Quasi-Conformal Convolution : A Learnable Convolution for Deep Learning on Riemann Surfaces

2025-02-03

AI Summary

To address the lack of geometrically adaptive convolution operators on non-Euclidean surfaces (e.g., 3D facial meshes), this paper proposes Learnable Quasi-Conformal Convolution (QCC), the first deep learning framework integrating quasi-conformal mapping theory. QCC enables differentiable, task-driven adaptation of convolution kernels on Riemannian manifolds via learnable geometric parameterization. It unifies diverse spatial convolution paradigms and ensures numerically stable, fully differentiable operations on discrete surfaces. Leveraging QCC, we design QCCNN—a novel architecture that achieves significant improvements in surface image classification, 3D craniofacial analysis, and facial lesion segmentation. Notably, QCCNN demonstrates superior accuracy and robustness on irregular, metric-deficient geometric data prevalent in clinical applications, where conventional Euclidean or graph-based convolutions fail to capture intrinsic surface geometry effectively.

Technology Category

• Machine Learning: Learning with Manifolds
• Computer Vision: Learning & Optimization for CV
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

Deep learning on non-Euclidean domains is important for analyzing complex geometric data that lacks common coordinate systems and familiar Euclidean properties. A central challenge in this field is to define convolution on domains, which inherently possess irregular and non-Euclidean structures. In this work, we introduce Quasi-conformal Convolution (QCC), a novel framework for defining convolution on Riemann surfaces using quasi-conformal theories. Each QCC operator is linked to a specific quasi-conformal mapping, enabling the adjustment of the convolution operation through manipulation of this mapping. By utilizing trainable estimator modules that produce Quasi-conformal mappings, QCC facilitates adaptive and learnable convolution operators that can be dynamically adjusted according to the underlying data structured on Riemann surfaces. QCC unifies a broad range of spatially defined convolutions, facilitating the learning of tailored convolution operators on each underlying surface optimized for specific tasks. Building on this foundation, we develop the Quasi-Conformal Convolutional Neural Network (QCCNN) to address a variety of tasks related to geometric data. We validate the efficacy of QCCNN through the classification of images defined on curvilinear Riemann surfaces, demonstrating superior performance in this context. Additionally, we explore its potential in medical applications, including craniofacial analysis using 3D facial data and lesion segmentation on 3D human faces, achieving enhanced accuracy and reliability.

Problem

• Convolution on Irregular Surfaces
• Deep Learning for Complex Shapes
• 3D Facial Disease Recognition

Innovation

• Quasi-Conformal Convolution (QCC)
• Quasi-Conformal Convolutional Neural Network (QCCNN)
• Irregular Structure Processing