My current research interests include:

- Computational Differential Geometry,
- Geometric Morphometrics,
- Medical Imaging,
- Scientific Computing, and
- Geometry Processing.

Over the past few years, I have developed several efficient conformal/quasi-conformal parameterization methods for meshes and point clouds wth applications to Computer Graphics. My current research focuses on computational differential geometry with applications to Biology, Physics and Medicine.

Finished projects:

- Fast Spherical Conformal Parameterization of Genus-0 Closed Meshes
- FLASH: Fast Landmark Aligned Spherical Harmonic Parameterization of Genus-0 Closed Surfaces
- Fast Disk Conformal Parameterization of Simply-Connected Open Surfaces
- Spherical Conformal Parameterization of Genus-0 Point Clouds for Meshing
- A Linear Formulation for Disk Conformal Parameterization of Simply-connected Open Surfaces
- TEMPO: Feature-Endowed Teichmuller Extremal Mappings of Point Clouds
- TRIM: Triangulating Images for Efficient Registration
- Fast Spherical Quasiconformal Parameterization of Genus-0 Closed Surfaces with Application to Adaptive Remeshing
- Density-Equalizing Maps for Simply-Connected Open Surfaces

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**Fast Spherical Conformal Parameterization of Genus-0 Closed Meshes**

Surface parameterization refers to the process of bijectively mapping a complicated surface to a simple canonical domain. The use of parameterizations has been widespread in computer graphics and geometry processing for tasks such as surface registration, mesh editing and medical visualization. In particular, surface conformal parameterization is usually preferred since conformal maps preserve angles and hence the local geometry. By the uniformization theorem, genus-0 closed surfaces are conformally equivalent to the unit sphere. Based on this theoretical guarantee, many algorithms have been proposed by different researchers for computing the spherical conformal parameterization of genus-0 closed surfaces. However, most conventional approaches are highly nonlinear and hence expensive. This hinders practical applications of the parameterization.

In this work, we develop a fast algorithm for the spherical conformal parameterizations of genus-0 closed surfaces. The main idea of the algorithm is to formulate the optimization problem on the complex plane so as to linearize the problem. For genus-0 closed surfaces, conformal maps are equivalent to harmonic maps. Hence, instead of directly solving for a conformal map, we can solve for a harmonic map \(f: M \to \mathbb{S}^2\) by minimizing the harmonic energy functional

\[ E(f) = \int_M \| \nabla f\|^2 dv_M \]

subject to \( \| f \| = 1\), which is nonlinear. We linearize this problem by solving the harmonic equation \(\Delta f = 0\) on the complex plane. Then, using the stereographic projection, we map the result back onto the unit sphere. The conformality distortion caused by the stereographic projection near the North Pole is then corrected using quasi-conformal theories. Specifically, we compose the map with another quasi-conformal map that is associated with a specific Beltrami coefficient, in order to cancel out the distortion. The bijectivity of the parameterization is also guaranteed by quasi-conformal theories.

Experimental results show that the computation of the spherical conformal parameterization is significantly accelerated by over 5000 times using our proposed algorithm. For meshes of moderate sizes (50K vertices), the computation takes only around 1 second. Besides, our parameterization results are with lower distortion than the conventional approaches. It is also noteworthy that all of our parameterization results are bijective, i.e. folding-free. The results demonstrate the strengths of our proposed algorithm.

**Reference:**

[1] 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, Volume 8, Issue 1, 67-94, 2015.

[Software]

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**FLASH: Fast Landmark Aligned Spherical Harmonic Parameterization of Genus-0 Closed Surfaces**

In human brain mapping, neuroscientists aim to study the anomaly of human brains in order to analyze certain diseases. To achieve this, surface registration, the process of finding a meaningful 1-1 correspondence between two cortical surfaces, is crucial for performing systematic comparisons between healthy and unhealthy brains. In particular, landmark-matching registrations that match anatomical features, called the sulcal landmarks, is usually preferred to ensure the accuracy of the registration. However, the highly convoluted structure of genus-0 human cerebral cortices is an obstacle in performing the registration and viewing the functional activity on them.

One common approach to analyze and compare the brain surfaces is to map them to a simple canonical space while preserving their geometric information as complete as possible. In this case, it is desirable to map the brain surfaces to the unit sphere with their sulcal landmarks 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. More specifically, conventional approaches aim to minimize the following combined energy

\[E(f) = \int \| \nabla f\|^2 + \lambda \int \|f(p_i)-q_i\|^2\]

where \(p_i, q_i\) are the sulcal landmark curves and \(\lambda\) is a weighting factor balancing the two terms. The conventional approaches were limited by the loss of bijectivity under large deformations and the slow computation, since the problem is nonlinear.

In this work, we propose *FLASH*, a fast algorithm to compute the optimized spherical harmonic parameterization with consistent landmark alignments. Given two brain surfaces, we first compute their spherical conformal parameterizations using our proposed algorithm mentioned above. We then solve for a landmark constrained harmonic map by formulating the problem to the complex plane, using the stereographic projection. This linearizes the optimization problem and hence the computation is accelerated. Also, we develop an iterative scheme to ensure the bijectivity of the mapping by adjusting the Beltrami differential. The proposed algorithm has been tested on 38 human brain surfaces of older adult subjects, where half of them are with dementia of the Alzheimer's type. The following figure shows the landmark aligned spherical harmonic parameterizations of several pairs of brain surfaces using our proposed FLASH algorithm. Experimental results demonstrate that the computation of the landmark aligned spherical harmonic parameterization and the registration is significantly accelerated by 100 times using the proposed algorithm. For meshes of moderate sizes (around 50K vertices), the computation usually completes within 20 seconds.

**Reference:**

[1] 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, Volume 8, Issue 1, 67-94, 2015.

[Software]

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**Fast Disk Conformal Parameterization of Simply-Connected Open Surfaces**

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 work, we propose a novel algorithm for the conformal parameterization of a simply-connected open surface onto the unit disk \(D\). First, we apply the conventional disk harmonic map as an initial map by solving the following Laplace equation:

\[ \Delta_M f(u) = 0 \]

with the boundary constraint \(f_{\partial{M}} = g\), where \(g: \partial M \to \partial D\) is given by the arc-length parameterization. This initial step is linear but the conformality distortion is quite large. 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. More specifically, to improve the conformality of the inner region, we apply the Cayley transform

\[ W(z) = i \frac{1+z}{1-z} \]

to map the disk to the upper half plane. Fixing the outermost part on the upper half plane, we compose the map with another quasi-conformal map and thus eliminates the distortion of the inner region. Then in the second step, we aim to correct the distortion near the boundary of the disk. To achieve this, we extend the map from the unit disk D to the extended complex plane by a reflection:

\[ z \in D - \partial D \to \frac{1}{\bar{z}} \in \bar{\mathbb{C}} - D. \]

After the extension, the boundary region of the unit disk becomes an inner region of a larger domain. Hence, we can fix the outermost part of the new domain and compose the extended map with a quasi-conformal map so as to eliminate the conformality distortion. In the discrete case, under the composition, the boundary of the original unit disk may be a bit different from a perfect circle. Therefore, we apply a projection to normalize the boundary and repeat the reflection step again until convergence.

Our proposed algorithm significantly accelerates the computation of disk conformal parameterizations. Experimental results show that our proposed method is faster than the conventional approaches by over 60%. Our proposed algorithm also enhances the conformality and stability, and guarantees the bijectivity of the parameterizations.

**Reference:**

[2] P. T. Choi and L. M. Lui, Fast Disk Conformal Parameterization of Simply-connected Open Surfaces. Journal of Scientific Computing, Volume 65, Issue 3, 1065-1090, 2015.

[Software]

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**Spherical Conformal Parameterization of Genus-0 Point Clouds for Meshing**

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.

**Reference:**

[3] 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, Volume 9, Issue 4, pp. 1582-1618, 2016.

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**A Linear Formulation for Disk Conformal Parameterization of Simply-Connected Open Surfaces**

In the previous work, we proposed a fast two-step iterative scheme for the disk conformal parameterization of simply-connected open surfaces. In this work, we further enhance the computational efficiency of disk conformal parameterizations. We propose to apply a double covering technique to turn a simply-connected open surface into a genus-0 closed surface, and then use a fast spherical parameterization algorithm to obtain a parameter sphere. 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 Mobius 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. It should be noticed that the entire algorithm involves only a few sparse symmetric positive definite linear systems, which can be efficiently solved by the conjugate gradient method. Experimental results show that our new approach further reduces the computational cost by over 60%, making the parameterization method more practical in real applications. At the same time, our proposed method retains comparable accuracy, bijectivity and robustness when compared with the state-of-the-art approaches.

Because of its efficiency and accuracy, our proposed algorithm is highly applicable for texture mapping in computer graphics and fashion design. A figure below shows two examples of texture mapping using our proposed algorithm. The left one is a simply-connected open bunny surface with a rainbow checkerboard texture mapped onto it using our new parameterization algorithm. The preservation of the orthogonality of the squares illustrates that our proposed parameterization algorithm is with very low angular distortion. The right one is a T-shirt mesh with a 2D flower pattern design mapped onto it. It is noteworthy that the patterns are well-preserved on the 3D T-shirt mesh. Unlike using the traditional 2D images of clothes,use of 3D models with our texture mapping technique provides a better and more realistic preview of the clothes for the designers. In addition, our proposed algorithm can be used in developing virtual dressing rooms. In virtual dressing rooms, it is desirable to have an efficient and accurate way for shoppers virtually try on clothes. Our proposed algorithm can help creating a virtual dressing experience for online customers.

**Reference:**

[4] G. P. T. Choi and L. M. Lui, A Linear Formulation for Disk Conformal Parameterization of Simply-connected Open Surfaces. Accepted by Advances in Computational Mathematics.

[Software]

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**TEMPO: Feature-Endowed Teichmüller Extremal Mappings of Point Clouds**

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 Teichmüller 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 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.

**Reference:**

[5] 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, Volume 9, Issue 4, pp. 1922-1962, 2016.

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**TRIM: Triangulating Images for Efficient Registration**

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 *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.

**Reference:**

[6] C. P. Yung, G. P. T. Choi, K. Chen, and L. M. Lui, TRIM: Triangulating Images for Efficient Registration. Submitted for publication. Preprint, arXiv:1605.06215.

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**Fast Spherical Quasiconformal Parameterization of Genus-0 Closed Surfaces with Application to Adaptive Remeshing**

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.

**Reference:**

[7] 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. Submitted for publication. Preprint, arXiv:1608.01140.

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**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.

**Reference:**

[8] G. P. T. Choi and C. H. Rycroft, Density-Equalizing Maps for Simply-Connected Open Surfaces. Submitted for publication. Preprint, arXiv:1704.02525.