DDM: A Metric for Comparing 3D Shapes Using Directional Distance Fields

TL;DR

The paper addresses the challenge of quantifying discrepancies between 3D geometric models without relying on costly explicit point correspondences. It introduces DDM, a differentiable distance metric built on Directional Distance Fields (DDFs) evaluated over a shared reference domain, enabling robust and efficient comparisons for both point clouds and triangle meshes. The authors prove basic metric properties, show that CD and P2F are special cases, and demonstrate DDM’s effectiveness across template fitting, rigid/non-rigid registration, scene flow estimation, and SMPL pose optimization, with extensive ablations. The approach yields superior accuracy and practical performance gains, highlighting the potential of implicit, reference-domain-based metrics to advance 3D geometry processing in real-world tasks.

Abstract

Qualifying the discrepancy between 3D geometric models, which could be represented with either point clouds or triangle meshes, is a pivotal issue with board applications. Existing methods mainly focus on directly establishing the correspondence between two models and then aggregating point-wise distance between corresponding points, resulting in them being either inefficient or ineffective. In this paper, we propose DDM, an efficient, effective, robust, and differentiable distance metric for 3D geometry data. Specifically, we construct DDM based on the proposed implicit representation of 3D models, namely directional distance field (DDF), which defines the directional distances of 3D points to a model to capture its local surface geometry. We then transfer the discrepancy between two 3D geometric models as the discrepancy between their DDFs defined on an identical domain, naturally establishing model correspondence. To demonstrate the advantage of our DDM, we explore various distance metric-driven 3D geometric modeling tasks, including template surface fitting, rigid registration, non-rigid registration, scene flow estimation and human pose optimization. Extensive experiments show that our DDM achieves significantly higher accuracy under all tasks. As a generic distance metric, DDM has the potential to advance the field of 3D geometric modeling. The source code is available at https://github.com/rsy6318/DDM.

Figures (15)

• Figure 1: Visual illustration of different distance metrics for 3D geometry data. For convenience, we use 2D illustration. The yellow points in (a), (b), (c), and (d) refer to 3D points located on 3D surfaces indicated by curves, and the brown points in (e) represent the generated reference points. The blue arrows represent the established correspondence. The color in (e) changes from green to red indicating the distance fields of the two surfaces indicated by curves (i.e., the set of the distances of arbitrary points in 3D space to the surfaces).
• Figure 2: (a) Direct correspondence establishment between 2D RGB images through the pixel locations uniformly distributed on a regular 2D grid (the blue points). (b) Indirect correspondence establishment between 3D geometry shapes through a set of additional reference points highlighted in blue distributed near the surfaces. Note that the two 3D shapes share an identical set of reference points.
• Figure 3: Illustration of the procedure for generating reference points. Here, reference points in blue are generated by adding offsets to the points sampled in orange on the surface.
• Figure 4: Visualization of the closest point estimation on the surface, depicted as (a) point clouds and (b) triangle meshes.
• Figure 5: Visual illustration of the confidence scores for reference points in both overlapping and non-overlapping regions. Here, $\mathbf{q}_1$ resides in the overlapping region, while $\mathbf{q}_2$ is situated in the non-overlapping region, leading to $s(\mathbf{q}_1)>s(\mathbf{q}_2)$.

Theorems & Definitions (6)

• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3
• proof