[Paper Review] Learning shape correspondence with anisotropic convolutional neural networks

Davide Boscaini, Jonathan Masci+2 more|arXiv (Cornell University)|May 20, 2016

Image Retrieval and Classification Techniques176 citations

TL;DR

The paper introduces Anisotropic Convolutional Neural Networks (ACNN) to learn dense intrinsic shape correspondences on non-Euclidean domains using anisotropic heat kernels as spatial patches, achieving state-of-the-art results on challenging benchmarks.

ABSTRACT

Establishing correspondence between shapes is a fundamental problem in geometry processing, arising in a wide variety of applications. The problem is especially difficult in the setting of non-isometric deformations, as well as in the presence of topological noise and missing parts, mainly due to the limited capability to model such deformations axiomatically. Several recent works showed that invariance to complex shape transformations can be learned from examples. In this paper, we introduce an intrinsic convolutional neural network architecture based on anisotropic diffusion kernels, which we term Anisotropic Convolutional Neural Network (ACNN). In our construction, we generalize convolutions to non-Euclidean domains by constructing a set of oriented anisotropic diffusion kernels, creating in this way a local intrinsic polar representation of the data (`patch'), which is then correlated with a filter. Several cascades of such filters, linear, and non-linear operators are stacked to form a deep neural network whose parameters are learned by minimizing a task-specific cost. We use ACNNs to effectively learn intrinsic dense correspondences between deformable shapes in very challenging settings, achieving state-of-the-art results on some of the most difficult recent correspondence benchmarks.

Motivation & Objective

• Motivate and address dense intrinsic shape correspondence under non-rigid deformations and data imperfections.
• Develop an intrinsic, deformation-invariant CNN framework applicable to meshes and point clouds.
• Leverage anisotropic diffusion to form directional local patches that enable robust learning of correspondences.

Proposed method

• Define anisotropic heat kernel based patch operator D(x)f with orientation ( θ) and scale t using h_{\,αθ t}(x,ξ).
• Construct intrinsic convolution using patches: (f * a)(x) = ∫ a(θ,t) (D(x)f)(θ,t) dtdθ.
• Assemble ACNN with layers: Fully Connected, Intrinsic Convolution, Dropout, Batch Normalization, and Softmax for dense correspondence.
• Train by minimizing multinomial regression loss to produce a soft correspondence f_Θ(x,y) between shapes.
• Refine correspondences via a functional map approach using confident points to solve Φ_I C = Ψ_I and re-derive point-wise matches.

Experimental results

Research questions

• RQ1 How can intrinsic, direction-aware convolutions be defined on non-Euclidean surfaces to learn dense correspondences?
• RQ2 Can anisotropic diffusion-based patches improve robustness to non-isometric deformations and topological noise compared to prior intrinsic CNNs?
• RQ3 Does the ACNN framework generalize across shape representations (meshes, point clouds) and outperform existing intrinsic methods on benchmarks?
• RQ4 Can soft correspondences learned by ACNN be refined into accurate dense mappings using functional maps?

Key findings

• ACNN enables intrinsic, deformation-invariant learning on non-Euclidean domains using anisotropic heat kernels for patch construction.
• The framework combines advantages of prior intrinsic models while avoiding their drawbacks, including dependence on injectivity radius and mesh-specific charting.
• ACNN achieves state-of-the-art results on challenging dense correspondence benchmarks relative to GCNN and ADD variants.
• The method yields reliable soft correspondences that can be refined into denser, more accurate point-wise mappings via functional map-based refinement.