep generative network for predicting shape. First, we present 3D-People, a large-scale synthetic dataset with 2 Million photo-realistic images of 80 subjects performing 70 activities and wearing diverse outfits. Besides providing textured 3D meshes for clothes and body we annotated the dataset with segmentation masks, skeletons, depth, normal maps and optical flow. All this together makes 3D-People suitable for a plethora of tasks. We then represent the 3D shapes using 2D geometry images. To build these images we propose a novel spherical area-preserving parameterization algorithm based on the optimal mass transportation method. We show this approach to improve existing spherical maps which tend to shrink the elongated parts of the full body models such as the arms and legs, making the geometry images incomplete. Finally, we design a multi-resolution deep generative network that, given an input image of a dressed human, predicts his/her geometry image (and thus the clothed body shape) in an end-to-end manner. We obtain very promising results in jointly capturing body pose and clothing shape, both for synthetic validation and on the wild images.

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 maps with any prescribed density distribution. By varying the initial density distribution, a large variety of flattening maps 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.

The point cloud is the most fundamental representation of three-dimensional geometric objects. Analyzing and processing point cloud surfaces is important in computer graphics and computer vision. However, most of the existing algorithms for surface analysis require connectivity information. Therefore, it is desirable to develop a mesh structure on point clouds. This task can be simplified with the aid of a parameterization. In particular, conformal parameterizations are advantageous in preserving the geometric information of the point cloud data. In this paper, we extend a state-of-the-art spherical conformal parameterization algorithm for genus-0 closed meshes to the case of point clouds, using an improved approximation of the Laplace--Beltrami operator on data points. Then, we propose an iterative scheme called the north-south reiteration for achieving a spherical conformal parameterization. A balancing scheme is introduced to enhance the distribution of the spherical parameterization. High-quality triangulations and quadrangulations can then be built on the point clouds with the aid of the parameterizations. Also, the meshes generated are guaranteed to be genus-0 closed meshes. Moreover, using our proposed spherical conformal parameterization, multilevel representations of point clouds can be easily constructed. Experimental results demonstrate the effectiveness of our proposed framework.

In recent decades, the use of three-dimensional point clouds has been widespread in the computer industry. The development of techniques for 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 Teichmüller extremal mappings, which guarantees uniform conformality distortions on point cloud surfaces. Based on the discrete analogue, we propose a novel method called TEMPO for computing Teichmüller extremal mappings between feature-endowed point clouds. Using our proposed method, the Teichmüller 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.

Surface registration between cortical surfaces is crucial in medical imaging for performing systematic comparisons between brains. Landmark-matching registration that matches anatomical features, called the sulcal landmarks, is often required to obtain a meaningful 1-1 correspondence between brain surfaces. This is commonly done by parameterizing the surface onto a simple parameter domain, such as the unit sphere, in which the sulcal landmarks are consistently aligned. Landmark-matching surface registration can then be obtained from the landmark aligned parameterizations. For genus-0 closed brain surfaces, the optimized spherical harmonic parameterization, which aligns landmarks to consistent locations on the sphere, has been widely used. This approach is limited by the loss of bijectivity under large deformations and the slow computation. In this paper, we propose FLASH, a fast algorithm to compute the optimized spherical harmonic parameterization with consistent landmark alignment. This is achieved by formulating the optimization problem to $\overline{\mathbb{C}}$ and thereby linearizing the problem. Errors introduced near the pole are corrected using quasi-conformal theories. Also, by adjusting the Beltrami differential of the mapping, a diffeomorphic (1-1, onto) spherical parameterization can be effectively obtained. The proposed algorithm has been tested on 38 human brain surfaces. Experimental results demonstrate that the computation of the landmark aligned spherical harmonic parameterization is significantly accelerated using the proposed algorithm.