A convex-nonconvex variational method for the additive decomposition of functions on surfaces

We present a convex-nonconvex variational approach for the additive decomposition of noisy scalar fields defined over triangulated surfaces into piecewise constant and smooth components. The energy functional to be minimized is defined by the weighted sum of three terms, namely an fidelity term for the noise component, a Tikhonov regularization term for the smooth component and a total variation (TV)-like non-convex term for the piecewise constant component. The last term is parametrized such that the free scalar parameter allows to tune its degree of non-convexity and, hence, to separate the piecewise constant component more effectively than by using a classical convex TV regularizer without renouncing to convexity of the total energy functional. A method is also presented for selecting the two regularization parameters. The unique solution of the proposed variational model is determined by means of an efficient ADMM-based minimization algorithm. Numerical experiments show a nearly perfect separation of the different components.