SIAM Journal on Imaging Sciences, Volume 14, Issue 2, Page 717-748, January 2021.
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 multichannel 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 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 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 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 nonlinearity 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 subproblem 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.

中文翻译：

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用于矢量值图像正则化的彩色 Elastica 模型

SIAM 成像科学杂志，第 14 卷，第 2 期，第 717-748 页，2021 年 1 月。
与欧拉弹性能量相关的模型已被证明可用于包括图像处理在内的许多应用。将 elastica 模型扩展到彩色图像和多通道数据是一项具有挑战性的任务，因为这些几何模型的稳定且一致的数值求解器通常涉及高阶导数。与单通道欧拉弹性模型和总变差模型一样，涉及高阶导数的几何度量在考虑最小化弹性特性的图像形成模型时可能会有所帮助。过去，来自高能物理学的波利亚科夫作用已成功应用于彩色图像处理。在这里，我们为彩色图像引入了 Polyakov 动作的附加功能，以最小化颜色流形曲率。彩色图像曲率是通过将 Laplace--Beltrami 算子应用于彩色图像通道来计算的。当减少到灰度图像时，在空间和颜色之间选择适当的缩放比例时，所提出的模型最小化了对图像级别集的欧拉弹性操作。为所提出的非线性几何模型寻找极小值是我们在本文中解决的一个挑战。具体来说，我们提出了一种算子分裂方法来最小化建议的功能。通过引入三个向量值和矩阵值变量来解耦非线性。然后将问题转换为求解相关初始值问题的稳态。初值问题被时间分割成三个分数步骤，这样每个子问题都有一个封闭形式的解决方案，或者可以通过快速算法解决。