Spectral Processing of Tangential Vector Fields

We propose a framework for the spectral processing of tangential vector fields on surfaces. The basis is a Fourier-type representation of tangential vector fields that associates frequencies with tangential vector fields. To implement the representation for piecewise constant tangential vector fields on triangle meshes, we introduce a discrete Hodge-Laplace operator that fits conceptually to the prominent cotan discretization of the Laplace-Beltrami operator. Based on the Fourier representation, we introduce schemes for spectral analysis, filtering and compression of tangential vector fields. Moreover, we introduce a spline-type editor for modelling of tangential vector fields with interpolation constraints for the field itself and its divergence and curl. Using the spectral representation, we propose a numerical scheme that allows for real-time modelling of tangential vector fields.