Total Variation Regularization of Shape Signals

This paper introduces the concept of shape signals, i.e., series of shapes which have a natural temporal or spatial ordering, as well as a variational formulation for the regularization of these signals. The proposed formulation can be seen as the shape-valued generalization of the Rudin-Osher-Fatemi (ROF) functional for intensity images. We derive a variant of the classical finite-dimensional representation of Kendall, but our framework is generic in the sense that it can be combined with any shape space. This representation allows for the explicit computation of geodesics and thus facilitates the efficient numerical treatment of the variational formulation by means of the cyclic proximal point algorithm. Similar to the ROF-functional, we demonstrate experimentally that $ℓ _{ 1 }$ -type penalties both for data fidelity term and regularizer perform best in regularizing shape signals. Finally, we show applications of our method to shape signals obtained from synthetic, photometric, and medical data sets.