# A Color Elastica Model for Vector-Valued Image Regularization

arXiv: https://arxiv.org/abs/2008.08255 · HTML: https://mlsanity.com/paper/2008.08255/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Aug 2020.

## Built on by (most cited first, 11 of 11)

- [PottsMGNet: A Mathematical Explanation of Encoder-Decoder Based Neural Networks](https://mlsanity.com/paper/2307.09039/) (2023)
- [An Operator-Splitting Method for the Gaussian Curvature Regularization Model with Applications to Surface Smoothing and Imaging](https://mlsanity.com/paper/2108.01914/) (2021)
- [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)
- [Euler's Elastica Based Cartoon-Smooth-Texture Image Decomposition](https://mlsanity.com/paper/2407.02794/) (2024)
- [A Mathematical Explanation of UNet](https://mlsanity.com/paper/2410.04434/) (2024)
- [Connections between Operator-splitting Methods and Deep Neural Networks with Applications in Image Segmentation](https://mlsanity.com/paper/2307.09052/) (2023)
- [A Fast Minimization Algorithm for the Euler Elastica Model Based on a Bilinear Decomposition](https://mlsanity.com/paper/2308.13471/) (2023)
- [A New Cross-Space Total Variation Regularization Model for Color Image Restoration with Quaternion Blur Operator](https://mlsanity.com/paper/2405.12114/) (2024)
- [Total Normal Curvature Regularization and its Minimization for Surface and Image Smoothing](https://mlsanity.com/paper/2512.18968/) (2025)
- [A PCA Based Model for Surface Reconstruction from Incomplete Point Clouds](https://mlsanity.com/paper/2509.15675/) (2025)

## Abstract (arXiv)

Models related to the Euler's elastica energy have proven to be useful for many applications including image processing. Extending elastica models to color images and multi-channel data is a challenging task, as stable and consistent numerical solvers for these geometric models often involve high order derivatives. Like the single channel Euler's elastica model and the total variation (TV) models, geometric measures that involve high order derivatives could help when considering image formation models that minimize elastic properties. In the past, the Polyakov action from high energy physics has been successfully applied to color image processing. Here, we introduce an addition to the Polyakov action for color images that minimizes the color manifold curvature. The color image curvature is computed by applying of the Laplace-Beltrami operator to the color image channels. When reduced to gray-scale images, while selecting appropriate scaling between space and color, the proposed model minimizes the Euler's elastica operating on the image level sets. Finding a minimizer for the proposed nonlinear geometric model is a challenge we address in this paper. Specifically, we present an operator-splitting method to minimize the proposed functional. The non-linearity is decoupled by introducing three vector-valued and matrix-valued variables. The problem is then converted into solving for the steady state of an associated initial-value problem. The initial-value problem is time-split into three fractional steps, such that each sub-problem has a closed form solution, or can be solved by fast algorithms. The efficiency and robustness of the proposed method are demonstrated by systematic numerical experiments.