Complex Functional Maps: A Conformal Link Between Tangent Bundles

Abstract

AbstractIn 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 theirorientation awareness. More specifically, we demonstrate that unlike regular functional maps that linkfunctional spacesof two manifolds, our complex functional maps establish a link betweenoriented 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 orientation‐aware, thus favouringorientation and angle preserving correspondenceacross 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.