Complex Functional Maps : a Conformal Link Between Tangent Bundles

TL;DR

This paper introduces complex functional maps that extend the functional map framework to conformal maps between tangent vector fields, enabling orientation-aware shape correspondence without extra regularization.

Contribution

It presents a novel complex structure on tangent bundles to achieve orientation-preserving conformal maps, improving robustness and efficiency in shape analysis.

Findings

Enables orientation-aware tangent bundle mappings

Improves robustness in shape correspondence tasks

Reduces orientation-reversing errors in functional map pipelines

Abstract

In this paper, we introduce complex functional maps, which extend the functional map framework to conformal maps between tangent vector fields on surfaces. A key property of these maps is their orientation awareness. More specifically, we demonstrate that unlike regular functional maps that link functional spaces of two manifolds, our complex functional maps establish a link between oriented tangent bundles, thus permitting robust and efficient transfer of tangent vector fields. By first endowing and then exploiting the tangent bundle of each shape with a complex structure, the resulting operations become naturally orientationaware, thus favoring orientation and angle preserving correspondence across shapes, without relying on descriptors or extra regularization. Finally, and perhaps more importantly, we demonstrate how these objects enable several practical applications within the functional map framework. We show that functional maps and their complex counterparts can be estimated jointly to promote orientation preservation, regularizing pipelines that previously suffered from orientation-reversing symmetry errors.