L₀ Gradient-Regularization and Scale Space Representation Model for Cartoon and Texture Decomposition

In this paper, we consider decomposing an image into its cartoon and texture components. Traditional methods, which mainly rely on the gradient amplitude of images to distinguish between these components, often show limitations in decomposing small-scale, high-contrast texture patterns and large-scale, low-contrast structural components. Specifically, these methods tend to decompose the former to the cartoon image and the latter to the texture image, neglecting the scale features inherent in both components. To overcome these challenges, we introduce a new variational model which incorporates an $L_{0}$ -based total variation norm for the cartoon component and an $L_{2}$ norm for the scale space representation of the texture component. We show that the texture component has a small $L_{2}$ norm in the scale space representation. We apply a quadratic penalty function to handle the non-separable $L_{0}$ norm minimization problem. Numerical experiments are given to illustrate the efficiency and effectiveness of our approach.