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LISR: Learning Linear 3D Implicit Surface Representation Using Compactly Supported Radial Basis Functions

Atharva Pandey, Vishal Yadav, Rajendra Nagar, Santanu Chaudhury

TL;DR

This work addresses implicit 3D surface reconstruction from partial, noisy point clouds by introducing a memory-efficient Linear Implicit Surface Representation (LISR) based on compactly supported radial basis functions (CSRBF). A neural-network-driven framework learns the linear coefficients for the CSRBF expansion, guided by ground-truth signed-distance fields (SDF), and is designed to satisfy a convergence condition on the associated linear system to ensure reliable optimization. The key contributions are the CSRBF-based LISR, a convergence-aware basis/query-point design, and empirical validation showing improved Chamfer distance and competitive F-scores, with demonstrated applicability to shape completion. The approach offers a scalable, topology-agnostic representation suitable for high-resolution 3D reconstruction tasks from partial scans.

Abstract

Implicit 3D surface reconstruction of an object from its partial and noisy 3D point cloud scan is the classical geometry processing and 3D computer vision problem. In the literature, various 3D shape representations have been developed, differing in memory efficiency and shape retrieval effectiveness, such as volumetric, parametric, and implicit surfaces. Radial basis functions provide memory-efficient parameterization of the implicit surface. However, we show that training a neural network using the mean squared error between the ground-truth implicit surface and the linear basis-based implicit surfaces does not converge to the global solution. In this work, we propose locally supported compact radial basis functions for a linear representation of the implicit surface. This representation enables us to generate 3D shapes with arbitrary topologies at any resolution due to their continuous nature. We then propose a neural network architecture for learning the linear implicit shape representation of the 3D surface of an object. We learn linear implicit shapes within a supervised learning framework using ground truth Signed-Distance Field (SDF) data for guidance. The classical strategies face difficulties in finding linear implicit shapes from a given 3D point cloud due to numerical issues (requires solving inverse of a large matrix) in basis and query point selection. The proposed approach achieves better Chamfer distance and comparable F-score than the state-of-the-art approach on the benchmark dataset. We also show the effectiveness of the proposed approach by using it for the 3D shape completion task.

LISR: Learning Linear 3D Implicit Surface Representation Using Compactly Supported Radial Basis Functions

TL;DR

This work addresses implicit 3D surface reconstruction from partial, noisy point clouds by introducing a memory-efficient Linear Implicit Surface Representation (LISR) based on compactly supported radial basis functions (CSRBF). A neural-network-driven framework learns the linear coefficients for the CSRBF expansion, guided by ground-truth signed-distance fields (SDF), and is designed to satisfy a convergence condition on the associated linear system to ensure reliable optimization. The key contributions are the CSRBF-based LISR, a convergence-aware basis/query-point design, and empirical validation showing improved Chamfer distance and competitive F-scores, with demonstrated applicability to shape completion. The approach offers a scalable, topology-agnostic representation suitable for high-resolution 3D reconstruction tasks from partial scans.

Abstract

Implicit 3D surface reconstruction of an object from its partial and noisy 3D point cloud scan is the classical geometry processing and 3D computer vision problem. In the literature, various 3D shape representations have been developed, differing in memory efficiency and shape retrieval effectiveness, such as volumetric, parametric, and implicit surfaces. Radial basis functions provide memory-efficient parameterization of the implicit surface. However, we show that training a neural network using the mean squared error between the ground-truth implicit surface and the linear basis-based implicit surfaces does not converge to the global solution. In this work, we propose locally supported compact radial basis functions for a linear representation of the implicit surface. This representation enables us to generate 3D shapes with arbitrary topologies at any resolution due to their continuous nature. We then propose a neural network architecture for learning the linear implicit shape representation of the 3D surface of an object. We learn linear implicit shapes within a supervised learning framework using ground truth Signed-Distance Field (SDF) data for guidance. The classical strategies face difficulties in finding linear implicit shapes from a given 3D point cloud due to numerical issues (requires solving inverse of a large matrix) in basis and query point selection. The proposed approach achieves better Chamfer distance and comparable F-score than the state-of-the-art approach on the benchmark dataset. We also show the effectiveness of the proposed approach by using it for the 3D shape completion task.
Paper Structure (4 sections, 1 theorem, 3 equations, 1 figure)

This paper contains 4 sections, 1 theorem, 3 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Related Works
  3. Methodology
  4. Linear Implicit Surface Representation

Key Result

Theorem 1

The solution $\boldsymbol{\alpha}^*$ will be equal to the optimal solution of the linear system $\mathbf{V}^\top\boldsymbol{\alpha} = \mathbf{s_{\text{gt}}}$ if and only if $\mathbf{VV}^\top$ is a full rank matrix.

Figures (1)

  • Figure :

Theorems & Definitions (2)

  • Theorem 1
  • proof