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

arXiv: https://arxiv.org/abs/2407.02794 · HTML: https://mlsanity.com/paper/2407.02794/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 61 references that are in the map.
- First posted on arXiv 3 Jul 2024.

## Builds on (most cited first, 7 of 7)

- [PottsMGNet: A Mathematical Explanation of Encoder-Decoder Based Neural Networks](https://mlsanity.com/paper/2307.09039/) (2023)
- [Curvature Regularized Surface Reconstruction from Point Cloud](https://mlsanity.com/paper/2001.07884/) (2020)
- [A Color Elastica Model for Vector-Valued Image Regularization](https://mlsanity.com/paper/2008.08255/) (2020)
- [An Operator-Splitting Method for the Gaussian Curvature Regularization Model with Applications to Surface Smoothing and Imaging](https://mlsanity.com/paper/2108.01914/) (2021)
- [DOSnet as a Non-Black-Box PDE Solver: When Deep Learning Meets Operator Splitting](https://mlsanity.com/paper/2212.05571/) (2022)
- [Double-well Net for Image Segmentation](https://mlsanity.com/paper/2401.00456/) (2023)
- [Elastica Models for Color Image Regularization](https://mlsanity.com/paper/2203.09995/) (2022)

## Built on by (most cited first, 4 of 4)

- [Image Decomposition with G-norm Weighted by Total Symmetric Variation](https://mlsanity.com/paper/2503.22560/) (2025)
- [A PCA Based Model for Surface Reconstruction from Incomplete Point Clouds](https://mlsanity.com/paper/2509.15675/) (2025)
- [From level set evolution to threshold optimization: A grayscale level set framework for image segmentation](https://mlsanity.com/paper/2607.22255/) (2026)
- [A Globally Convergent Algorithm for Total Scaled-Gradient Variation via Cone-Constrained Bilinear Decomposition](https://mlsanity.com/paper/2609.00036/) (2026)

## Abstract (arXiv)

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 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.