Euler's Elastica-Based Cartoon-Smooth-Texture Image Decomposition

We propose a novel model for decomposing grayscale images into three distinct components: the structural part, representing sharp boundaries and regions with strong light-to-dark transitions; the smooth part, capturing soft shadows and shades; and the oscillatory part, characterizing textures and noise. To capture the homogeneous structures, we introduce a combination of L⁰-gradient and curvature regularization on level lines. This new regularization term enforces strong sparsity on the image gradient while reducing the undesirable staircase effects as well as preserving the geometry of contours. For the smoothly varying component, we utilize the L²-norm of the Laplacian that favors isotropic smoothness. To capture the oscillation, we use the inverse Sobolev seminorm. To solve the associated minimization problem, we design an efficient operator-splitting algorithm. Our algorithm effectively addresses the challenging nonconvex nonsmooth problem by separating it into subproblems. Each subproblem can be solved either directly using closed-form solutions or efficiently using the fast Fourier transform. We provide systematic experiments, including ablation and comparison studies, to analyze our model's behaviors and demonstrate its effectiveness as well as efficiency.