Hyperbolic Harmonic Mapping for Surface Registration

Manifold-Constrained Geometric Optimization via Local Parameterizations

Many geometric optimization problems contain manifold constraints that restrict the optimized vertices on some specified manifold surface. The constraints are highly nonlinear and non-convex, therefore existing methods usually suffer from a breach of condition or low optimization quality. In this article, we present a novel divide-and-conquer methodology for manifold-constrained geometric optimization problems. Central to our methodology is to use local parameterizations to decouple the optimization with hard constraints, which transforms nonlinear constraints into linear constraints. We decompose the input mesh into a set of developable or nearly-developable overlapping patches with disc topology, then flatten each patch into the planar domain with very low isometric distortion, optimize vertices with linear constraints and recover the patch. Finally, we project it onto the constrained manifold surface. We demonstrate the applicability and robustness of our methodology through a variety of geometric optimization tasks. Experimental results show that our method performs much better than existing methods.

A learning-based harmonic mapping: Framework, assessment, and case study of human-to-robot hand pose mapping

Harmonic mapping provides a natural way of mapping two manifolds by minimizing distortion induced by the mapping. However, most applications are limited to mapping between 2D and/or 3D spaces owing to the high computational cost. We propose a novel approach, the harmonic autoencoder (HAE), by approximating a harmonic mapping in a data-driven way. The HAE learns a mapping from an input domain to a target domain that minimizes distortion and requires only a small number of input–target reference pairs. The HAE can be applied to high-dimensional applications, such as human-to-robot hand pose mapping. Our method can map from the input to the target domain while minimizing distortion over the input samples, covering the target domain, and satisfying the reference pairs. This is achieved by extending an existing neural network method called the contractive autoencoder. Starting from a contractive autoencoder, the HAE takes into account a distance function between point clouds within the input and target domains, in addition to a penalty for estimation error on reference points. For efficiently selecting a set of input–target reference pairs during the training process, we introduce an adaptive optimization criterion. We demonstrate that pairs selected in this way yield a higher-performance mapping than pairs selected randomly, and the mapping is comparable to that from pairs selected heuristically by the experimenter. Our experimental results with synthetic data and human-to-robot hand pose data demonstrate that our method can learn an effective mapping between the input and target domains.

Optimizing Visual Cortex Parameterization with Error-Tolerant Teichmüller Map in Retinotopic Mapping

The mapping between the visual input on the retina to the cortical surface, i.e., retinotopic mapping, is an important topic in vision science and neuroscience. Human retinotopic mapping can be revealed by analyzing cortex functional magnetic resonance imaging (fMRI) signals when the subject is under specific visual stimuli. Conventional methods process, smooth, and analyze the retinotopic mapping based on the parametrization of the (partial) cortical surface. However, the retinotopic maps generated by this approach frequently contradict neuropsychology results. To address this problem, we propose an integrated approach that parameterizes the cortical surface, such that the parametric coordinates linearly relates the visual coordinate. The proposed method helps the smoothing of noisy retinotopic maps and obtains neurophysiological insights in human vision systems. One key element of the approach is the Error-Tolerant Teichmüller Map, which uniforms the angle distortion and maximizes the alignments to self-contradicting landmarks. We validated our overall approach with synthetic and real retinotopic mapping datasets. The experimental results show the proposed approach is superior in accuracy and compatibility. Although we focus on retinotopic mapping, the proposed framework is general and can be applied to process other human sensory maps.

Hyperbolic Wasserstein Distance for Shape Indexing

Shape space is an active research topic in computer vision and medical imaging fields. The distance defined in a shape space may provide a simple and refined index to represent a unique shape. This work studies the Wasserstein space and proposes a novel framework to compute the Wasserstein distance between general topological surfaces by integrating hyperbolic Ricci flow, hyperbolic harmonic map, and hyperbolic power Voronoi diagram algorithms. The resulting hyperbolic Wasserstein distance can intrinsically measure the similarity between general topological surfaces. Our proposed algorithms are theoretically rigorous and practically efficient. It has the potential to be a powerful tool for 3D shape indexing research. We tested our algorithm with human face classification and Alzheimer's disease (AD) progression tracking studies. Experimental results demonstrated that our work may provide a succinct and effective shape index.

Hierarchical spherical deformation for cortical surface registration

We present hierarchical spherical deformation for a group-wise shape correspondence to address template selection bias and to minimize registration distortion. In this work, we aim at a continuous and smooth deformation field to guide accurate cortical surface registration. In conventional spherical registration methods, a global rigid alignment and local deformation are independently performed. Motivated by the composition of precession and intrinsic rotation, we simultaneously optimize global rigid rotation and non-rigid local deformation by utilizing spherical harmonics interpolation of local composite rotations in a single framework. To this end, we indirectly encode local displacements by such local composite rotations as functions of spherical locations. Furthermore, we introduce an additional regularization term to the spherical deformation, which maximizes its rigidity while reducing registration distortion. To improve surface registration performance, we employ the second order approximation of the energy function that enables fast convergence of the optimization. In the experiments, we validate our method on healthy normal subjects with manual cortical surface parcellation in registration accuracy and distortion. We show an improved shape correspondence with high accuracy in cortical surface parcellation and significantly low registration distortion in surface area and edge length. In addition to validation, we discuss parameter tuning, optimization, and implementation design with potential acceleration.

Predicting Imminent Progression to Clinically Significant Memory Decline Using Volumetric MRI and FDG PET

BACKGROUND

Brain imaging measurements can provide evidence of possible preclinical Alzheimer's disease (AD). Their ability to predict individual imminent clinical conversion remains unclear.

OBJECTIVE

To investigate the ability of pre-specified volumetric magnetic resonance imaging (MRI) and fluorodeoxyglucose positron emission tomography (FDG-PET) measurements to predict which cognitively unimpaired older participants would subsequently progress to amnestic mild cognitive impairment (aMCI) within 2 years.

METHODS

From an apolipoprotein E4 (APOE4) enriched prospective cohort study, 18 participants subsequently progressed to the clinical diagnosis of aMCI or probable AD dementia within 1.8±0.8 years (progressors); 20 participants matched for sex, age, education, and APOE allele dose remained cognitively unimpaired for at least 4 years (nonprogressors). A complementary control group not matched for APOE allele dose included 35 nonprogressors. Groups were compared on baseline FDG-PET and MRI measures known to be preferentially affected in the preclinical and clinical stages of AD and by voxel-wise differences in regional gray matter volume and glucose metabolism. Receiver Operating Characteristic, binary logistic regression, and leave-one-out procedures were used to predict clinical outcome for the a priori measures.

RESULTS

Compared to non-progressors and regardless of APOE-matching, progressors had significantly reduced baseline MRI and PET measurements in brain regions preferentially affected by AD and reduced hippocampal volume was the strongest predictor of an individual's imminent progression to clinically significant memory decline (79% sensitivity/78% specificity among APOE-matched cohorts).

CONCLUSION

Regional MRI and FDG-PET measurements may be useful in predicting imminent progression to clinically significant memory decline.

Intrinsic 3D Dynamic Surface Tracking based on Dynamic Ricci Flow and Teichmüller Map

3D dynamic surface tracking is an important research problem and plays a vital role in many computer vision and medical imaging applications. However, it is still challenging to efficiently register surface sequences which has large deformations and strong noise. In this paper, we propose a novel automatic method for non-rigid 3D dynamic surface tracking with surface Ricci flow and Teichmüller map methods. According to quasi-conformal Teichmüller theory, the Techmüller map minimizes the maximal dilation so that our method is able to automatically register surfaces with large deformations. Besides, the adoption of Delaunay triangulation and quadrilateral meshes makes our method applicable to low quality meshes. In our work, the 3D dynamic surfaces are acquired by a high speed 3D scanner. We first identified sparse surface features using machine learning methods in the texture space. Then we assign landmark features with different curvature settings and the Riemannian metric of the surface is computed by the dynamic Ricci flow method, such that all the curvatures are concentrated on the feature points and the surface is flat everywhere else. The registration among frames is computed by the Teichmüller mappings, which aligns the feature points with least angle distortions. We apply our new method to multiple sequences of 3D facial surfaces with large expression deformations and compare them with two other state-of-the-art tracking methods. The effectiveness of our method is demonstrated by the clearly improved accuracy and efficiency.

Towards a Holistic Cortical Thickness Descriptor: Heat Kernel-Based Grey Matter Morphology Signatures

In this paper, we propose a heat kernel based regional shape descriptor that may be capable of better exploiting volumetric morphological information than other available methods, thereby improving statistical power on brain magnetic resonance imaging (MRI) analysis. The mechanism of our analysis is driven by the graph spectrum and the heat kernel theory, to capture the volumetric geometry information in the constructed tetrahedral meshes. In order to capture profound brain grey matter shape changes, we first use the volumetric Laplace-Beltrami operator to determine the point pair correspondence between white-grey matter and CSF-grey matter boundary surfaces by computing the streamlines in a tetrahedral mesh. Secondly, we propose multi-scale grey matter morphology signatures to describe the transition probability by random walk between the point pairs, which reflects the inherent geometric characteristics. Thirdly, a point distribution model is applied to reduce the dimensionality of the grey matter morphology signatures and generate the internal structure features. With the sparse linear discriminant analysis, we select a concise morphology feature set with improved classification accuracies. In our experiments, the proposed work outperformed the cortical thickness features computed by FreeSurfer software in the classification of Alzheimer's disease and its prodromal stage, i.e., mild cognitive impairment, on publicly available data from the Alzheimer's Disease Neuroimaging Initiative. The multi-scale and physics based volumetric structure feature may bring stronger statistical power than some traditional methods for MRI-based grey matter morphology analysis.