Complex Functional Maps : a Conformal Link Between Tangent Bundles
This work addresses orientation preservation in shape correspondence for computer graphics and geometry processing, representing an incremental improvement within the functional map framework.
The paper tackles the problem of orientation-reversing symmetry errors in shape correspondence by introducing complex functional maps, which extend the functional map framework to conformal maps between tangent vector fields on surfaces, enabling robust and efficient transfer of tangent vector fields without relying on descriptors or extra regularization.
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.