Non-linearity to improve and analyze functional maps for geometric data.

ended

The functional maps (FM) is an effective and established method for shape matching in several applications such as computer graphics, medical imaging, computer-aided design and biomedical data analysis among many others. In the last few years, a large quantity of research has been devoted to this topic, giving rise to new analysis and implementation of innovative methods. The interest in FM arises from its efficiency, its effectiveness and the several applications in which FM can be involved. Given two 3D shapes, FM focus on the correspondence between the functional spaces defined on two surfaces rather than between the points of their 3D embeddings. A small matrix can compactly encode the FM representing the functional spaces on a fixed basis. A common choice is to approximate the functional spaces as a linear vector space through a truncated subset of their Fourier basis. In most cases, only the vector space structure of functional spaces is exploited, but not their algebra, i.e., the ability to take pointwise products of functions. With NON-LINFMAPS, we aim to reinforce the FM injecting the non-linearity in the framework through new representations of the functional spaces and innovative techniques for the conversion of the FM in a point-to-point matching. The main motivations of this proposal are twofold. First, there is solid evidence that non-linearity encodes essential map properties. Second, the non-linearity of maps between embedded surfaces makes non-linearity more suitable to extract high-quality correspondences from FM. Through the MSCA, we plan to create an entirely novel computational framework for FM that directly exploits the algebra structure of functional spaces integrating non-linearity. Based on these new insights, we will design efficient algorithms and entirely new applications for different shape representations, such as graphs, point clouds and volumetric data confirming the multi-disciplinary potential of our proposal.