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 $L^0$-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 $L^2$-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 non-convex non-smooth problem by separating it into sub-problems.
Each sub-problem can be solved either directly using closed-form solutions or
efficiently using the Fast Fourier Transform (FFT). We provide systematic
experiments, including ablation and comparison studies, to analyze our model's
behaviors and demonstrate its effectiveness as well as efficiency.

我们提出了一种新颖模型，用于将灰度图像分解为三个不同的组成部分：结构部分，表示具有锐利边界和强烈光暗转换的区域；平滑部分，捕捉柔和阴影和色调；以及振荡部分，描述纹理和噪声。