Scale-space characterization of higher-order continuous TV optimization in maximal slope space

The novel scale-space-based higher-order continuous TV optimization in the maximal slope space has been proposed. The maximal slope space equipped with the relaxed upper gradient is a problem that arises in the time-dependent evolutionary systems, such as the mean-curvature evolution. We formulate the continuous TV optimization as the topological completion over level sets where the smooth transitions between the curvatures have been expressed by the quadratic modulation operator. Then, the Lagrangian multiplier acts as the scale-space parameter depending the intrinsic geometry of the image. To express the multi-scale characteristics of the proposed TV optimization, we define the higher-order BV space restricted to a certain energy locally on a level set. The existence of the Bochner integral then is satisfied in the length space equipped with sequence of bounded divergence functions of geodesics. Also, we leverage the symmetry between higher-order divergence and gradient to propose a family of approximated differential of the functions. The proposed multi-scale framework improves the lower bound of the proposed functional decomposition by reconstructing the smaller oscillating patterns. The proposed methodology also allows to predict the behavior of the scale-space parameter iteratively. Numerical examples have been reported.