Publications

Submitted
Density-Equalizing Maps for Simply-Connected Open Surfaces
G. P. T. Choi and C. H. Rycroft, “Density-Equalizing Maps for Simply-Connected Open Surfaces,” Submitted. PreprintAbstract

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.

TRIM: Triangulating Images for Efficient Registration
C. P. Yung, G. P. - T. Choi, K. Chen, and L. M. Lui, “TRIM: Triangulating Images for Efficient Registration,” Submitted. PreprintAbstract

With the advancement in the digital camera technology, the use of high resolution images and videos has been widespread in the modern society. In particular, image and video frame registration is frequently applied in computer graphics and film production. However, the conventional registration approaches usually require long computational time for high quality images and video frames. This hinders the applications of the registration approaches in the modern industries. In this work, we propose a novel approach called {\em TRIM} to accelerate the computations of the registration by triangulating the images. More specifically, given a high resolution image or video frame, we compute an optimal coarse triangulation which captures the important features of the image. Then, the computation of the registration can be simplified with the aid of the coarse triangulation. Experimental results suggest that the computational time of the registration is significantly reduced using our triangulation-based approach, meanwhile the accuracy of the registration is well retained when compared with the conventional grid-based approach.

Forthcoming
Fast Spherical Quasiconformal Parameterization of Genus-0 Closed Surfaces with Application to Adaptive Remeshing
G. P. - T. Choi, M. H. - Y. Man, and L. M. Lui, “Fast Spherical Quasiconformal Parameterization of Genus-0 Closed Surfaces with Application to Adaptive Remeshing,” Geometry, Imaging and Computing, Forthcoming. PreprintAbstract

In this work, we are concerned with the spherical quasiconformal parameterization of genus-0 closed surfaces. Given a genus-0 closed triangulated surface and an arbitrary user-defined quasiconformal distortion, we propose a fast algorithm for computing a spherical parameterization of the surface that satisfies the prescribed distortion. The proposed algorithm can be effectively applied to adaptive surface remeshing for improving the visualization in computer graphics and animations. Experimental results are presented to illustrate the effectiveness of our algorithm.

2017
Conformal Mapping of Carotid Vessel Wall and Plaque Thickness Measured from Three-Dimensional Ultrasound Images
G. P. T. Choi, Y. Chen, L. M. Lui, and B. Chiu, “Conformal Mapping of Carotid Vessel Wall and Plaque Thickness Measured from Three-Dimensional Ultrasound Images,” Medical & Biological Engineering & Computing, 2017. Publisher's VersionAbstract
Measurements of vessel-wall-plus-plaque thickness (VWT) from 3D carotid ultrasound have been shown to be sensitive to the effect of pharmaceutical interventions. Since the geometry of carotid arteries is highly subject-specific, quantitative comparison of the distributions of point-wise VWT measured for different patients or for the same patients at different ultrasound scanning sessions requires the development of a mapping strategy to adjust for the geometric variability of different carotid surface models. In this paper, we present an algorithm mapping each 3D carotid surface to a 2D carotid template with an emphasis on preserving the local geometry of the carotid surface by minimizing local angular distortion. The previously described arc-length scaling (AL) approach was applied to generate an initial 2D VWT map. Using results established in the quasi-conformal theory, a new map was computed to compensate for the angular distortion incurred in AL mapping. As the 2D carotid template lies on an L-shaped non-convex domain, one-to-one correspondence of the mapping operation was not guaranteed. To address this issue, an iterative Beltrami differential chopping and smoothing procedure was developed to enforce bijectivity. Evaluations performed in the 20 carotid surface models showed that the reduction in average angular distortion made by the proposed algorithm was highly significant (P = 2.06 × 10−5). This study is the first study showing that a bijective conformal map to a non-convex domain can be obtained using the iterative Beltrami differential chopping and smoothing procedure. The improved consistency exhibited in the 2D VWT map generated by the proposed algorithm will allow for unbiased quantitative comparisons of VWT as well as local geometric and hemodynamic quantities in population studies.
A Linear Formulation for Disk Conformal Parameterization of Simply-connected Open Surfaces
G. P. - T. Choi and L. M. Lui, “A Linear Formulation for Disk Conformal Parameterization of Simply-connected Open Surfaces,” Advances in Computational Mathematics, 2017. Publisher's VersionAbstract

Surface parameterization is widely used in computer graphics and geometry processing. It simplifies challenging tasks such as surface registrations, morphing, remeshing and texture mapping. In this paper, we present an efficient algorithm for computing the disk conformal parameterization of simply-connected open surfaces. A double covering technique is used to turn a simply-connected open surface into a genus-0 closed surface, and then a fast algorithm for parameterization of genus-0 closed surfaces can be applied. The symmetry of the double covered surface preserves the efficiency of the computation. A planar parameterization can then be obtained with the aid of a M\"obius transformation and the stereographic projection. After that, a normalization step is applied to guarantee the circular boundary. Finally, we achieve a bijective disk conformal parameterization by a composition of quasi-conformal mappings. Experimental results demonstrate a significant improvement in the computational time by over 60%. At the same time, our proposed method retains comparable accuracy, bijectivity and robustness when compared with the state-of-the-art approaches. Applications for texture mapping are considered for illustrating the effectiveness of our proposed algorithm.

2016
P. T. Choi, “Surface Conformal/Quasi-conformal Parameterization with Applications,” 2016.Abstract

 

Surface conformal and quasi-conformal parameterizations are important in computer graphics and medical imaging. In this thesis, we develop efficient and practical algorithms for mesh and point cloud parameterizations with various applications. Firstly, we propose a linear algorithm for the spherical conformal parameterizations and an efficient algorithm called FLASH for landmark-aligned spherical optimized conformal mappings of genus-0 closed triangular meshes. The algorithms are applied for the registration of human cortical surfaces, and the shape analysis of the carotid arteries and the hippocampal surfaces in medical imaging. Secondly, we propose two fast disk conformal parameterization algorithms for simply-connected open triangular meshes and apply the parameterizations for texture mapping. Thirdly, we develop a linear algorithm for computing spherical quasi-conformal parameterization and Teichmüller parameterization of genus-0 closed meshes. Fourthly, we develop an iterative scheme for the spherical conformal parameterizations of genus-0 point clouds with applications in meshing and multi-level representation. Fifthly, we propose a novel algorithm called TEMPO for the landmark aligned Teichmüller parameterization of disk-type point clouds with theoretical guarantee. The algorithm is applied for developing a dissimilarity metric of point clouds. Experimental results are presented for demonstrating the effectiveness of our proposed algorithms.

 

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Spherical Conformal Parameterization of Genus-0 Point Clouds for Meshing
G. P. - T. Choi, K. T. Ho, and L. M. Lui, “Spherical Conformal Parameterization of Genus-0 Point Clouds for Meshing,” SIAM Journal on Imaging Sciences, vol. 9, no. 4, pp. 1582-1618, 2016. Publisher's VersionAbstract

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.

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TEMPO: Feature-Endowed Teichmüller Extremal Mappings of Point Clouds
T. W. Meng, G. P. - T. Choi, and L. M. Lui, “TEMPO: Feature-Endowed Teichmüller Extremal Mappings of Point Clouds,” SIAM Journal on Imaging Sciences, vol. 9, no. 4, pp. 1922-1962, 2016. Publisher's VersionAbstract

In recent decades, the use of 3D point clouds has been widespread in computer industry. The development of techniques in 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 Teichm\"{u}ller extremal mappings, which guarantee uniform conformality distortions, on point cloud surfaces. Based on the discrete analogue, we propose a novel method called TEMPO for computing Teichm\"{u}ller extremal mappings between feature-endowed point clouds. Using our proposed method, the Teichm\"{u}ller metric is introduced for evaluating the dissimilarity of point clouds. Consequently, our algorithm enables accurate recognitions and classifications of point clouds. Experimental results demonstrate the effectiveness of our proposed method.

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2015
Fast Disk Conformal Parameterization of Simply-connected Open Surfaces
P. T. Choi and L. M. Lui, “Fast Disk Conformal Parameterization of Simply-connected Open Surfaces,” Journal of Scientific Computing, vol. 65, no. 3, pp. 1065-1090, 2015. Publisher's VersionAbstract

Surface parameterizations have been widely used in computer graphics and geometry processing. In particular, as simply-connected open surfaces are conformally equivalent to the unit disk, it is desirable to compute the disk conformal parameterizations of the surfaces. In this paper, we propose a novel algorithm for the conformal parameterization of a simply-connected open surface onto the unit disk, which significantly speeds up the computation, enhances the conformality and stability, and guarantees the bijectivity. The conformality distortions at the inner region and on the boundary are corrected by two steps, with the aid of an iterative scheme using quasi-conformal theories. Experimental results demonstrate the effectiveness of our proposed method.

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FLASH: Fast Landmark Aligned Spherical Harmonic Parameterization for Genus-0 Closed Brain Surfaces
P. T. Choi, K. C. Lam, and L. M. Lui, “FLASH: Fast Landmark Aligned Spherical Harmonic Parameterization for Genus-0 Closed Brain Surfaces,” SIAM Journal on Imaging Sciences, vol. 8, no. 1, pp. 67-94, 2015. Publisher's VersionAbstract
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.
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