Teichmuller Mapping (T-Map) and Its Applications to Landmark Matching Registration

Lui, Lok Ming; Lam, Ka Chun; Yau, Shing–Tung; Gu, Xianfeng

doi:10.1137/120900186

Cited by 84 publications

(58 citation statements)

References 28 publications

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“…Here, a BC is called admissible if it is associated to a bijective quasi-conformal map subject to the landmark constraints. In [32], the surface Teichüller map (T-Map) was computed through adjusting BCs. The BC is normalized in each iteration to an adjusted BC with a constant norm.…”

Section: Contributionsmentioning

confidence: 99%

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Landmark constrained registration of high-genus surfaces applied to vestibular system morphometry

Wen

Wang

Shi

et al. 2015

Computerized Medical Imaging and Graphics

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Section: Contributionsmentioning

confidence: 99%

“…2. Second, we propose a modified iterative scheme, which extends the scheme in [32], to search for an admissible BC under landmark constraints with a natural distribution for general high-genus surfaces. The key idea is to find an admissible BC under landmark constraints between the Poincaré disks.…”

Section: Contributionsmentioning

confidence: 99%

Landmark constrained registration of high-genus surfaces applied to vestibular system morphometry

Wen

Wang

Shi

et al. 2015

Computerized Medical Imaging and Graphics

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“…Recently, various algorithms have been proposed to compute the quasi-conformal map from a prescribed BC efficiently. For example, Lui et al [24,37] proposed to compute quasi-conformal maps by solving a generalized Laplace's equation. The algorithm, which is called the Linear Beltrami Solver (LBS), can be discretized into a sparse symmetric positive definite linear system.…”

Section: Beltrami Holomorphic Flowmentioning

confidence: 99%

“…The computation is therefore quite time-consuming. Various efficient algorithms have been introduced to compute the quasiconformal map [23,24,[36][37][38]41]. In this paper, we approximate the quasi-conformal map f with a Beltrami coefficient μ by directy solving the Beltrami's equation (35) as in [38].…”

Section: Minimization Of Subproblem (26) Involving Fmentioning

confidence: 99%

A Splitting Method for Diffeomorphism Optimization Problem Using Beltrami Coefficients

Lui

2014

J Sci Comput

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Finding a meaningful 1-1 correspondence between different data, such as images or surface data, has important applications in various fields. It involves the optimization of certain energy functionals over the space of all diffeomorphisms. This type of optimization problems (called the diffeomorphism optimization problems, DOPs) is especially challenging, since the bijectivity of the mapping has to be ensured. Recently, a method, called the Beltrami holomorphic flow (BHF), has been proposed to solve the DOP using quasi-conformal theories (Lui et al. in J Sci Comput 50(3):557-585, 2012). The optimization problem is formulated over the space of Beltrami coefficients (BCs), instead of the space of all diffeomorphisms. BHF iteratively finds a sequence of BCs associated with a sequence of diffeomorphisms, using the gradient descent method, to minimize the energy functional. The use of BCs effectively controls the smoothness and bijectivity of the mapping, and hence makes it easier to handle the constrained optimization problem. However, the algorithm is computationally expensive. In this paper, we propose an efficient splitting algorithm, based on the classical alternating direction method of multiplier (ADMM), to solve the DOP. The basic idea is to split the energy functional into two energy terms: one involves the BC whereas the other involves the quasi-conformal map. Alternating minimization scheme can then be applied to minimize the energy functional. The proposed method significantly speeds up the previous BHF approach. It also extends the previous BHF algorithm to Riemann surfaces of arbitrary topologies, such as multiply-connected shapes. Experiments have been carried out on synthetic together with real surface data, which demonstrate the efficiency and efficacy of the proposed algorithm to solve the DOP.

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“…Then, they use regression forest to simultaneously learn the optimal set of features to best characterize each landmark and the non-linear mappings from local patch appearances of image points to their displacements towards each landmark. (Lui et al 2014) proposed an efficient iterative algorithm, called the quasi-conformal (QC) iteration, to compute the T-Map. The basic idea is to represent the set of diffeomorphisms using Beltrami coefficients (BCs) and look for an optimal BC associated to the desired T-Map.…”

Section: Geometric Transformationmentioning

confidence: 99%

Automated detection and classification of brain injury lesions on structural magnetic resonance imaging

Serrano¹

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Teichmuller Mapping (T-Map) and Its Applications to Landmark Matching Registration

Cited by 84 publications

References 28 publications

Landmark constrained registration of high-genus surfaces applied to vestibular system morphometry

Landmark constrained registration of high-genus surfaces applied to vestibular system morphometry

A Splitting Method for Diffeomorphism Optimization Problem Using Beltrami Coefficients

Automated detection and classification of brain injury lesions on structural magnetic resonance imaging

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