Implicit Neural Surface Deformation with Explicit Velocity Fields

TL;DR

The paper presents an unsupervised framework that jointly learns time-varying neural implicit surfaces and velocity fields to interpolate deformations between two point clouds. It keyizes a Modified Level-Set Equation to deform the implicit field directly, while enforcing Eikonal constraints and volume preservation via a divergence-free velocity field, enabling physically plausible intermediate shapes without ground-truth intermediates. The method uses two MLPs (Implicit-Net and Velocity-Net) and a compact loss combining MLSE, smoothness, and matching terms, trained end-to-end with forward Euler integration. Experiments across multiple datasets show competitive or superior interpolation quality and robustness to incomplete or sparse inputs, with favorable runtime compared to prior approaches and applicability to both rigid and non-rigid deformations.

Abstract

In this work, we introduce the first unsupervised method that simultaneously predicts time-varying neural implicit surfaces and deformations between pairs of point clouds. We propose to model the point movement using an explicit velocity field and directly deform a time-varying implicit field using the modified level-set equation. This equation utilizes an iso-surface evolution with Eikonal constraints in a compact formulation, ensuring the integrity of the signed distance field. By applying a smooth, volume-preserving constraint to the velocity field, our method successfully recovers physically plausible intermediate shapes. Our method is able to handle both rigid and non-rigid deformations without any intermediate shape supervision. Our experimental results demonstrate that our method significantly outperforms existing works, delivering superior results in both quality and efficiency.

Figures (24)

• Figure 1: Given two point clouds $\mathcal{P}_0$ and $\mathcal{P}_1$, our method predicts a time-varying neural implicit surface that represents a smooth and physically plausible deformation from $\mathcal{P}_0$ to $\mathcal{P}_1$. To ensure physical plausibility, we utilize a velocity network that leverages smoothness and divergence-free constraints.
• Figure 2: Pipeline of our method: given two point cloud $\mathcal{P}_0$ and $\mathcal{P}_1$, we train a time-varying Implicit-Net to predict SDF in different time steps and Velocity-Net to predict the velocity of the point at each time step. We directly deform the implicit field using MLSE loss.
• Figure 3: Experiment on extrinsic deformation. LipMLP liu2022learning and NISE Novello2023neural fail to estimate the physically plausible intermediate shapes. NFGP yang2021geometry can recover reasonable meshes but it is trained separately for each time step. Our method can recover realistic intermediate shapes in one model.
• Figure 4: Experiment involves both extrinsic and intrinsic deformation. While LipMLP( liu2022learning) and NISE( Novello2023neural) fail to create reasonable middle-step meshes, our method generates appropriate transition meshes from two given point clouds.
• Figure 5: Quantitative evaluation of the deformed shapes. Chamfer Distance (CD) scaled by $10^3$ and Hausdorff Distance (HD) scaled by $10^2$ for the $55$ intermediate shapes.