Ricci Flow Method Research Articles

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.

Conformal freeform surfaces

The conformality of freeform surfaces highly affects their rendering and tessellation results. To improve the conformality of freeform surfaces, a novel freeform surface representation named hierarchical freeform surfaces is presented in this paper. The conformality energy of hierarchical freeform surfaces is first formulated and its numerical approximation is then constructed using the composite Simpson’s rule. By constructing the parameterization of the initial freeform transformation using the Ricci flow method, the optimal freeform transformation is obtained by the Levenberg–Marquardt method, which is further interleaved with the freeform refinement procedure to generate a hierarchical freeform surface with bounded conformality deviations. Examples are given to show the performance of our algorithm for rendering and tessellation applications.

Conformal invariants for multiply connected surfaces: Application to landmark curve-based brain morphometry analysis.

Landmark curves were widely adopted in neuroimaging research for surface correspondence computation and quantified morphometry analysis. However, most of the landmark based morphometry studies only focused on landmark curve shape difference. Here we propose to compute a set of conformal invariant-based shape indices, which are associated with the landmark curve induced boundary lengths in the hyperbolic parameter domain. Such shape indices may be used to identify which surfaces are conformally equivalent and further quantitatively measure surface deformation. With the surface Ricci flow method, we can conformally map a multiply connected surface to the Poincaré disk. Our algorithm provides a stable method to compute the shape index values in the 2D (Poincaré Disk) parameter domain. The proposed shape indices are succinct, intrinsic and informative. Experimental results with synthetic data and 3D MRI data demonstrate that our method is invariant under isometric transformations and able to detect brain surface abnormalities. We also applied the new shape indices to analyze brain morphometry abnormalities associated with Alzheimer' s disease (AD). We studied the baseline MRI scans of a set of healthy control and AD patients from the Alzheimer' s Disease Neuroimaging Initiative (ADNI: 30 healthy control subjects vs. 30 AD patients). Although the lengths of the landmarks in Euclidean space, cortical surface area, and volume features did not differ between the two groups, our conformal invariant based shape indices revealed significant differences by Hotelling' s T2 test. The novel conformal invariant shape indices may offer a new sensitive biomarker and enrich our brain imaging analysis toolset for studying diagnosis and prognosis of AD.

OnionMap: A Scalable Geometric Addressing and Routing Scheme for 3D Sensor Networks

Geometric routing or geo-routing has been shown as a promising approach to scalable routing in sensor networks. Despite its success in 2-D networks, very few designs are available for 3-D networks that can ensure short routes using only small per-node state, without incurring high load imbalance on the nodes. In this paper, we propose a novel addressing and routing scheme, i.e., OnionMap, for 3-D sensor networks that achieve the above goals, using solely connectivity information and at a linear message cost. The key idea is to decompose a 3-D network into a set of connected layers, which are then mapped to a set of concentric sphere structures (similar to an onion). On each sphere, a discrete Ricci flow method is used to assign each node a set of coordinates that permits purely greedy routing within that sphere; across the different spheres, a layer alignment algorithm helps rotate and scale the spheres, to form a coherent global coordinate system that guides global routing. Theoretical analysis and simulation show OnionMap's advantages over state-of-the-art solutions in path stretch, per-node storage, and load balance.

Studying ventricular abnormalities in mild cognitive impairment with hyperbolic Ricci flow and tensor-based morphometry

Mild Cognitive Impairment (MCI) is a transitional stage between normal aging and dementia and people with MCI are at high risk of progression to dementia. MCI is attracting increasing attention, as it offers an opportunity to target the disease process during an early symptomatic stage. Structural magnetic resonance imaging (MRI) measures have been the mainstay of Alzheimer's disease (AD) imaging research, however, ventricular morphometry analysis remains challenging because of its complicated topological structure. Here we describe a novel ventricular morphometry system based on the hyperbolic Ricci flow method and tensor-based morphometry (TBM) statistics. Unlike prior ventricular surface parameterization methods, hyperbolic conformal parameterization is angle-preserving and does not have any singularities. Our system generates a one-to-one diffeomorphic mapping between ventricular surfaces with consistent boundary matching conditions. The TBM statistics encode a great deal of surface deformation information that could be inaccessible or overlooked by other methods. We applied our system to the baseline MRI scans of a set of MCI subjects from the Alzheimer's Disease Neuroimaging Initiative (ADNI: 71 MCI converters vs. 62 MCI stable). Although the combined ventricular area and volume features did not differ between the two groups, our fine-grained surface analysis revealed significant differences in the ventricular regions close to the temporal lobe and posterior cingulate, structures that are affected early in AD. Significant correlations were also detected between ventricular morphometry, neuropsychological measures, and a previously described imaging index based on fluorodeoxyglucose positron emission tomography (FDG-PET) scans. This novel ventricular morphometry method may offer a new and more sensitive approach to study preclinical and early symptomatic stage AD.

Brain morphometry on congenital hand deformities based on Teichmüller space theory

Congenital Hand Deformities (CHD) usually occurred between the fourth and the eighth week after the embryo is formed. Failure of the transformation from arm bud cells to upper limb can lead to an abnormal appearing/functioning upper extremity which is presented at birth. Some causes are linked to genetics while others are affected by the environment, and the rest have remained unknown. CHD patients develop prehension through the use of their hands, which affects the brain as time passes. In recent years, CHD have gained increasing attention and researches have been conducted on CHD, both surgically and psychologically. However, the impacts of CHD on the brain structure are not well-understood so far. Here, we propose a novel approach to apply Teichmüller space theory and conformal welding method to study brain morphometry in CHD patients. Conformal welding signature reflects the geometric relations among different functional areas on the cortex surface, which is intrinsic to the Riemannian metric, invariant under conformal deformation, and encodes complete information of the functional area boundaries. The computational algorithm is based on discrete surface Ricci flow, which has theoretic guarantees for the existence and uniqueness of the solutions. In practice, discrete Ricci flow is equivalent to a convex optimization problem, therefore has high numerically stability. In this paper, we compute the signatures of contours on general 3D surfaces with the surface Ricci flow method, which encodes both global and local surface contour information. Then we evaluated the signatures of pre-central and post-central gyrus on healthy control and CHD subjects for analyzing brain cortical morphometry. Preliminary experimental results from 3D MRI data of CHD/control data demonstrate the effectiveness of our method. The statistical comparison between left and right brain gives us a better understanding on brain morphometry of subjects with Congenital Hand Deformities, in particular, missing the distal part of the upper limb.

Ricci flow-based spherical parameterization and surface registration

This paper presents an improved Euclidean Ricci flow method for spherical parameterization. We subsequently invent a scale space processing built upon Ricci energy to extract robust surface features for accurate surface registration. Since our method is based on the proposed Euclidean Ricci flow, it inherits the properties of Ricci flow such as conformality, robustness and intrinsicalness, facilitating efficient and effective surface mapping. Compared with other surface registration methods using curvature or sulci pattern, our method demonstrates a significant improvement for surface registration. In addition, Ricci energy can capture local differences for surface analysis as shown in the experiments and applications.

Teichmüller Shape Descriptor and Its Application to Alzheimer’s Disease Study

We propose a novel method to apply Teichmüller 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 signature—Teichmüller shape descriptor—is 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 Alzheimer’s disease neuroimaging initiative (ADNI) dataset [152 healthy control subjects versus 169 Alzheimer’s disease (AD) patients] demonstrate the effectiveness of our method and illustrate its potential as a novel surface-based cortical morphometry measurement in AD research.

Manifold splines with a single extraordinary point

This paper develops a novel computational technique to define and construct manifold splines with only one singular point by employing the rigorous mathematical theory of Ricci flow. The central idea and new computational paradigm of manifold splines are to systematically extend the algorithmic pipeline of spline surface construction from any planar domain to an arbitrary topology. As a result, manifold splines can unify planar spline representations as their special cases. Despite its earlier success, the existing manifold spline framework is plagued by the topology-dependent, large number of singular points (i.e., | 2 g − 2 | for any genus- g surface), where the analysis of surface behaviors such as continuity remains extremely difficult. The unique theoretical contribution of this paper is that we devise new mathematical tools so that manifold splines can now be constructed with only one singular point, reaching their theoretic lower bound of singularity for real-world applications. Our new algorithm is founded upon the concept of discrete Ricci flow and associated techniques. First, Ricci flow is employed to compute a special metric of any manifold domain (serving as a parametric domain for manifold splines), such that the metric becomes flat everywhere except at one point. Then, the metric naturally induces an affine atlas covering the entire manifold except this singular point. Finally, manifold splines are defined over this affine atlas. The Ricci flow method is theoretically sound, and practically simple and efficient. We conduct various shape experiments and our new theoretical and algorithmic results alleviate the modeling difficulty of manifold splines, and hence, promote the widespread use of manifold splines in surface and solid modeling, geometric design, and reverse engineering.

Variational principles for branched circle patterns

We are concerned with variational principles for branched circle patterns. We use critical point theory to establish existence and uniqueness. Furthermore, we apply the Ricci flow method to give an approximation scheme which converges exponentially to the branched circle pattern metric.

Exotic Structures and the Limitations of Certain Analytical Methods in Geometry

In this survey we review some results concerning negatively curved exotic structures (DIFF and PL) and its (unexpected) implications on the limitations of some analytic methods in geometry. Among these methods are the harmonic map method and the Ricci flow method. First in section 1 we mention certain results about the rigidity of negatively curved manifolds. In section 2 and 3 we survey some results concerning the limitations of the harmonic map technique and the natural map technique for negatively curved manifolds. Finally,in section 4, we mention some limitations of the Ricci flow method for pinched negatively curved manifolds. We are grateful to J-F. Lafont and R. Spatzier for the useful information they provided to us and to E.Gasparim for suggesting some improvements in the text. We are also grateful to the referee for pointing out certain inaccuracies.