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

TL;DR

This work tackles convolution on non-Euclidean domains by introducing Quasi-Conformal Convolution (QCC), a learnable framework that defines the convolution operator on Riemann surfaces through quasi-conformal mappings. A trainable estimator generates data-dependent mappings, enabling adaptive, task-specific convolutions and the formation of the Quasi-Conformal Convolutional Neural Network (QCCNN) for classification and segmentation on curved geometries. The authors provide a theoretical foundation linking manifold convolutions to parameterized, quasi-conformal ones, and implement a QC mapping estimator with regularization to ensure bijectivity and smoothness. Empirical results across MNIST-on-surfaces, craniofacial analysis, and facial lesion segmentation show that QCCNN surpasses state-of-the-art geometric convnet baselines, demonstrating the practical value of geometry-aware, learnable convolutions for real-world curved-data problems.

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.

Figures (12)

• Figure 1: Illustration of how the Beltrami coefficient measures the conformality distortion of a quasi-conformal map
• Figure 1: Illustration of a conventional convolutional neural network with predefined, untrainable convolution operations (top) compared to our proposed model (bottom) featuring learnable, data-driven convolution operations.
• Figure 1: MNIST images printed on simple surfaces (left) and complex surfaces (right)
• Figure 2: Illustration of quasi-conformal mapping between Riemann surfaces.
• Figure 2: Illustration of the Quasi-conformal Convolutional layer: The process begins by parameterizing the manifold to establish a common planar domain for computations. A learnable, adaptive convolution is done through a data-driven quasi-conformal mapping. The dashed box highlights the equivalence between deformable convolution on the manifold, deformable convolution on the parameterized domain, and standard 2D convolution on the reparameterized domain via the quasi-conformal mapping.

Theorems & Definitions (19)

• Definition 3.1: Quasi-conformal map
• Theorem 3.2
• Theorem 3.3: Measurable Riemann mapping theorem gardiner2000quasiconformal
• Definition 4.1: Convolution
• Definition 4.2: Displacement function and displacement vector
• Definition 4.3: Manifold convolution
• Remark 4.4
• Definition 4.5: Parametrized manifold convolution
• Lemma 4.6
• Proof 1