A Variational Approach to Additive Image Decomposition into Structure, Harmonic, and Oscillatory Components

We propose a nonconvex variational decomposition model which separates a given image into piecewise-constant, smooth, and oscillatory components. This decomposition is motivated not only by image denoising and structure separation, but also by shadow and spot light removal. The proposed model clearly separates the piecewise-constant structure and smoothly varying harmonic part, thanks to having a separated oscillatory component. The piecewise-constant part is captured by TV-like nonconvex regularization, harmonic term via second-order regularization, and oscillatory (noise and texture) term via a H^{-1}-norm penalty. There are interesting interactions between these three regularization terms. We explore the effects of each regularization and the choice of parameters carefully. We propose an efficient alternating direction method of multipliers based minimization for fast numerical computation of the optimization problem. Various experiments are presented to show the robustness against a high level of noise, applications to soft spotlight and shadow removal, and the comparisons with other methods.