Spherical Parameterization Research Articles

Riemannian metric optimization on surfaces (RMOS) for intrinsic brain mapping in the Laplace-Beltrami embedding space.

Surface mapping methods play an important role in various brain imaging studies from tracking the maturation of adolescent brains to mapping gray matter atrophy patterns in Alzheimer's disease. Popular surface mapping approaches based on spherical registration, however, have inherent numerical limitations when severe metric distortions are present during the spherical parameterization step. In this paper, we propose a novel computational framework for intrinsic surface mapping in the Laplace-Beltrami (LB) embedding space based on Riemannian metric optimization on surfaces (RMOS). Given a diffeomorphism between two surfaces, an isometry can be defined using the pullback metric, which in turn results in identical LB embeddings from the two surfaces. The proposed RMOS approach builds upon this mathematical foundation and achieves general feature-driven surface mapping in the LB embedding space by iteratively optimizing the Riemannian metric defined on the edges of triangular meshes. At the core of our framework is an optimization engine that converts an energy function for surface mapping into a distance measure in the LB embedding space, which can be effectively optimized using gradients of the LB eigen-system with respect to the Riemannian metrics. In the experimental results, we compare the RMOS algorithm with spherical registration using large-scale brain imaging data, and show that RMOS achieves superior performance in the prediction of hippocampal subfields and cortical gyral labels, and the holistic mapping of striatal surfaces for the construction of a striatal connectivity atlas from substantia nigra.

Three-dimensional modeling and assessment of cardiac adipose tissue distribution.

Objective: The layer of fat that accumulates around the heart, called cardiac adipose tissue (CAT), can influence the development of coronary disease and is indicative of cardiovascular risk. While volumetric assessment of magnetic resonance imaging (MRI) can quantify CAT, volume alone gives no information about its distribution across the myocardial surface, which may be an important factor in risk assessment. In this study, a three-dimensional (3D) modeling technique is developed and used to quantify the distribution of the CAT across the surface of the heart.Methods: Dixon MRI scans, which produce a registered 3D set of fat-only and water-only images, were acquired in 10 subjects for a study on exercise intervention. A previously developed segmentation algorithm was used to identify the heart and CAT. Extracted contours were used to build 3D models. Procrustes analysis was used to register the heart models and an iterative closest point algorithm was used to register and align the CAT models for calculation of CAT thickness. Rays were cast in directions specified by a spherical parameterization of elevation and azimuthal angles, and intersections of the ray with the CAT surface were used to calculate the thickness at each location. To evaluate the effects of the spherical parameterization on the thickness estimates, a set of synthetic models were created with increasing major-to-minor axis ratios.Results: Based on the validation in the synthetic models, the average error in CAT thickness ranged from 1.25% to 17.3% for increasing major-to-minor axis ratio.Conclusions: A process was developed, based on Dixon MRI data, to provide 3D models of the myocardial surface and the cardiac fat. The models can be used in future segmentation algorithm development and for studies on changes in cardiac fat as a result of various interventions.

A Preferred Curvature-Based Continuum Mechanics Framework for Modeling Embryogenesis

Mechanics plays a key role in the development of higher organisms. However, understanding this relationship is complicated by the difficulty of modeling the link between local forces generated at the subcellular level and deformations observed at the tissue and whole-embryo levels. Here we propose an approach first developed for lipid bilayers and cell membranes, in which force-generation by cytoskeletal elements enters a continuum mechanics formulation for the full system in the form of local changes in preferred curvature. This allows us to express and solve the system using only tissue strains. Locations of preferred curvature are simply related to products of gene expression. A solution, in that context, means relaxing the system’s mechanical energy to yield global morphogenetic predictions that accommodate a tendency toward the local preferred curvature, without a need to explicitly model force-generation mechanisms at the molecular level. Our computational framework, which we call SPHARM-MECH, extends a 3D spherical harmonics parameterization known as SPHARM to combine this level of abstraction with a sparse shape representation. The integration of these two principles allows computer simulations to be performed in three dimensions on highly complex shapes, gene expression patterns, and mechanical constraints. We demonstrate our approach by modeling mesoderm invagination in the fruit-fly embryo, where local forces generated by the acto-myosin meshwork in the region of the future mesoderm lead to formation of a ventral tissue fold. The process is accompanied by substantial changes in cell shape and long-range cell movements. Applying SPHARM-MECH to whole-embryo live imaging data acquired with light-sheet microscopy reveals significant correlation between calculated and observed tissue movements. Our analysis predicts the observed cell shape anisotropy on the ventral side of the embryo and suggests an active mechanical role of mesoderm invagination in supporting the onset of germ-band extension.

A novel local/global approach to spherical parameterization

This paper proposes a novel local/global spherical parameterization for the genus-zero triangular mesh, which naturally extends the planar approach to the spherical case. In our method, we derive two fitting matrices (conformal and isometric) in 3D space. By optimizing the so-called spring energy, the spherical results are achieved by solving a nonlinear system with spherical constraints. Intuitively, it represents the stitching together of the 1-ring patches to form a unit sphere. Moreover, the derivation of the 3D fitting matrices can also be applied to planar triangles directly, so that we can obtain a class of novel planar approaches (conformal, isometric, authalic) to the problem of flattening triangular meshes. In order to enhance robustness of the proposed spherical method, a stretch operator is introduced for dealing with high-curvature models. Numerical results demonstrate that our method is simple, efficient and convergent, and it outperforms several state-of-the-art methods in terms of trading-off the distortions of angle, area and stretch. Furthermore, it achieves better visualization in texture mapping.

Advanced Hierarchical Spherical Parameterizations.

Computing spherical parameterizations for genus-zero closed surfaces is a fundamental task for geometric processing and computer graphics. Existing methods usually suffer from a lack of practical robustness or poor quality. In this paper, we present a practically robust method to compute high-quality spherical parameterizations with bijection and low isometric distortion. Our method is based on the hierarchical scheme containing mesh decimation and parameterization refinement. The practical robustness of our method relies on two novel techniques. The first one is a flat-to-extrusive decimation strategy, which contains two decimation error metrics to alleviate the difficulty of further mesh refinement. The second is a flexible group refinement technique that consists of flexible vertex insertion and efficient volumetric distortion minimization to control the maximum distortion. We convert the task of volumetric distortion minimization to one of tetrahedral mesh improvement to make the vertices distribute uniformly for efficient refinement. Compared with state-of-the-art methods, our method is more practically robust and possesses better mapping qualities. We demonstrate the efficacy of our method in spherical parameterization computations on a data set containing over five thousand complex models.

EMPOWERING CORTICAL THICKNESS MEASURES IN CLINICAL DIAGNOSIS OF ALZHEIMER'S DISEASE WITH SPHERICAL SPARSE CODING.

Cortical thickness estimation performed in vivo via magnetic resonance imaging (MRI) is an important technique for the diagnosis and understanding of the progression of Alzheimer's disease (AD). Directly using raw cortical thickness measures as features with Support Vector Machine (SVM) for clinical group classification only yields modest results since brain areas are not equally atrophied during AD progression. Therefore, feature reduction is generally required to retain only the most relevant features for the final classification. In this paper, a spherical sparse coding and dictionary learning method is proposed and it achieves relatively high classification results on publicly available data from the Alzheimer's Disease Neuroimaging Initiative (ADNI) 2 dataset (N = 201) which contains structural MRI data of four clinical groups: cognitive unimpaired (CU), early mild cognitive impairment (EMCI), later MCI (LMCI) and AD. The proposed framework takes the estimated cortical thickness and the spherical parameterization computed by FreeSurfer as inputs and constructs weighted patches in the spherical parameter domain of the cortical surface. Then sparse coding is applied to the resulting surface patch features, followed by max-pooling to extract the final feature sets. Finally, SVM is employed for binary group classifications. The results show the superiority of the proposed method over other cortical morphometry systems and offer a different way to study the early identification and prevention of AD.

Spherical parameterization based on planar ARAP+ method

This paper proposes a novel spherical parameterization to process the genus zero models. The method is based on the planar ARAP+ method and further research of the local/global parameterization. It consists of two main steps, local phase and global phase. In the local phase, we employ the planar ARAP+ method to optimize the Spring energy, and achieve the displacement of planar vertex around its 1-ring neighborhood, then map the new vertex to the 3D 1-ring neighborhood on the sphere. In the global phase, we add the spherical constrains to the planar ARAP+ method, and then obtain the final result iteratively according to the Newton method. Numerical results demonstrate that our method is efficient and convergent, and outperforms several popular methods in term of controlling the distortion measures (angle, area, rigidity). Furthermore, it achieves a better visualization performance in texture mapping.

Holistic Mapping of Striatum Surfaces in the Laplace-Beltrami Embedding Space.

In brain shape analysis, the striatum is typically divided into three parts: the caudate, putamen, and accumbens nuclei for its analysis. Recent connectivity and animal studies, however, indicate striatum-cortical inter-connections do not always follow such subdivisions. For the holistic mapping of striatum surfaces, conventional spherical registration techniques are not suitable due to the large metric distortions in spherical parameterization of striatal surfaces. To overcome this difficulty, we develop a novel striatal surface mapping method using the recently proposed Riemannian metric optimization techniques in the Laplace-Beltrami (LB) embedding space. For the robust resolution of sign ambiguities in the LB spectrum, we also devise novel anatomical contextual features to guide the surface mapping in the embedding space. In our experimental results, we compare with spherical registration tools from FreeSurfer and FSL to demonstrate that our novel method provides a superior solution to the striatal mapping problem. We also apply our method to map the striatal surfaces from 211 subjects of the Human Connectome Project (HCP), and use the surface maps to construct a cortical connectivity atlas. Our atlas results show that the striato-cortical connectivity is not distinctive according to traditional structural subdivision of the striatum, and further confirms the holistic approach for mapping striatal surfaces.

3D High Resolution Mesh Deformation Based on Multi Library Wavelet Neural Network Architecture

This paper deals with the features of a novel technique for large Laplacian boundary deformations using estimated rotations. The proposed method is based on a Multi Library Wavelet Neural Network structure founded on several mother wavelet families (MLWNN). The objective is to align features of mesh and minimize distortion with a fixed feature that minimizes the sum of the distances between all corresponding vertices. New mesh deformation method worked in the domain of Region of Interest (ROI). Our approach computes deformed ROI, updates and optimizes it to align features of mesh based on MLWNN and spherical parameterization configuration. This structure has the advantage of constructing the network by several mother wavelets to solve high dimensions problem using the best wavelet mother that models the signal better. The simulation test achieved the robustness and speed considerations when developing deformation methodologies. The Mean-Square Error and the ratio of deformation are low compared to other works from the state of the art. Our approach minimizes distortions with fixed features to have a well reconstructed object.

Multi-layer compression algorithm for 3D deformed mesh based on multi library wavelet neural network architecture

We propose in this paper a 3D mesh compression algorithm for 3D deformation objects to facilitate the transmission of deformed object to another. This algorithm allows eliminating an object in the sequence of deformed objects and reducing the information needed to represent the geometry of a mesh sequence. In our approach, we used Multi Library Wavelet Neural Network architecture (MLWNN) to align features of mesh and minimize distortion with fixed features. The introduced method minimizes the sum of the distances between all the corresponding vertices. It computes deformed ROI (Region Of Interest), updates and optimizes it to align the mesh features. First, our compression was performed using spherical geometrical image obtained by our trust region spherical parameterization. Geometrical images also facilitate compression and level-of-detail control. Second, the spherical wavelet transformation was used to decompose the geometrical image into multi-resolution sub-images characterizing the underlying functions in a local fashion in both spatial and frequency domains. Experimental results show that the progressive compression algorithm yields efficient compression capabilities with minimal set of features used to have good deformation scheme.

Fast mapping and morphing for genus-zero meshes with cross spherical parameterization

We introduce an efficient approach to establish surface correspondences between genus-zero triangle meshes and animate a morph between them. We employ the concept of mesh simplification and refinement to progressively parameterize meshes using a novel kernel sampling algorithm. Our two-step feature alignment method optimizes the distribution of vertices in a parametric domain. A new mesh is generated after merging mesh topologies, and it represents the shapes of all input meshes and their transitions in-between. Our approach merges topological structures rather than mapping surface properties. Complex meshes with tens of thousands of vertices can be parameterized and merged in a few minutes. The advantages of our approach are demonstrated through 3D morphing applications. Comparing to existing approaches, our solution advances in performance while maintaining high-quality parameterization and morphing.

Spherical Parameterization Balancing Angle and Area Distortions.

This work presents a novel framework for spherical mesh parameterization. An efficient angle-preserving spherical parameterization algorithm is introduced, which is based on dynamic Yamabe flow and the conformal welding method with solid theoretic foundation. An area-preserving spherical parameterization is also discussed, which is based on discrete optimal mass transport theory. Furthermore, a spherical parameterization algorithm, which is based on the polar decomposition method, balancing angle distortion and area distortion is presented. The algorithms are tested on 3D geometric data and the experiments demonstrate the efficiency and efficacy of the proposed methods.

Spherical Conformal Parameterization of Genus-0 Point Clouds for Meshing

Point cloud is the most fundamental representation of 3D geometric objects. Analyzing and processing point cloud surfaces is important in computer graphics and computer vision. However, most of the existing algorithms for surface analysis require connectivity information. Therefore, it is desirable to develop a mesh structure on point clouds. This task can be simplified with the aid of a parameterization. In particular, conformal parameterizations are advantageous in preserving the geometric information of the point cloud data. In this paper, we extend a state-of-the-art spherical conformal parameterization algorithm for genus-0 closed meshes to the case of point clouds, using an improved approximation of the Laplace-Beltrami operator on data points. Then, we propose an iterative scheme called the North-South reiteration for achieving a spherical conformal parameterization. A balancing scheme is introduced to enhance the distribution of the spherical parameterization. High quality triangulations and quadrangulations can then be built on the point clouds with the aid of the parameterizations. Also, the meshes generated are guaranteed to be genus-0 closed meshes. Moreover, using our proposed spherical conformal parameterization, multilevel representations of point clouds can be easily constructed. Experimental results demonstrate the effectiveness of our proposed framework.

Fast spherical quasiconformal parameterization of genus-$0$ closed surfaces with application to adaptive remeshing

In this work, we are concerned with the spherical quasiconformal parameterization of genus-0 closed surfaces. Given a genus-0 closed triangulated surface and an arbitrary user-defined quasiconformal distortion, we propose a fast algorithm for computing a spherical parameterization of the surface that satisfies the prescribed distortion. The proposed algorithm can be effectively applied to adaptive surface remeshing for improving the visualization in computer graphics and animations. Experimental results are presented to illustrate the effectiveness of our algorithm.

Source-Type-Specific Inversion of Moment Tensors

The mapping of the fit of seismic moment tensor (MT) solutions in source-type space helps to characterize uncertainty and solution uniqueness. Current practice relies on the forward testing of a distribution of randomly generated MTs in source-type space, which is slow and does not necessarily recover the true maximum fit surface. We design an iterative damped least-squares inversion scheme to invert waveforms and/or P -wave first motions for best-fitting MT solutions for specific source types. An event associated with the sinkhole at the Napoleonville salt dome, Louisiana, an industrial quarry explosion, and an earthquake at The Geysers geothermal field, northern California, are presented as examples. We find that the inversion method is more accurate and successful than the random-search approach in recovering the region of best-fitting MT solutions or source types and is substantially faster. The approach also enables the determination of the best-fitting MT for specified source types such as pure double couples, tensile cracks, or explosions, as well as compound mechanisms in a single numerical framework. Online Material: Description and figures comparing inversion performance using spherical and Cartesian eigenvector parameterizations; figures showing the revised full moment tensor solution of event TE1 and additional network sensitivity solutions of event TE3.

Towards Quantitative Optical Cross Sections in Entomological Laser Radar - Potential of Temporal and Spherical Parameterizations for Identifying Atmospheric Fauna.

In recent years, the field of remote sensing of birds and insects in the atmosphere (the aerial fauna) has advanced considerably, and modern electro-optic methods now allow the assessment of the abundance and fluxes of pests and beneficials on a landscape scale. These techniques have the potential to significantly increase our understanding of, and ability to quantify and manage, the ecological environment. This paper presents a concept whereby laser radar observations of atmospheric fauna can be parameterized and table values for absolute cross sections can be catalogued to allow for the study of focal species such as disease vectors and pests. Wing-beat oscillations are parameterized with a discrete set of harmonics and the spherical scatter function is parameterized by a reduced set of symmetrical spherical harmonics. A first order spherical model for insect scatter is presented and supported experimentally, showing angular dependence of wing beat harmonic content. The presented method promises to give insights into the flight heading directions of species in the atmosphere and has the potential to shed light onto the km-range spread of pests and disease vectors.

An analytical representation of conformal mapping for genus-zero implicit surfaces and its application to surface shape similarity assessment

This paper develops an analytical representation of conformal mapping for genus-zero implicit surfaces based on algebraic polynomial functions, and its application to surface shape similarity assessment. Generally, the conformal mapping often works as a tool of planar or spherical parameterization for triangle mesh surfaces. It is further exploited for implicit surface matching in this study. The method begins with discretizing one implicit surface by triangle mesh, where a discrete harmonic energy model related to both the mesh and the other implicit surface is established based on a polynomial-function mapping. Then both the zero-center constraint and the landmark constraints are added to the model to ensure the uniqueness of mapping result with the Möbius transformation. By searching optimal polynomial coefficients with the Lagrange–Newton method, the analytical representation of conformal mapping is obtained, which reveals all global and continuous one-to-one correspondent point pairs between two implicit surfaces. Finally, a shape similarity assessment index for (two) implicit surfaces is proposed through calculating the differences of all the shape index values among those corresponding points. The proposed analytical representation method of conformal mapping and the shape assessment index are both verified by the simulation cases for the closed genus-zero implicit surfaces. Experimental results show that the method is effective for genus-zero implicit surfaces, which will offer a new way for object retrieval and manufactured surface inspection.

Predicting anatomical landmarks and bone morphology of the femur using local region matching.

Anatomical landmarks and bony features are frequently used in biomechanical and surgical applications. The purpose of this study was to develop a local region matching-based anatomical landmark prediction method. A reference femur model with anatomical landmarks and a surface division map was prepared. Initial registration between the reference femur model and a target femur model was performed in three-dimensional Cartesian space, and closest point pairs were determined by the initial surface correspondence. The models were mapped to unit spheres through spherical parameterization. Spherical registration using the closest point pairs in the spherical parametric space enabled the application of a division map from the reference model to the target model. The reference and target models were divided into local regions defined in the division map, and the corresponding regions were again registered in Cartesian space. Anatomical landmarks in the local regions were identified in the target model. The accuracy of the proposed method was tested for anatomical landmarks marked by a clinician on 35 femoral models. The effectiveness of local region matching was demonstrated by automatic measurements of the femoral neck-shaft angle. The average prediction error for all eight anatomical landmarks of the femur was 2.74 (±1.78) mm. The average of the predicted neck-shaft angle for our Korean subjects was 126.41° (±4.92°), which was comparable to previous studies in Japanese and Chinese populations. Anatomical landmarks and features could be accurately predicted using the proposed local region matching method. This method offers robustness and accuracy in determining anatomical bony landmarks and bone morphology for clinical and biomechanical applications.

FLASH: Fast Landmark Aligned Spherical Harmonic Parameterization for Genus-0 Closed Brain Surfaces

Surface registration between cortical surfaces is crucial in medical imaging for performing systematic comparisons between brains. Landmark-matching registration that matches anatomical features, called the sulcal landmarks, is often required to obtain a meaningful 1-1 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. Landmark-matching surface registration can then be obtained from the landmark aligned parameterizations. For genus-0 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 quasi-conformal theories. Also, by adjusting the Beltrami differential of the mapping, a diffeomorphic (1-1, 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.

Iris Compression and Recognition using Spherical Geometry Image

this research is considered to be a research to attract attention to the 3D iris compression to store the database of the iris. Actually, the 3D iris database cannot be found and in trying to solve this problem 2D iris database images are converted to 3D images just to implement the compression techniques used in 3D domain to test it and give an approximation results or to focus on this new direction in research. In this research a fully automated 3D iris compression and recognition system is presented. We use spherical based wavelet coefficients for efficient representation of the 3D iris. The spherical wavelet transformation is used to decompose the iris image into multi-resolution sub images. The representation of features based on spherical wavelet parameterization of the iris image was proposed for the 3D iris compression system. To evaluate the performance of the proposed approach, experiments were performed on the CASIA Iris database. Experimental results show that the spherical wavelet coefficients yield excellent compression capabilities with minimal set of features. Haar wavelet coefficients extracted from the iris image was found to generate good recognition results.