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

Abstract

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.

Topology-preserving smoothing of retinotopic maps

“…To ensure topological condition, for any mapping whose |μ| is greater than 1, we scale its magnitude by μ 0 = μ/(|μ| +�). The procedure is called "topological projection" [48,49] or "chopping" [50]. The goal is to find a topological mapping that is close to the original non-topological mapping.…”

Ricci Planner: Zero-Shot Transfer for Goal-Conditioned Reinforcement Learning via Geometric Flow

“…Therefore, directly applying gradient-based optimization methods to this problem can result in suboptimal or infeasible plan. To address these issues, we applied a transformation process called the Ricci flow, which has been consistently used in the field of machine learning ( [8], [13], [25], [48], [49]). In this work, we utilized the Ricci flow to transform the target manifold into a simpler manifold.…”

Differential and Integral Invariants Under Möbius Transformation

“…The descriptions of conformal mapping contain angle preservation [12,26,5], metric rescaling [21,27], preservation of circles [14,28], etc. Some key ideas reside in the conformal surface geometry are Dirac equation [6], Cauchy-Riemann equation [22], Möbius transformations [27,28], Riemann mapping [10,9,35,33], Ricci flow [34], etc. The conformal geometry lies between the topology geometry and the Riemannian geometry, it studies the invariants of the conformal transformation group.…”