Projects
SUBMITTED/PREPRINT

Abstract
In this paper, we propose an adaptive fast solver for a general class of symmetric positive definite (SPD) matrices which include the wellknown graph Laplacian. We achieve this by developing an adaptive operator compression scheme and a multiresolution matrix factorization algorithm which achieve nearly optimal performance on both complexity and wellposedness. To develop our adaptive operator compression and multiresolution matrix factorization methods, we first introduce a novel notion of energy decomposition for SPD matrix \(A\) using the representation of energy elements. The interaction between these energy elements depicts the underlying topological structure of the operator. This concept of decomposition naturally reflects the hidden geometric structure of the operator which inherits the localities of the structure. By utilizing the intrinsic geometric information under this Energy framework, we propose a systematic operator compression scheme for the inverse operator \(A^{1}\). In particular, with an appropriate partition of the underlying geometric structure, we can construct localized basis by using the concept of interior and closed energy. Meanwhile, two important localized quantities are introduced, namely the error factor and the condition factor. Our error analysis results show that these two factors will be the guidelines for finding the appropriate partition of the basis functions such that prescribed compression error and acceptable condition number can be achieved. By virtue of this insight, we propose the Patch Pairing algorithm to realize our energy partition framework for operator compression with controllable compression error and condition number.
BibTeX Citation Copy
@article{hou2017fast,
title={An adaptive fast solver for a general class of positive definite matrices via energy decomposition},
author={Hou, Thomas Y and Huang, D and Lam, KC and Zhang, P},
journal={arXiv preprint arXiv:1707.08277v2},
year={2017}
}
PUBLICATIONS

Abstract
Analyzing the deformation pattern of an object is crucial in various fields, such as in computer visions and medical imaging. A deformation can be considered as a combination of local and global deformations at different locations. To fully understand and analyze the deformation pattern, extracting deformation components of various scales and locations is necessary. We propose an algorithm for the multiscale decomposition of a bijective deformation using quasiconformal theories. A deformation of an object can be described as a orientationpreserving homeomorphism of a two dimensional domain. The mapping is then represented by its associated Beltrami coefficient (BC), which measures the local geometric (conformality) distortion of the deformation. The BC is a complexvalued function defined on the source domain. By applying the wavelet transform on the BC, the BC can be decomposed into different components of different frequencies compactly supported in different subdomains. Quasiconformal mappings associated to different components of the BC can be reconstructed by solving the Beltrami's equation. A multiscale decomposition of the deformation can then be constructed. To validate our proposed algorithm, we test it on synthetic examples as well as real medical data. Experimental results show the efficacy of our proposed model to decompose deformations at multiple scales and locations.
BibTeX Citation Copy
@article{lam2017multiscale,
title={Multiscale Representation of Deformation via Beltrami Coefficients},
author={Lam, Ka Chun and Ng, Tsz Ching and Lui, Lok Ming},
journal={Multiscale Modeling \& Simulation},
volume={15},
number={2},
pages={864891},
year={2017},
publisher={SIAM}
} 
Abstract
Parameterization, a process of mapping a complicated domain onto a simple canonical domain, is crucial in different areas such as computer graphics, medical imaging and scientific computing. Conformal parameterization has been widely used since it preserves the local geometry well. However, a major drawback is the area distortion introduced by the conformal parameterization, causing inconvenience in many applications such as texture mapping in computer graphics or visualization in medical imaging. This work proposes a remedy to construct a parameterization that balances between conformality and area distortions. We present a variational algorithm to compute the optimized quasiconformal parameterization with controllable area distortions. The distribution of the area distortion can be prescribed by users according to the application. The main strategy is to minimize a combined energy functional consisting of an area mismatching term and a regularization term involving the Beltrami coefficient of the map. The Beltrami coefficient controls the conformality of the parameterization. Landmark constraints can be incorporated into the model to obtain landmarkaligned parameterization. Experiments have been carried out on both synthetic and real data. Results demonstrate the efficacy of the proposed algorithm to compute the optimized parameterization with controllable area distortion while preserving the local geometry as good as possible.
BibTeX Citation Copy
@article{lam2017optimized,
title={Optimized conformal parameterization with userdefined area distortions},
author={Lam, Ka Chun and Lui, Lok Ming},
journal={Communications in Mathematical Sciences},
volume={15},
number={7},
pages={20272054},
year={2017},
publisher={International Press}
} 
Abstract
We propose a new method to obtain landmarkmatching transformations between \(n\)dimensional Euclidean spaces with large deformations. Given a set of feature correspondences, our algorithm searches for an optimal foldingfree mapping that satisfies the prescribed landmark constraints. The standard conformality distortion defined for mappings between 2dimensional spaces is first generalized to the \(n\)dimensional conformality distortion \(K(f)\) for a mapping \(f\) between \(n\)dimensional Euclidean spaces \((n≥3)\) . We then propose a variational model involving \(K(f)\) to tackle the landmarkmatching problem in higher dimensional spaces. The generalized conformality term \(K(f)\) enforces the bijectivity of the optimized mapping and minimizes its local geometric distortions even with large deformations. Another challenge is the high computational cost of the proposed model. To tackle this, we have also proposed a numerical method to solve the optimization problem more efficiently. Alternating direction method with multiplier is applied to split the optimization problem into two subproblems. Preconditioned conjugate gradient method with multigrid preconditioner is applied to solve one of the subproblems, while a fixedpoint iteration is proposed to solve another subproblem. Experiments have been carried out on both synthetic examples and lung CT images to compute the diffeomorphic landmarkmatching transformation with different landmark constraints. Results show the efficacy of our proposed model to obtain a foldingfree landmarkmatching transformation between ndimensional spaces with large deformations.
BibTeX Citation Copy
@article{lee2016landmark,
title={Landmarkmatching Transformation with Large Deformation via ndimensional Quasiconformal Maps},
author={Lee, Yin Tat and Lam, Ka Chun and Lui, Lok Ming},
journal={Journal of Scientific Computing},
volume={67},
number={3},
pages={926954},
year={2016},
publisher={Springer}
} 
Abstract
Fusion of images with same or different modalities has been conquering medical imaging field more rapidly due to the presence of highly accessible patients’ information in recent years. For example, cross platform nonrigid registration of CT with MRI images has found a significant role in different clinical application. In some instances labelling of anatomical features by medical experts are also involved to further improve the accuracy and authenticity of the registration. Being motivated by these, we propose a new algorithm to compute diffeomorphic hybrid multimodality registration with large deformations. Our iterative scheme consists of mainly two steps. First, we obtain the optimal Beltrami coefficient corresponding to the diffeomorphic mapping that exactly superimposes the feature points. The second step detects the intensity difference in the framework of mutual information. A nonrigid deformation which minimizes the intensity difference is then obtained. Experiments have been carried out on both synthetic and real data. Results demonstrate the stability and efficacy of the proposed algorithm to obtain diffeomorphic image registration.
BibTeX Citation Copy
@inproceedings{lam2015quasi,
title={QuasiConformal Hybrid Multimodality Image Registration and its Application to Medical Image Fusion},
author={Lam, Ka Chun and Lui, Lok Ming}, booktitle={International Symposium on Visual Computing},
pages={809818},
year={2015},
organization={Springer}
} 
Abstract
We address the registration problem of genusone surfaces (such as vertebrae bones) with prescribed landmark constraints. The highgenus topology of the surfaces makes it challenging to obtain a unique and bijective surface mapping that matches landmarks consistently. This work proposes to tackle this registration problem using a special class of quasiconformal maps called Teichmüller maps (TMaps). A landmark constrained TMap is the unique mapping between genus1 surfaces that minimizes the maximal conformality distortion while matching the prescribed feature landmarks. Existence and uniqueness of the landmark constrained TMap are theoretically guaranteed. This work presents an iterative algorithm to compute the TMap. The main idea is to represent the set of diffeomorphism using the Beltrami coefficients (BC). The BC is iteratively adjusted to an optimal one, which corresponds to our desired TMap that matches the prescribed landmarks and satisfies the periodic boundary condition on the universal covering space. Numerical experiments demonstrate the effectiveness of our proposed algorithm. The method has also been applied to register vertebrae bones with prescribed landmark points and curves, which gives accurate surface registrations.
BibTeX Citation Copy
@article{lam2015landmark,
title={Landmark constrained genusone surface Teichmüller map applied to surface registration in medical imaging},
author={Lam, Ka Chun and Gu, Xianfeng and Lui, Lok Ming},
journal={Medical image analysis},
volume={25},
number={1},
pages={4555},
year={2015},
publisher={Elsevier}
} 
Abstract
This paper presents a novel algorithm to obtain landmarkbased genus1 surface registration via a special class of quasiconformal maps called the Teichmüller maps. Registering shapes with important features is an important process in medical imaging. However, it is challenging to obtain a unique and bijective genus1 surface matching that satisfies the prescribed landmark constraints. In addition, as suggested by [11], conformal transformation provides the most natural way to describe the deformation or growth of anatomical structures. This motivates us to look for the unique mapping between genus1 surfaces that matches the features while minimizing the maximal conformality distortion. Existence and uniqueness of such optimal diffeomorphism is theoretically guaranteed and is called the Teichmüller extremal mapping. In this work, we propose an iterative algorithm, called the Quasiconformal (QC) iteration, to find the Teichmüller extremal mapping between the covering spaces of genus1 surfaces. By representing the set of diffeomorphisms using Beltrami coefficients (BCs), we look for an optimal BC which corresponds to our desired diffeomorphism that matches prescribed features and satisfies the periodic boundary condition on the covering space. Numerical experiments show that our proposed algorithm is efficient and stable for registering genus1 surfaces even with large amount of landmarks. We have also applied the algorithm on registering vertebral bones with prescribed feature curves, which demonstrates the usefulness of the proposed algorithm.
BibTeX Citation Copy
@inproceedings{lam2014genus,
title={Genusone surface registration via teichm{\"u}ller extremal mapping},
author={Lam, Ka Chun and Gu, Xianfeng and Lui, Lok Ming},
booktitle={International Conference on Medical Image Computing and ComputerAssisted Intervention}, pages={2532},
year={2014},
organization={Springer}
} 
Abstract
Registration, which aims to find an optimal 11 correspondence between shapes, is an important process in different research areas. Landmarkbased surface registration has been widely studied to obtain a mapping between shapes that matches important features. Obtaining a unique and bijective surface registration that matches features consistently is generally challenging, especially when a large number of landmark constraints are enforced. This motivates us to search for a unique landmark matching surface diffeomorphism, which minimizes the local geometric distortion. For this purpose, we propose a special class of diffeomorphisms called the Teichmüller mappings (TMaps). Under suitable conditions on the landmark constraints, a unique TMap between two surfaces can be obtained, which minimizes the maximal conformality distortion. The conformality distortion measures how far the mapping deviates from a conformal mapping, and hence it measures the local geometric distortion. In this paper, we propose an efficient iterative algorithm, called the quasiconformal (QC) iteration, to compute the TMap. The basic idea is to represent the set of diffeomorphisms using Beltrami coefficients (BCs) and look for an optimal BC associated to the desired TMap. The associated diffeomorphism can be efficiently reconstructed from the optimal BC using the linear Beltrami solver (LBS). Using BCs to represent diffeomorphisms guarantees the diffeomorphic property of the registration, even with very large deformation. Using our proposed method, the TMap can be accurately and efficiently computed. The obtained registration is guaranteed to be bijective. The proposed algorithm can also be extended to compute TMap with soft landmark constraints. We applied the proposed algorithm to real applications, such as brain landmark matching registration, constrained texture mapping, and human face registration. Experimental results shows that our method is both effective and efficient in computing a nonoverlap landmark matching registration with the least amount of conformality distortion.
BibTeX Citation Copy
@article{lui2014teichmuller,
title={Teichmuller mapping (tmap) and its applications to landmark matching registration},
author={Lui, Lok Ming and Lam, Ka Chun and Yau, ShingTung and Gu, Xianfeng},
journal={SIAM Journal on Imaging Sciences},
volume={7},
number={1},
pages={391426},
year={2014},
publisher={SIAM}
} 
Abstract
This paper presents two algorithms, based on conformal geometry, for the multiscale representations of geometric shapes and surface morphing. A multiscale surface representation aims to describe a 3D shape at different levels of geometric detail, which allows analyzing or editing surfaces at the global or local scales effectively. Surface morphing refers to the process of interpolating between two geometric shapes, which has been widely applied to estimate or analyze deformations in computer graphics, computer vision and medical imaging. In this work, we propose two geometric models for surface morphing and multiscale representation for 3D surfaces. The basic idea is to represent a 3D surface by its mean curvature function, \(H\), and conformal factor function \(\lambda\), which uniquely determine the geometry of the surface according to Riemann surface theory. Once we have the \((\lambda,H)\) parameterization of the surface, postprocessing of the surface can be done directly on the conformal parameter domain. In particular, the problem of multiscale representations of shapes can be reduced to the signal filtering on the \(\lambda\) and \(H\) parameters. On the other hand, the surface morphing problem can be transformed to an interpolation process of two sets of \((\lambda,H)\) parameters. We test the proposed algorithms on 3D human face data and MRIderived brain surfaces. Experimental results show that our proposed methods can effectively obtain multiscale surface representations and give natural surface morphing results.
BibTeX Citation Copy
@article{lam2014conformal,
title={Conformalbased surface morphing and multiscale representation},
author={Lam, Ka Chun and Wen, Chengfeng and Lui, Lok Ming},
journal={Axioms},
volume={3},
number={2},
pages={222243},
year={2014},
publisher={Multidisciplinary Digital Publishing Institute}
} 
Abstract
Surface registration between cortical surfaces is crucial in medical imaging for performing systematic comparisons between brains. Landmarkmatching registration that matches anatomical features, called the sulcal landmarks, is often required to obtain a meaningful 11 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. Landmarkmatching surface registration can then be obtained from the landmark aligned parameterizations. For genus0 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 quasiconformal theories. Also, by adjusting the Beltrami differential of the mapping, a diffeomorphic (11, 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.
BibTeX Citation Copy
@article{choi2015flash,
title={FLASH: Fast landmark aligned spherical harmonic parameterization for genus0 closed brain surfaces},
author={Choi, Pui Tung and Lam, Ka Chun and Lui, Lok Ming},
journal={SIAM Journal on Imaging Sciences},
volume={8},
number={1},
pages={6794},
year={2015},
publisher={SIAM}
} 
Abstract
Registration, which aims to find an optimal onetoone correspondence between different data, is an important problem in various fields. This problem is especially challenging when large deformations occur. In this paper, we present a novel algorithm to obtain diffeomorphic image or surface registrations with large deformations via quasiconformal maps. The basic idea is to minimize an energy functional involving a Beltrami coefficient term, which measures the distortion of the quasiconformal map. The Beltrami coefficient effectively controls the bijectivity and smoothness of the registration, even with very large deformations. Using the proposed algorithm, landmarkbased registration between images or surfaces can be effectively computed. The obtained registration is guaranteed to be diffeomorphic (11 and onto), even with a large deformation or large number of landmark constraints. The proposed algorithm can also be combined with matching intensity (such as image intensity or surface curvature) to improve the accuracy of the registration. Experiments have been carried out on both synthetic and real data. Results demonstrate the efficacy of the proposed algorithm to obtain diffeomorphic registration between images or surfaces.
BibTeX Citation Copy
@article{lam2014landmark,
title={Landmarkand intensitybased registration with large deformations via quasiconformal maps},
author={Lam, Ka Chun and Lui, Lok Ming},
journal={SIAM Journal on Imaging Sciences},
volume={7},
number={4},
pages={23642392},
year={2014},
publisher={SIAM}
} 
Abstract
Surface parameterizations and registrations are important in computer graphics and imaging, where 11 correspondences between meshes are computed. In practice, surface maps are usually represented and stored as threedimensional 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 complexvalued function defined on triangular faces of the surface mesh with supreme norm strictly less than 1. Under suitable normalization, there is a 11 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.
BibTeX Citation Copy
@article{lui2013texture,
title={Texture map and video compression using Beltrami representation},
author={Lui, Lok Ming and Lam, Ka Chun and Wong, Tsz Wai and Gu, Xianfeng},
journal={SIAM Journal on Imaging Sciences},
volume={6},
number={4},
pages={18801902},
year={2013},
publisher={SIAM}
}
THESIS

Abstract
Quasiconformal mapping, which was introduced by Grötzsch (1928) and named by Ahlfors (1935), is a homeomorphism between plane domains which can be viewed as a generalization of conformal mapping. Intuitively, a quasiconformal mapping takes small circles to small ellipses of bounded eccentricity. Mathematically, a function or mapping \(f: \Omega \rightarrow \Omega'\), where \(\Omega\) and \(\Omega'\) are two domains in \(\mathbb{C}\), is quasiconformal if it satisfies the Beltrami equation \(\frac{\partial f}{\partial \overline{z}} = \mu(z) \frac{\partial f}{\partial z}\) for some complexvalued Lebesgue measurable function \(\mu\) satisfying \(\Vert \mu \Vert_{\infty} < 1\). Over the last 90 years, numerous important theories of quasiconformal mappings have been discovered. Yet, computation of these mappings had not been developed or utilized in real applications. We therefore propose to explore the numerical techniques for quasiconformal mappings with the applications to medical imaging.
This thesis presents various computational methods for quasiconformal mappings. In particular, variational approaches for registering/parameterizing images/surfaces with prescribed feature correspondences are proposed. We consider the set of all quasiconformal mappings to be the search space for matching predefined features. The intrinsic idea is to represent the set of all landmark aligned orientation preserving diffeomorphisms by smooth Beltrami coefficients. Variational models are then built upon this representation and the registration/parameterization problem is transformed to an optimization problem with respect to the Beltrami coefficient.
The outline of this thesis is as follows: In the Part I, we first introduce the QCLR and QCHR algorithms for general medical image registration. Secondly, we present the QCiteration to register genusone surfaces within the space of all Teichmüller maps with applications to vertebral bones shape analysis. Thirdly, we propose a generalization of \(n\)dimensional conformality distortion for registering volumetric medical images. In Part II, an algorithm, called FLASH, for genuszero cortical surface registration is reported. Following that, we introduce an algorithm to obtained an optimized conformal parameterization with controllable area distortion. Lastly, we present a multiscale representation of diffeomorphic deformation for morphometry. The thesis concludes with a discussion on the advantages of applying quasiconformal theory to tackle registration and parameterization problems originated from medical image analysis.