Min ZhangStony Brook UniversityRen GuoOregon State UniveristyWei ZengSchool of Computing and Information Sciences, Florida International UniversityFeng LuoRutgers UniversityShing Tung YauHarvard UniversityXianfeng GuStony Brook Univerisity
Computational GeometryDifferential GeometryGeometric Modeling and ProcessingConvex and Discrete Geometry mathscidoc:1612.01001
Graphical Models/Geometric Modeling and Processing 2014, 76, (5), 321-339, 2014.9
Ricci ﬂow deformsthe Riemannian metric proportionallyto the curvature, such that the curvatureevolves accordingto a heat diffusion process and eventually becomes constant everywhere. Ricci ﬂow has demonstrated its great potential by solving various problems in many ﬁelds, which can be hardly handled by alternative methods so far. This work introduces the uniﬁed theoretic framework for discrete Surface Ricci Flow, including all the common schemes: Tangential Circle Packing, Thurston’s Circle Packing, Inversive Distance Circle Packing and Discrete Yamabe Flow. Furthermore, this work also introduces a novel schemes, Virtual Radius Circle Packing and the Mixed Type schemes, under the uniﬁed framework. This work gives explicit geometric interpretation to the discrete Ricci energies for all the schemes with all back ground geometries, and the corresponding Hessian matrices. The uniﬁed frame work deepens our understanding to the the discrete surface Ricci ﬂow theory, and has inspired us to discover the new schemes, improved the ﬂexibility and robustness of the algorithms, greatly simpliﬁed the implementation and improved the efﬁciency. Experimental results show the uniﬁed surface Ricci ﬂow algorithms can handle general surfaces with different topologies, and is robust to meshes with different qualities, and is effective for solving real problems.
Ronald Lok Ming LuiUniversity of California, Los AngelesSheshadri ThiruvenkadamUniversity of California, Los AngelesYalin WangUniversity of California, Los AngelesPaul M. ThompsonUniversity of California, Los AngelesTony F. ChanUniversity of California, Los Angeles
SIAM Journal of Imaging Sciences, 3, (1), 52-78, 2010
Surface registration, which transforms different sets of surface data into one common reference space, is an important process which allows us to compare or integrate the surface data effectively. If a nonrigid transformation is required, surface registration is commonly done by parameterizing the surfaces onto a simple parameter domain, such as the unit square or sphere. In this work, we are interested in looking for meaningful registrations between surfaces through parameterizations, using prior features in the form of landmark curves on the surfaces. In particular, we generate optimized conformal parameterizations which match landmark curves exactly with shape-based correspondences between them. We propose a variational method to minimize a compound energy functional that measures the harmonic energy of the parameterization maps and the shape dissimilarity between mapped points on the landmark curves. The novelty is that the computed maps are guaranteed to align the landmark features consistently and give a shape-based diffeomorphism between the landmark curves. We achieve this by intrinsically modeling our search space of maps as flows of smooth vector fields that do not flow across the landmark curves. By using the local surface geometry on the curves to define a shape measure, we compute registrations that ensure consistent correspondences between anatomical features. We test our algorithm on synthetic surface data. An application of our model to medical imaging research is shown, using experiments on brain cortical surfaces, with anatomical (sulcal) landmarks delineated, which show that our computed maps give a shape-based alignment of the sulcal curves without significantly impairing conformality. This ensures correct averaging and comparison of data across subjects.
Ronald Lok Ming LuiHarvard UniversityTsz Wai WongUniversity of California, Los AngelesWei ZengStony Brook UniversityXianfeng GuStony Brook UniversityPaul M. ThompsonUniversity of California, Los AngelesTony F. ChanThe Hong Kong University of Science and TechnologyShing-Tung YauHarvard University
Journal of Inverse Problem and Imaging, 4, (2), 311 - 333, 2010.5
We address the problem of detecting deformities on elastic surfaces. This is of great importance for shape analysis, with applications such as detecting abnormalities in biological shapes (e.g., brain structures). We propose an effective algorithm to detect abnormal deformations by generating quasi-conformal maps between the original and deformed surfaces. We firstly flatten the 3D surfaces conformally onto 2D rectangles using the discrete Yamabe flow and use them to compute a quasi-conformal map that matches internal features lying within the surfaces. The deformities on the elastic surface are formulated as non-conformal deformations, whereas normal deformations that preserve local geometry are formulated as conformal deformations. We then detect abnormalities by computing the Beltrami coefficient associated uniquely with the quasi-conformal map. The Beltrami coefficient is a complex-valued function defined on the surface. It describes the deviation of the deformation from conformality at each point. By considering the norm of the Beltrami coefficient, we can effectively segment the regions of abnormal changes, which are invariant under normal (non-rigid) deformations that preserve local geometry. Furthermore, by considering the argument of the Beltrami coefficient, we can capture abnormalities induced by local rotational changes. We tested the algorithm by detecting abnormalities on synthetic surfaces, 3D human face data and MRI-derived brain surfaces. Experimental results show that our algorithm can effectively detect abnormalities and capture local rotational alterations. Our method is also more effective than other existing methods, such as the isometric indicator, for locating abnormalities.
Ronald Lok Ming LuiHarvard UniversityTsz Wai WongUniversity of California, Los AngelesPaul M. ThompsonUniversity of California, Los AngelesTony F. ChanThe Hong Kong University of Science and TechnologyXianfeng GuStony Brook UniversityShing-Tung YauHarvard University
IEEE Committee on Computer Vision and Pattern Recognition, 2839-2846, 2010
Surface registration is widely used in machine vision and medical imaging, where 1-1 correspondences between surfaces are computed to study their variations. Surface maps are usually stored as the 3D coordinates each vertex is mapped to, which often requires lots of storage memory. This causes inconvenience in data transmission and data storage, especially when a large set of surfaces are analyzed. To tackle this problem, we propose a novel representation of surface diffeomorphisms using Beltrami coefficients, which are complex-valued functions defined on surfaces with supreme 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 Beltrami coefficients on the source domain. Hence, every bijective surface map can be represented by a unique Beltrami
coefficient. Conversely, given a Beltrami coefficient, we can reconstruct the unique surface map associated to it using the Beltrami Holomorphic flow (BHF) method introduced in this paper. Using this representation, 1/3 of the storage space is saved. We can further reduce the storage requirement by 90% by compressing the Beltrami coefficients using Fourier approximations. We test our algorithm on synthetic data, real human brain and hippocampal surfaces. Our results show high accuracy in the reconstructed data, while the amount of storage is greatly reduced. Our approach is compared with the Fourier compression of the coordinate functions using the same amount of data. The latter approach often shows jaggy results and cannot guarantee
to preserve diffeomorphisms.
Wei ZengWayne State UniversityRonald Lok Ming LuiHarvard UniversityLin ShiThe Chinese University of Hong KongDefeng WangThe Chinese University of Hong KongWinnie C.W. ChuThe Chinese University of Hong KongJack C.Y. ChuThe Chinese University of Hong KongJing HuaWayne State UniversityShing-Tung YauHarvard UniversityXianfeng GuStony Brook University
Medical Image Computing and Computer Assisted Intervention, 538-546, 2010
Hawasli A H, Hullar T E, Dorward I G, et al. Idiopathic scoliosis and the vestibular system[J]. European Spine Journal, 2015, 24(2): 227-233.
Shiqing Xin · Ying He · Chiwing Fu · Defeng Wang · Shi Lin · Winnie C W Chu · Jack C Y Cheng · Xianfeng Gu · Lok Ming Lui. Euclidean geodesic loops on high-genus surfaces applied to the morphometry of vestibular systems. 2011.
Catanzariti J F, Agnani O, Guyot M A, et al. Does adolescent idiopathic scoliosis relate to vestibular disorders? A systematic review.[J]. Annals of Physical and Rehabilitation Medicine, 2014, 57(6): 465-479.
Tsz Ching Ng · Xianfeng Gu · Lok Ming Lui. Computing Extremal Teichmüller Map of Multiply-Connected Domains Via Beltrami Holomorphic Flow. 2014.
Combes B, Fournier M, Kennedy D N, et al. EM-ICP strategies for joint mean shape and correspondences estimation: Applications to statistical analysis of shape and of asymmetry[C]. international symposium on biomedical imaging, 2011: 1257-1263.
Chengfeng Wen · Defeng Wang · Lin Shi · Winnie C W Chu · Jack C Y Cheng · Lok Ming Lui. Landmark constrained registration of high-genus surfaces applied to vestibular system morphometry.. 2015.
Wei Zeng · Rui Shi · Zhengyu Su · David Xianfeng Gu. Colon Surface Registration Using Ricci Flow. 2014.
Jiaxi Hu · Hajar Hamidian · Zichun Zhong · Jing Hua. Visualizing Shape Deformations with Variation of Geometric Spectrum. 2016.
Tsz Wai Wong · Hongkai Zhao. Computing Surface Uniformization Using Discrete Beltrami Flow. 2015.
Kin Tat Ho · Lok Ming Lui. QCMC: quasi-conformal parameterizations for multiply-connected domains. 2016.
Adolescent Idiopathic Scoliosis (AIS) characterized by the 3D spine deformity affects about 4% schoolchildren worldwide. One of the prominent theories of the etiopathogenesis of AIS was proposed to be the poor postural balance control due to the impaired vestibular function. Thus, the morphometry of the vestibular system (VS) is of great importance for studying AIS. The VS is a genus-3 structure situated in the inner ear and consists of three semicircular canals lying perpendicular to each other. The high-genus topology of the surface poses great challenge for shape analysis. In this work, we propose an effective method to analyze shapes of high-genus surfaces by considering their geodesic spectra. The key is to compute the canonical hyperbolic geodesic loops of the surface, using the Ricci flow method. The Fuchsian group generators are then computed which can be used to determine the geodesic spectra. The geodesic spectra effectively measure shape differences between high-genus surfaces up to the hyperbolic isometry. We applied the proposed algorithm to the VS of 12 normal and 15 AIS subjects. Experimental results show the effectiveness of our algorithm and reveal statistical shape difference in the VS between right-thoracic AIS and normal subjects.