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

TL;DR

This paper introduces a new image decomposition model based on Euler's Elastica that separates images into structural, smooth, and oscillatory components, using advanced regularization and an efficient algorithm.

Contribution

It presents a novel multi-component image decomposition model with specialized regularizations and an efficient operator-splitting algorithm for solving the non-convex optimization problem.

Findings

Effective separation of image components demonstrated.

Model reduces staircase artifacts and preserves contours.

Algorithm achieves high efficiency with FFT-based solutions.

Abstract

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.