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A numerical method for computing the extremal Teichmüller map between multiply-connected domains is presented. Given two multiply-connected domains, there exists a unique Teichmüller map (T-Map) between them minimizing the conformality distortion. The extremal T-Map can be considered as the ‘most conformal’ map between multiply-connected domains. In this paper, we propose an iterative algorithm to compute the extremal T-Map using the Beltrami holomorphic flow (BHF). The BHF procedure iteratively adjusts the initial map based on a sequence of Beltrami coefficients, which are complex-valued functions defined on the source domain. It produces a sequence of quasi-conformal maps, which converges to the T-Map minimizing the conformality distortion. We test our method on synthetic data together with real human face data. Results show that our algorithm computes the extremal T-Map between two multiply-connected domains of the same topology accurately and efficiently.

Surface parameterizations and registrations are important in computer graphics and imaging, where 1-1 correspondences between meshes are computed. In practice, surface maps are usually represented and stored as three-dimensional coordinates each vertex is mapped to, which often requires lots of memory. This causes inconvenience in data transmission and data storage. To tackle this problem, we propose an effective algorithm for compressing surface homeomorphisms using Fourier approximation of the Beltrami representation. The Beltrami representation is a complex-valued function defined on triangular faces of the surface mesh with supreme norm strictly less than 1. Under suitable normalization, there is a 1-1 correspondence between the set of surface homeomorphisms and the set of Beltrami representations. Hence, every bijective surface map is associated with a unique Beltrami representation. Conversely, given a Beltrami representation, the corresponding bijective surface map can be exactly reconstructed using the linear Beltrami solver introduced in this paper. Using the Beltrami representation, the surface homeomorphism can be easily compressed by Fourier approximation, without distorting the bijectivity of the map. The storage requirement can be effectively reduced, which is useful for many practical problems in computer graphics and imaging. In this paper, we propose applying the algorithm to texture map compression and video compression. With our proposed algorithm, the storage requirement for the texture properties of a textured surface can be significantly reduced. Our algorithm can further be applied to compressing motion vector fields for video compression, which effectively improves the compression ratio.

Surface conformal maps between genus-0 surfaces play important roles in applied mathematics and engineering, with applications in medical image analysis and computer graphics. Previous work (Gu and Yau in Commun Inf Syst 2(2):121–146, 2002) introduces a variational approach, where global conformal parameterization of genus-0 surfaces was addressed through minimizing the harmonic energy, with two weaknesses: its gradient descent iteration is slow, and its solutions contain undesired parameterization foldings when the underlying surface has long sharp features. In this paper, we propose an algorithm that significantly accelerates the harmonic energy minimization and a method that iteratively removes foldings by taking advantages of the weighted Laplace–Beltrami eigen-projection. Experimental results show that the proposed approaches compute genus-0 surface harmonic maps much faster than the existing algorithm in Gu and Yau (Commun Inf Syst 2(2):121–146, 2002) and the new results contain no foldings.

Shape analysis is a central problem in the field of computer vision. In 2D shape analysis, classification and recognition of objects from their observed silhouettes are extremely crucial but difficult. It usually involves an efficient representation of 2D shape space with a metric, so that its mathematical structure can be used for further analysis. Although the study of 2D simply-connected shapes has been subject to a corpus of literatures, the analysis of multiply-connected shapes is comparatively less studied. In this work, we propose a representation for general 2D multiply-connected domains with arbitrary topologies using conformal welding. A metric can be defined on the proposed representation space, which gives a metric to measure dissimilarities between objects. The main idea is to map the exterior and interior of the domain conformally to unit disks and circle domains (unit disk with several inner disks removed), using holomorphic 1-forms. A set of diffeomorphisms of the unit circle S1 can be obtained, which together with the conformal modules are used to define the shape signature. A shape distance between shape signatures can be defined to measure dissimilarities between shapes. We prove theoretically that the proposed shape signature uniquely determines the multiply-connected objects under suitable normalization. We also introduce a reconstruction algorithm to obtain shapes from their signatures. This completes our framework and allows us to move back and forth between shapes and signatures. With that, a morphing algorithm between shapes can be developed through the interpolation of the Beltrami coefficients associated with the signatures. Experiments have been carried out on shapes extracted from real images. Results demonstrate the efficacy of our proposed algorithm as a stable shape representation scheme.

We address the problem of surface inpainting, which aims to fill in holes or missing regions on a Riemann surface based on its surface geometry. In practical situation, surfaces obtained from range scanners often have holes where the 3D models are incomplete. In order to analyze the 3D shapes effectively, restoring the incomplete shape by filling in the surface holes is necessary. In this paper, we propose a novel conformal approach to inpaint surface holes on a Riemann surface based on its surface geometry. The basic idea is to represent the Riemann surface using its conformal factor and mean curvature. According to Riemann surface theory, a Riemann surface can be uniquely determined by its conformal factor and mean curvature up to a rigid motion. Given a Riemann surface S, its mean curvature H and conformal factor λ can be computed easily through its conformal parameterization. Conversely, given λ and H, a Riemann surface can be uniquely reconstructed by solving the Gauss-Codazzi equation on the conformal parameter domain. Hence, the conformal factor and the mean curvature are two geometric quantities fully describing the surface. With this λ-H representation of the surface, the problem of surface inpainting can be reduced to the problem of image inpainting of λ and H on the conformal parameter domain. Once λ and H are inpainted, a Riemann surface can be reconstructed which effectively restores the 3D surface with missing holes. Since the inpainting model is based on the geometric quantities λ and H, the restored surface follows the surface geometric pattern. We test the proposed algorithm on synthetic data as well as real surface data. Experimental results show that our proposed method is an effective surface inpainting algorithm to fill in surface holes on an incomplete 3D models based their surface geometry.

Adolescent Idiopathic Scoliosis (AIS) characterized by the 3D spine deformity affects about 4% schoolchildren worldwide. Several studies have demonstrated the malfunctioning of postural balance, proprioception, and equilibrium control in patients with AIS. Since these functions are closely related to structures in and around the brainstem, the morphometry of the brainstem surface is of utmost importance. In this paper, we propose an effective method to accurately compute the registration between brainstem surfaces. Four consistent features, which describe the global geometry of the brainstem, are automatically extracted to guide the surface registration. Using the discrete Ricci flow method, brainstem surfaces are parameterized conformally onto the quadrilaterally-faced hexahedron, through which the brainstem registration can be obtained. Our registration algorithm can guarantee the exact landmark correspondence between brainstem surfaces. With the obtained registration, a shape energy can be defined to measure the local shape difference between different brainstem surfaces. We have tested our algorithms on 30 real brainstem surfaces extracted from MRIs of 15 normal subjects and 15 AIS patients. Experimental results show the efficacy of the proposed algorithm to register brainstem surfaces, which matches landmark features consistently. The computed registration can be used for the morphometry of brainstems.

This paper proposes a novel approach for extracting two intrinsic feature curves on hippocampal (HC) surfaces. The hippocampus is a key target of study in medical imaging, as it degenerates in conditions such as epilepsy and Alzheimer's disease (AD), but its structure is complex. To facilitate HC morphometry, we generate two intrinsic feature curves that describe their global geometries. For example, the separation of them captures thickness changes in HC surfaces, which can be used to effectively measure HC atrophy found in patients with AD. They also separate HC surfaces into upper and lower surface patches where intrinsic shape analysis using conformal modules can be carried out. Based on these curves, we further propose a parameterization of HC surfaces called the eigen-harmonic parameterization (EHP). EHP maps each HC surface onto a parameter domain and imposes longitudinal and azimuthal coordinates on each surface, which follow the gradient and level sets of its first nontrivial Laplace-Beltrami eigenfunction, respectively. Each tubular domain is constructed according to the geometry of an individual HC surface. This gives a parameter domain with much less geometric distortion compared to spherical parameterization. With EHP, all HC surfaces are automatically registered with intrinsic feature curves preserved and geometric distortions minimized. This allows shape analysis on any number of HC surfaces to be performed consistently. We studied geometric changes over time in 138 HC surfaces of patients with AD and normal subjects scanned at two different times. We successfully located areas with significantly different shape changes over time between the two groups.

Surface mapping plays an important role in geometric processing. They induce both area and angular distortions. If the angular distortion is bounded, the mapping is called a quasi-conformal map. Many surface maps in our physical world are quasi-conformal. The angular distortion of a quasi-conformal map can be represented by Beltrami differentials. According to quasi-conformal Teichmüller theory, there is an 1-1 correspondence between the set of Beltrami differentials and the set of quasi-conformal surface maps. Therefore, every quasi-conformal surface map can be fully determined by the Beltrami differential and can be reconstructed by solving the so-called Beltrami equation. In this work, we propose an effective method to solve the Beltrami equation on general Riemann surfaces. The solution is a quasi-conformal map associated with the prescribed Beltrami differential. We firstly formulate a discrete analog of quasi-conformal maps on triangular meshes. Then, we propose an algorithm to compute discrete quasi-conformal maps. The main strategy is to define a discrete auxiliary metric of the source surface, such that the original quasi-conformal map becomes conformal under the newly defined discrete metric. The associated map can then be obtained by using the discrete Yamabe flow method. Numerically, the discrete quasi-conformal map converges to the continuous real solution as the mesh size approaches to 0. We tested our algorithm on surfaces scanned from real life with different topologies. Experimental results demonstrate the generality and accuracy of our auxiliary metric method.

In shape analysis, finding an optimal 1-1 correspondence between 3D surfaces within a large class of admissible bijective mappings is of great importance. Such a process is called surface registration. The difficulty lies in the fact that the space of all surface diffeomorphisms is a complicated functional space, making it challenging to exhaustively search for the best mapping. To tackle this problem, we propose a simple representation of bijective surface maps using Beltrami coefficients (BCs)—complex-valued functions defined on surfaces with supremum norm less than 1. Fixing any 3 points on a pair of surfaces, there is a 1-1 correspondence between the set of surface diffeomorphisms between them and the set of BCs. Hence, every bijective surface map may be represented by a unique BC. Conversely, given a BC, we can reconstruct the unique surface map associated with it using the Beltrami Holomorphic flow (BHF) method. Using BCs to represent surface maps is advantageous because it is a much simpler functional space, which captures many essential features of a surface map. By adjusting BCs, we equivalently adjust surface diffeomorphisms to obtain the optimal map with desired properties. More specifically, BHF gives us the variation of the associated map under the variation of BC. Using this, a variational problem over the space of surface diffeomorphisms can be easily reformulated into a variational problem over the space of BCs. This makes the minimization procedure much easier. More importantly, the diffeomorphic property is always preserved. We test our method on synthetic examples and real medical applications. Experimental results demonstrate the effectiveness of our proposed algorithm for surface registration.

In this paper, we propose novel algorithms for inpainting and refinement of diffeomorphisms. We first represent a diffeomorphism by its Beltrami coefficient. Then it is possible to refine and inpaint the diffeomorphism by processing this Beltrami coefficient. With the inpainted/refined Beltrami coefficient, we construct a new diffeomorphism using the exact Beltrami holomorphic flow algorithm proposed in this paper. We apply our algorithms on several practical applications, which include the inpainting of a highly distorted diffeomorphism, the inpainting of image sequences of deforming shapes, the super-resolution of diffeomorphisms and the global parameterization of cortical surfaces by combining local parameterizations. Experiments show that our algorithm can solve these problems with natural and smooth results. We demonstrate how our proposed method can be widely applied in areas from texture mapping to video processing, and from computer graphics to medical imaging.