Here we propose a novel method to compute Teichmller shape space based shape index to study brain morphometry. Such a shape index is intrinsic, and invariant under conformal transformations, rigid motions and scaling. We conformally map a genus-zero open boundary surface to the Poincar disk with the Yamabe flow method. The shape indices that we compute are the lengths of a special set of geodesics under hyperbolic metric. Tests on longitudinal brain imaging data were used to demonstrate the stability of the derived feature vectors. In leave-one-out validation tests, we achieved 100% accurate classification (versus only 68% accuracy for volume measures) in distinguishing 11 HIV/AIDS individuals from 8 healthy control subjects, based on Teichmller coordinates for lateral ventricular surfaces extracted from their 3D MRI scans.

Superpixels are perceptually meaningful atomic regions that can effectively capture image features. Among various methods for computing uniform superpixels, simple linear iterative clustering (SLIC) is popular due to its simplicity and high performance. In this paper, we extend SLIC to compute content-sensitive superpixels, i.e., small superpixels in content-dense regions with high intensity or colour variation and large superpixels in content-sparse regions. Rather than using the conventional SLIC method that clusters pixels in R^5, we map the input image I to a 2-dimensional manifoldMR5, whose area elements are a good measure of the content density in I. We propose a simple method, called intrinsic manifold SLIC (IMSLIC), for computing a geodesic centroidal Voronoi tessellation (GCVT)—a uniform tessellation—onM, which induces the content-sensitive superpixels in I. In contrast to the existing algorithms, IMSLIC characterizes the content sensitivity by measuring areas of Voronoi cells onM. Using a simple and fast approximation to a closed-form solution, the method can compute the GCVT at a very low cost and guarantees that all Voronoi cells are simply connected. We thoroughly evaluate IMSLIC and compare it with eleven representative methods on the BSDS500 dataset and seven representative methods on the NYUV2 dataset. Computational results show that IMSLIC outperforms existing methods in terms of commonly used quality measures pertaining to superpixels such as compactness, adherence to boundaries, and achievable segmentation accuracy. We also evaluate IMSLIC and seven representative methods in an image contour closure application, and the results on two datasets, WHD and WSD, show that IMSLIC achieves the best foreground segmentation performance.

We address the registration problem of genus-one surfaces (such as vertebrae bones) with prescribed landmark constraints. The high-genus 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 quasi-conformal maps called Teichmüller maps (T-Maps). A landmark constrained T-Map is the unique mapping between genus-1 surfaces that minimizes the maximal conformality distortion while matching the prescribed feature landmarks. Existence and uniqueness of the landmark constrained T-Map are theoretically guaranteed. This work presents an iterative algorithm to compute the T-Map. 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 T-Map 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.

This paper presents two algorithms, based on conformal geometry, for the multi-scale representations of geometric shapes and surface morphing. A multi-scale 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 multi-scale representation for 3D surfaces. The basic idea is to represent a 3D surface by its mean curvature function, H, and conformal factor function λ, which uniquely determine the geometry of the surface according to Riemann surface theory. Once we have the (λ,H) parameterization of the surface, post-processing of the surface can be done directly on the conformal parameter domain. In particular, the problem of multi-scale representations of shapes can be reduced to the signal filtering on the  and H parameters. On the other hand, the surface morphing problem can be transformed to an interpolation process of two sets of (λ,H) parameters. We test the proposed algorithms on 3D human face data and MRI-derived brain surfaces. Experimental results show that our proposed methods can effectively obtain multi-scale surface representations and give natural surface morphing results.

The medial axis is an important shape representation that finds a wide range of applications in shape analysis. For large-scale shapes of high resolution, a progressive medial axis representation that starts with the lowest resolution and gradually adds more details is desired. In this paper, we propose a fast and robust geometric algorithm that computes progressive medial axes of a large-scale planar shape. The key ingredient of our method is a novel structural analysis of merging medial axes of two planar shapes along a shared boundary. Our method is robust by separating the analysis of topological structure from numerical computation. Our method is also fast and we show that the time complexity of merging two medial axes is O(n log nv), where n is the number of total boundary generators, nv is strictly smaller than n and behaves as a small constant in all our experiments. Experiments on large-scale polygonal data and comparison with state-of-the-art methods show the efficiency and effectiveness of the proposed method.

This paper presents a method to compute the quasi-conformal parameterization (QCMC) for a multiply-connected 2D domain or surface. QCMC computes a quasi-conformal map from a multiply-connected domain S onto a punctured disk D_S associated with a given Beltrami differential. The Beltrami differential, which measures the conformality distortion, is a complex-valued function μ: S→ℂ with supremum norm strictly less than 1. Every Beltrami differential gives a conformal structure of S. Hence, the conformal module of D_S, which are the radii and centers of the inner circles, can be fully determined by μ, up to a Möbius transformation. In this paper, we propose an iterative algorithm to simultaneously search for the conformal module and the optimal quasi-conformal parameterization. The key idea is to minimize the Beltrami energy subject to the boundary constraints. The optimal solution is our desired quasi-conformal parameterization onto a punctured disk. The parameterization of the multiply-connected domain simplifies numerical computations and has important applications in various fields, such as in computer graphics and vision. Experiments have been carried out on synthetic data together with real multiply-connected Riemann surfaces. Results show that our proposed method can efficiently compute quasi-conformal parameterizations of multiply-connected domains and outperforms other state-of-the-art algorithms. Applications of the proposed parameterization technique have also been explored.

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.

In this paper, we consider the problem of approximating the longest path in a grid graph that possesses a square-free 2-factor. By (1) analyzing several characteristics of four types of cross cells and (2) presenting three cycle-merging operations and one extension operation, we propose an algorithm that finds in an n-vertex grid graph G, a long path of length at least 5/6 n + 2. Our approximation algorithm runs in quadratic time. In addition to its theoretical value, our work has also an interesting application in image-guided maze generation.

We propose a novel method to apply Teichmller space theory to study the signature of a family of nonintersecting closed 3D curves on a general genus zero closed surface. Our algorithm provides an efficient method to encode both global surface and local contour shape information. The signatureTeichmller shape descriptoris computed by surface Ricci flow method, which is equivalent to solving an elliptic partial differential equation on surfaces and is numerically stable. We propose to apply the new signature to analyze abnormalities in brain cortical morphometry. Experimental results with 3D MRI data from Alzheimers disease neuroimaging initiative (ADNI) dataset [152 healthy control subjects versus 169 Alzheimers disease (AD) patients] demonstrate the effectiveness of our method and illustrate its potential as a novel surface-based cortical morphometry measurement in AD research.

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 non-rigid 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 multi-modality 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 non-rigid 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.

In medical imaging, parameterized 3D surface models are of great interest for anatomical modeling and visualization, statistical comparisons of anatomy, and surface-based registration and signal processing. Here we introduce a parameterization method based on algebraic functions. By solving the Yamabe equation with the Ricci flow method, we can conformally map a brain surface to a multi-hole disk. The resulting parameterizations do not have any singularities and are intrinsic and stable. To illustrate the technique, we computed parameterizations of several types of anatomical surfaces in MRI scans of the brain, including the hippocampi and the cerebral cortices with various landmark curves labeled. For the cerebral cortical surfaces, we show the parameterization results are consistent with selected landmark curves and can be matched to each other using constrained harmonic maps. Unlike previous

In this paper, we introduce a novel method that can generate a sequence of physical transformations between 3D models with different shape and topology. Feasible transformations are realized on a chain structure with connected components that are 3D printed. Collision-free motions are computed to transform between different configurations of the 3D printed chain structure. To realize the transformation between different 3D models, we first voxelize these input models into similar number of voxels. The challenging part of our approach is to generate a simple path --- as a chain configuration to connect most voxels. A layer-based algorithm is developed with theoretical guarantee of the existence and the path length. We find that collision-free motion sequence can always be generated when using a straight line as the intermediate configuration of transformation. The effectiveness of our method is demonstrated by both the simulation and the experimental tests taken on 3D printed chains.

A structured light system for three-dimensional shape measurement with single camera has the shortcoming of camera occlusion. To alleviate this problem, this paper introduces a structured light system with dual cameras for three-dimensional shape measurement. We discuss (1) system description, (2) system calibration, (3) three-dimensional data registration using the iterative closest-point (ICP) algorithm, and (4) three-dimensional data merging using holoimage. The principle of the system is introduced, and experiments are presented to verify its performance.

3D surface classification is a fundamental problem in computer vision and computational geometry. Surfaces can be classified by different transformation groups. Traditional classification methods mainly use topological transformation groups and Euclidean transformation groups. We introduce a novel method to classify surfaces by conformal transformation groups. Conformal equivalent class is refiner than topological equivalent class and coarser than isometric equivalent class, making it suitable for practical classification purposes. For general surfaces, the gradient fields of conformal maps form a vector space, which has a natural structure invariant under conformal transformations. We present an algorithm to compute this conformal structure, which can be represented as matrices, and use it to classify surfaces. The result is intrinsic to the geometry, invariant to triangulation and insensitive to resolution. To the best

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.

Brain Cortical surface registration is required for inter-subject studies of functional and anatomical data. Harmonic mapping has been applied for brain mapping, due to its existence, uniqueness, regularity and numerical stability. In order to improve the registration accuracy, sculcal landmarks are usually used as constraints for brain registration. Unfortunately, constrained harmonic mappings may not be diffeomorphic and produces invalid registration. This work conquer this problem by changing the Riemannian metric on the target cortical surface to a hyperbolic metric, so that the harmonic mapping is guaranteed to be a diffeomorphism while the landmark constraints are enforced as boundary matching condition. The computational algorithms are based on the Ricci flow method and hyperbolic heat diffusion. Experimental results demonstrate that, by changing the Riemannian metric, the registrations are

Computational conformal geometry focuses on developing the computational methodologies on discrete surfaces to discover conformal geometric invariants. In this work, we briefly summarize the recent developments for methods and related applications in computational conformal geometry. There are two major approaches, holomorphic differentials and curvature flow. The holomorphic differential method is a linear method, which is more efficient and robust to triangulations with lower quality. The curvature flow method is nonlinear and requires higher quality triangulations, but more flexible. The conformal geometric methods have been broadly applied in many engineering fields, such as computer graphics, vision, geometric modeling and medical imaging. The algorithms are robust for surfaces scanned from real life, general for surfaces with different topologies. The efficiency and efficacy of the algorithms

Can you capture the motion of a smile in 3-D? This paper presents a video acquisition system that measures 3-D geometry accurately. The data acquisition speed is 90 fps and over one quarter million points per frame. Acquisition, reconstruction, and display are simultaneously realized at 30 fps.

In medical imaging, parameterized 3-D surface models are useful for anatomical modeling and visualization, statistical comparisons of anatomy, and surface-based registration and signal processing. Here we introduce a parameterization method based on Riemann surface structure, which uses a special curvilinear net structure (conformal net) to partition the surface into a set of patches that can each be conformally mapped to a parallelogram. The resulting surface subdivision and the parameterizations of the components are intrinsic and stable (their solutions tend to be smooth functions and the boundary conditions of the Dirichlet problem can be enforced). Conformal parameterization also helps transform partial differential equations (PDEs) that may be defined on 3-D brain surface manifolds to modified PDEs on a two-dimensional parameter domain. Since the Jacobian matrix of a conformal parameterization is