Diffeomorphic Mesh Deformation via Efficient Optimal Transport for Cortical Surface Reconstruction

TL;DR

A novel metric is proposed for learning mesh deformation which is defined by sliced Wasserstein distance on meshes represented as probability measures that generalize the set-based approach and surpasses other competing methods in multiple datasets and metrics.

Abstract

Mesh deformation plays a pivotal role in many 3D vision tasks including dynamic simulations, rendering, and reconstruction. However, defining an efficient discrepancy between predicted and target meshes remains an open problem. A prevalent approach in current deep learning is the set-based approach which measures the discrepancy between two surfaces by comparing two randomly sampled point-clouds from the two meshes with Chamfer pseudo-distance. Nevertheless, the set-based approach still has limitations such as lacking a theoretical guarantee for choosing the number of points in sampled point-clouds, and the pseudo-metricity and the quadratic complexity of the Chamfer divergence. To address these issues, we propose a novel metric for learning mesh deformation. The metric is defined by sliced Wasserstein distance on meshes represented as probability measures that generalize the set-based approach. By leveraging probability measure space, we gain flexibility in encoding meshes using diverse forms of probability measures, such as continuous, empirical, and discrete measures via varifold representation. After having encoded probability measures, we can compare meshes by using the sliced Wasserstein distance which is an effective optimal transport distance with linear computational complexity and can provide a fast statistical rate for approximating the surface of meshes. To the end, we employ a neural ordinary differential equation (ODE) to deform the input surface into the target shape by modeling the trajectories of the points on the surface. Our experiments on cortical surface reconstruction demonstrate that our approach surpasses other competing methods in multiple datasets and metrics.

Figures (6)

• Figure 1: 2D deformation toy example. We deform a (a) template circle to a (b) target polygon via an optimization-based setting. (c) Points are uniformly sampled from the two contours. CD loss is easily trapped at local minima, i.e. the green points are not uniformly distributed on the target contour as desired. In contrast, (e) SWD loss can find the optimal transport plan among discrete probability measures, i.e. the resulting points are distributed more uniformly along the contour. More details about this toy example can be found in the Appendix \ref{['sec:toy-example']}.
• Figure 2: Qualitative results of white matter surface reconstruction. The color represents the point-to-face distance, i.e., the darker color is, the further the predicted mesh to the pseudo-ground truth. More visualization is given in Appendix \ref{['sec:visualizations']} and the video supplementary.
• Figure 3: Running time comparison. The diagram indicates the scalability of three presented losses. The lines imply the losses computed between two sets of points in 3D-coordinate, the dashed lines with dots represent the loss computed between two varifolds. OV denotes oriented varifold, Reg denotes regularization.
• Figure 4: Comparison between SWD loss (left) and CD loss (right). The mesh obtained through probability measure representation and SWD optimization exhibits a more uniformly surfaced appearance compared to the set-based approach that optimizes with CD loss.
• Figure 5: Visualization of the optimization process of 2D toy example. The set of green points, i.e. sampled points from the template circle, optimized by CD loss often concentrates around the acute region of the polygon and easily gets trapped at some local regions. Nonetheless, the set of points optimized by SWD loss distributes more uniformly along the edge of the polygon, thus making the optimization process more robust.

Theorems & Definitions (1)

• Theorem 1