Conformal Parameterization Research Articles

Surface Meshing with Curvature Convergence.

Surface meshing plays a fundamental role in graphics and visualization. Many geometric processing tasks involve solving geometric PDEs on meshes. The numerical stability, convergence rates and approximation errors are largely determined by the mesh qualities. In practice, Delaunay refinement algorithms offer satisfactory solutions to high quality mesh generations. The theoretical proofs for volume based and surface based Delaunay refinement algorithms have been established, but those for conformal parameterization based ones remain wide open. This work focuses on the curvature measure convergence for the conformal parameterization based Delaunay refinement algorithms. Given a metric surface, the proposed approach triangulates its conformal uniformization domain by the planar Delaunay refinement algorithms, and produces a high quality mesh. We give explicit estimates for the Hausdorff distance, the normal deviation, and the differences in curvature measures between the surface and the mesh. In contrast to the conventional results based on volumetric Delaunay refinement, our stronger estimates are independent of the mesh structure and directly guarantee the convergence of curvature measures. Meanwhile, our result on Gaussian curvature measure is intrinsic to the Riemannian metric and independent of the embedding. In practice, our meshing algorithm is much easier to implement and much more efficient. The experimental results verified our theoretical results and demonstrated the efficiency of the meshing algorithm.

A Novel Symmetric Skew-Hamiltonian Isotropic Lanczos Algorithm for Spectral Conformal Parameterizations

In the past decades, many methods for computing conformal mesh parameterizations have been developed in response to demand of numerous applications in the field of geometry processing. Spectral conformal parameterization (SCP) (Mullen et al. in Proceedings of the symposium on geometry processing, SGP '08. Eurographics Association, Aire-la-Ville, Switzerland, pp 1487---1494, 2008) is one of these methods used to compute a quality conformal parameterization based on the spectral techniques. SCP focuses on a generalized eigenvalue problem (GEP) $$L_{C}{\mathbf {f}} = \lambda B{\mathbf {f}}$$ L C f = ? B f whose eigenvector(s) associated with the smallest positive eigenvalue(s) provide the conformal parameterization result. This paper is devoted to studying a novel eigensolver for this GEP. Based on structures of the matrix pair $$(L_{C},B)$$ ( L C , B ) , we show that this GEP can be transformed into a small-scale compressed and deflated standard eigenvalue problem with a symmetric positive definite skew-Hamiltonian operator. We then propose a symmetric skew-Hamiltonian isotropic Lanczos algorithm ( $${\mathbb {S}}$$ S HILA) to solve the reduced problem. Numerical experiments show that our compressed deflating technique can exclude the impact of convergence from the kernel of $$L_{C}$$ L C and transform the original problem to a more robust system. The novel $${\mathbb {S}}$$ S HILA method can effectively avoid the disturbance of duplicate eigenvalues. As a result, based on the spectral model of SCP, our numerical eigensolver can compute the conformal parameterization accurately and efficiently.

Conformal parameterization for multiply connected domains: combining finite elements and complex analysis

Conformal parameterization plays an important role in isogeometric analysis. Genus zero surfaces with multiple boundary components (multiply connected domains) can be conformally mapped onto planar domains with circular holes (circle domains). This work introduces a novel method to compute such conformal mappings combining finite element and complex analysis methods. First, the surface is mapped to planar annulus with concentric circular slits using holomorphic differentials, which is carried out using a finite element method based on Hodge decomposition; second the slit domain is conformally mapped to a circle domain by a Laurent series method. Compared with existing algorithms, the proposed method is more efficient and robust. Numerical experiments demonstrate the efficiency and efficacy of the method.

Folding-Free Global Conformal Mapping for Genus-0 Surfaces by Harmonic Energy Minimization

Surface conformal maps between genus-0 surfaces play important roles in applied mathematics and engineering, with applications in medical image analysis and computer graphics. Previous work (Gu and Yau in Commun Inf Syst 2(2):121---146, 2002) introduces a variational approach, where global conformal parameterization of genus-0 surfaces was addressed through minimizing the harmonic energy, with two weaknesses: its gradient descent iteration is slow, and its solutions contain undesired parameterization foldings when the underlying surface has long sharp features. In this paper, we propose an algorithm that significantly accelerates the harmonic energy minimization and a method that iteratively removes foldings by taking advantages of the weighted Laplace---Beltrami eigen-projection. Experimental results show that the proposed approaches compute genus-0 surface harmonic maps much faster than the existing algorithm in Gu and Yau (Commun Inf Syst 2(2):121---146, 2002) and the new results contain no foldings.

Surface fluid registration of conformal representation: application to detect disease burden and genetic influence on hippocampus.

In this paper, we develop a new automated surface registration system based on surface conformal parameterization by holomorphic 1-forms, inverse consistent surface fluid registration, and multivariate tensor-based morphometry (mTBM). First, we conformally map a surface onto a planar rectangle space with holomorphic 1-forms. Second, we compute surface conformal representation by combining its local conformal factor and mean curvature and linearly scale the dynamic range of the conformal representation to form the feature image of the surface. Third, we align the feature image with a chosen template image via the fluid image registration algorithm, which has been extended into the curvilinear coordinates to adjust for the distortion introduced by surface parameterization. The inverse consistent image registration algorithm is also incorporated in the system to jointly estimate the forward and inverse transformations between the study and template images. This alignment induces a corresponding deformation on the surface. We tested the system on Alzheimer's Disease Neuroimaging Initiative (ADNI) baseline dataset to study AD symptoms on hippocampus. In our system, by modeling a hippocampus as a 3D parametric surface, we nonlinearly registered each surface with a selected template surface. Then we used mTBM to analyze the morphometry difference between diagnostic groups. Experimental results show that the new system has better performance than two publicly available subcortical surface registration tools: FIRST and SPHARM. We also analyzed the genetic influence of the Apolipoprotein E[element of]4 allele (ApoE4), which is considered as the most prevalent risk factor for AD. Our work successfully detected statistically significant difference between ApoE4 carriers and non-carriers in both patients of mild cognitive impairment (MCI) and healthy control subjects. The results show evidence that the ApoE genotype may be associated with accelerated brain atrophy so that our work provides a new MRI analysis tool that may help presymptomatic AD research.

A conformal approach for surface inpainting

We address the problem of surface inpainting, which aims to fill in holes or missing regions on a Riemann surface based on its surface geometry. In practical situation, surfaces obtained from range scanners often have holes where the 3D models are incomplete. In order to analyze the 3D shapes effectively, restoring the incomplete shape by filling in the surface holes is necessary. In this paper, we propose a novel conformal approach to inpaint surface holes on a Riemann surface based on its surface geometry. The basic idea is to represent the Riemann surface using its conformal factor and mean curvature. According to Riemann surface theory, a Riemann surface can be uniquely determined by its conformal factor and mean curvature up to a rigid motion. Given a Riemann surface $S$, its mean curvature $H$ and conformal factor $\lambda$ can be computed easily through its conformal parameterization. Conversely, given $\lambda$ and $H$, a Riemann surface can be uniquely reconstructed by solving the Gauss-Codazzi equation on the conformal parameter domain. Hence, the conformal factor and the mean curvature are two geometric quantities fully describing the surface. With this $\lambda$-$H$ representation of the surface, the problem of surface inpainting can be reduced to the problem of image inpainting of $\lambda$ and $H$ on the conformal parameter domain. Once $\lambda$ and $H$ are inpainted, a Riemann surface can be reconstructed which effectively restores the 3D surface with missing holes. Since the inpainting model is based on the geometric quantities $\lambda$ and $H$, the restored surface follows the surface geometric pattern. We test the proposed algorithm on synthetic data as well as real surface data. Experimental results show that our proposed method is an effective surface inpainting algorithm to fill in surface holes on an incomplete 3D models based their surface geometry.

Can Mean‐Curvature Flow be Modified to be Non‐singular?

AbstractThis work considers the question of whether mean‐curvature flow can be modified to avoid the formation of singularities. We analyze the finite‐elements discretization and demonstrate why the original flow can result in numerical instability due to division by zero. We propose a variation on the flow that removes the numerical instability in the discretization and show that this modification results in a simpler expression for both the discretized and continuous formulations. We discuss the properties of the modified flow and present empirical evidence that not only does it define a stable surface evolution for genus‐zero surfaces, but that the evolution converges to a conformal parameterization of the surface onto the sphere.

Mesh Parameterization with Generalized Discrete Conformal Maps

We introduce a new method to compute conformal parameterizations using a recent definition of discrete conformity, and establish a discrete version of the Riemann mapping theorem. Our algorithm can parameterize triangular, quadrangular and digital meshes. It can also be adapted to preserve metric properties. To demonstrate the efficiency of our method, many examples are shown in the experiment section.

Brain surface conformal parameterization with the Ricci flow.

In brain mapping research, parameterized 3-D surface models are of great interest for statistical comparisons of anatomy, surface-based registration, and signal processing. Here, we introduce the theories of continuous and discrete surface Ricci flow, which can create Riemannian metrics on surfaces with arbitrary topologies with user-defined Gaussian curvatures. The resulting conformal parameterizations have no singularities and they are intrinsic and stable. First, we convert a cortical surface model into a multiple boundary surface by cutting along selected anatomical landmark curves. Secondly, we conformally parameterize each cortical surface to a parameter domain with a user-designed Gaussian curvature arrangement. In the parameter domain, a shape index based on conformal invariants is computed, and inter-subject cortical surface matching is performed by solving a constrained harmonic map. We illustrate various target curvature arrangements and demonstrate the stability of the method using longitudinal data. To map statistical differences in cortical morphometry, we studied brain asymmetry in 14 healthy control subjects. We used a manifold version of Hotelling's T(2) test, applied to the Jacobian matrices of the surface parameterizations. A permutation test, along with the cumulative distribution of p-values, were used to estimate the overall statistical significance of differences. The results show our algorithm's power to detect subtle group differences in cortical surfaces.

Evaluation of fiber bundles across subjects through brain mapping and registration of diffusion tensor data

This paper presents a visualization and analysis framework for evaluating changes in structural organization of fiber bundles in human brain white matter. Statistical analysis of fiber bundle organization is conducted using an anisotropy measure, volume ratio (VR), which is ratio of anisotropic and isotropic components. Initially fiber bundles are tracked using a probabilistic algorithm starting from seed voxels. To ensure accurate selection of seed voxels and to prevent operator bias, a reference brain (MNI_152) is used when marking ROIs. Individual structural MRI brain scans are mapped to the reference using volumetric conformal parameterization. This mapping preserves topology and aligns features perfectly making it a robust and accurate registration technique. One-to-one mapping to the template allows ROI selection and subsequent transfer of ROI to structural MRI of subject. Affine registration coregisters structural MRI and DTI. Seed voxels are mapped to DTI using the resulting transformation parameters. To evaluate the proposed approach, MRI and DTI of 12 normal volunteers and 15 medial temporal lobe epilepsy patients are used. First, a statistical hypothesis testing is conducted to test for anisotropy changes in cingulum and fornix fiber bundles of epileptic patients. Experimental results reveal a 40% decrease in anisotropy levels of cingulum in patients compared to volunteers. They also show a 25% overall decrease in anisotropy of fornix. Secondly, shapes of the bundles are visualized in 3D illustrating that the bundles of epileptic patients are bumpy while those of normal volunteers are smooth.

Optimized Conformal Surface Registration with Shape-based Landmark Matching

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.

Multivariate tensor-based morphometry on surfaces: Application to mapping ventricular abnormalities in HIV/AIDS

Here we developed a new method, called multivariate tensor-based surface morphometry (TBM), and applied it to study lateral ventricular surface differences associated with HIV/AIDS. Using concepts from differential geometry and the theory of differential forms, we created mathematical structures known as holomorphic one-forms, to obtain an efficient and accurate conformal parameterization of the lateral ventricular surfaces in the brain. The new meshing approach also provides a natural way to register anatomical surfaces across subjects, and improves on prior methods as it handles surfaces that branch and join at complex 3D junctions. To analyze anatomical differences, we computed new statistics from the Riemannian surface metrics—these retain multivariate information on local surface geometry. We applied this framework to analyze lateral ventricular surface morphometry in 3D MRI data from 11 subjects with HIV/AIDS and 8 healthy controls. Our method detected a 3D profile of surface abnormalities even in this small sample. Multivariate statistics on the local tensors gave better effect sizes for detecting group differences, relative to other TBM-based methods including analysis of the Jacobian determinant, the largest and smallest eigenvalues of the surface metric, and the pair of eigenvalues of the Jacobian matrix. The resulting analysis pipeline may improve the power of surface-based morphometry studies of the brain.

Brain Surface Conformal Parameterization with Holomorphic Flow Method and Its Application to HIV/AIDS

Brain Surface Conformal Parameterization with Holomorphic Flow Method and Its Application to HIV/AIDS

Detecting corpus callosum abnormalities in autism subtype using planar conformal mapping

AbstractA number of studies have documented that autism has a neurobiological basis, but the anatomical extent of these neurobiological abnormalities is largely unknown. In this paper, we apply advanced computational techniques to extract 3D models of the corpus callosum (CC) and subsequently analyze local shape variations in a homogeneous group of autistic children. Besides the traditional volumetric analysis, we explore additional phenotypic traits based on the oriented bounding rectangle of the CC. In shape analysis, a new conformal parameterization is applied in our shape analysis work, which maps the surface onto a planar domain. Surface matching among different individual meshes is achieved by aligning the planar domains of individual meshes. Shape differences of the CC between autistic patients and the controls are computed using Hotelling T2 two‐sample metric followed by a permutation test. The raw and corrected p‐values are shown in the results. Additional visualization of the group difference is provided via mean difference map. Copyright © 2009 John Wiley & Sons, Ltd.

Optimal Surface Parameterization Using Inverse Curvature Map

Mesh parameterization is a fundamental technique in computer graphics. Our paper focuses on solving the problem of finding the best discrete conformal mapping that also minimizes area distortion. Firstly, we deduce an exact analytical differential formula to represent area distortion by curvature change in the discrete conformal mapping, giving a dynamic Poisson equation. Our result shows the curvature map is invertible. Furthermore, we give the explicit Jacobi matrix of the inverse curvature map. Secondly, we formulate the task of computing conformal parameterizations with least area distortions as a constrained nonlinear optimization problem in curvature space. We deduce explicit conditions for the optima. Thirdly, we give an energy form to measure the area distortions, and show it has a unique global minimum. We use this to design an efficient algorithm, called free boundary curvature diffusion, which is guaranteed to converge to the global minimum. This result proves the common belief that optimal parameterization with least area distortion has a unique solution and can be achieved by free boundary conformal mapping. Major theoretical results and practical algorithms are presented for optimal parameterization based on the inverse curvature map. Comparisons are conducted with existing methods and using different energies. Novel parameterization applications are also introduced.

Spectral Conformal Parameterization

AbstractWe present a spectral approach to automatically and efficiently obtain discrete free‐boundary conformal parameterizations of triangle mesh patches, without the common artifacts due to positional constraints on vertices and without undue bias introduced by sampling irregularity. High‐quality parameterizations are computed through a constrained minimization of a discrete weighted conformal energy by finding the largest eigenvalue/eigenvector of a generalized eigenvalue problem involving sparse, symmetric matrices. We demonstrate that this novel and robust approach improves on previous linear techniques both quantitatively and qualitatively.

Variational Method on Riemann Surfaces using Conformal Parameterization and its Applications to Image Processing

Variational method is a useful mathematical tool in various areas of research, espe- cially in image processing. Recently, solving image processing problems on general manifolds using variational techniques has become an important research topic. In this paper, we solve several image processing problems on general Riemann surfaces using variational models defined on surfaces. We propose an explicit method to solve variational problems on general Riemann surfaces, using the conformal parameterization and covariant derivatives defined on the surface. To simplify the com- putation, the surface is firstly mapped conformally to the two dimensional rectangular domains, by computing the holomorphic 1-form on the surface. It is well known that the Jacobian of a conformal map is simply the scalar multiplication of the conformal factor. Therefore with the conformal para- meterization, the covariant derivatives on the parameter domain are similar to the usual Euclidean differential operators, except for the scalar multiplication. As a result, any variational problem on the surface can be formulated to a 2D problem with a simple formula and efficiently solved by well developed numerical scheme on the 2D domain. With the proposed method, we solve various image processing problems on surfaces using different variational models, which include image segmen- tation, surface denoising, surface inpainting, texture extraction and automatic landmark tracking. Experimental results show that our method can effectively solve variational problems and tackle image processing tasks on general Riemann surfaces.

Computation of curvatures using conformal parameterization

Curvatures on the surface are important geometric invariants and are widely used in different area of research. Examples include feature recognition, segmentation, or shape analysis. Therefore, it is of interest to develop an effective algorithm to approximate the curvatures accurately. The classical methods to compute these quantities involve the estimation of the normal and some involve the computation of the second derivatives of the 3 coordinate functions under the param- eterization. Error is inevitably introduced because of the inaccurate approximation of the second derivatives and the normal. In this paper, we propose several novel methods to compute curva- tures on the surface using the conformal parameterization. With the conformal parameterization, the conformal factor function $\lambda$ can be defined on the surface. Mean curvature (H) and Gaussian curvatures (K) can then be computed with the conformal Factor ($\lambda$). It involves computing only the derivatives of the function $\lambda$, instead of the 3 coordinate functions and the normal. We also introduce a technique to compute H from K and vice versa, using the parallel surface.

Parameterization of discrete surfaces

Parameterization of surfaces, sometimes alled ”‘flattening”’ as it maps a surface embedded in 3D into its intrinsic 2D domain, is a powerfull tool for the analysis of surfaces. For the past years, there has been a growing interest in the Community that even lead to one implementation of one special type of parameterization in ITK [1,5,8,9]. We are providing here a more general framework for parameterization of single connected surfaces of any genus. It is based on a recent addition to ITK: itkQuadEdgeMesh [6] which allows an elegant an optimal implementation of algorithm for geometry and topology processing of discrete 2-manifolds. The 5 algorithms that we implemented map the meshes into a planar domain with fixed boundary leading to more stability and speed than mapping into the spherical domain. Each of them use different kind of parameterization with different properties. The conformal parameterization is usually used as it is intrinsic to the geometry of the mesh and thus allow shape analysis independently of the connectivity. However if flattening the mesh is your only goal, then the simplest parameterization algorithm (graph theory) will give you the best speed. This is the case when you want to further process the mesh in a lower dimension. Using specific solvers, the parameterization of meshes can be done sufficiently fast (couple of seconds) to consider this approach in interactive applications.

Brain anatomical feature detection by solving partial differential equations on general manifolds

One important problem in human brain mapping research is to locate the important anatomical features. Anatomical features on the cortical surface are usually represented by landmark curves, called sulci/gyri curves. These landmark curves are important information for neuroscientists to study brain disease and to match different cortical surfaces. Manual labelling of these landmark curves is time-consuming, especially when large sets of data have to be analyzed. In this paper, we present algorithms to automatically detect and match landmark curves on cortical surfaces to get an optimized brain conformal parametrization. First, we propose an algorithm to obtain a hypothesized landmark region/curves using the Chan-Vese segmentation method, which solves a Partial Differential Equation (PDE) on a manifold with global conformal parameterization. This is done by segmentating the high mean curvature region. Second, we propose an automatic landmark curve tracing method based on the principal directions of the local Weingarten matrix. Based on the global conformal parametrization of a cortical surface, our method adjusts the landmark curves iteratively on the spherical or rectangular parameter domain of the cortical surface along its principal direction field, using umbilic points of the surface as anchors. The landmark curves can then be mapped back onto the cortical surface. Experimental results show that the landmark curves detected by our algorithm closely resemble these manually labeled curves. Next, we applied these automatically labeled landmark curves to generate an optimized conformal parametrization of the cortical surface, in the sense that homologous features across subjects are caused to lie at the same parameter locations in a conformal grid. Experimental results show that our method can effectively help in automatically matching cortical surfaces across subjects.