SIAM Journal on Imaging Sciences, Volume 14, Issue 3, Page 1273-1305, January 2021.
In image processing, classical methods minimize a suitable functional that balances between computational feasibility (convexity of the functional is ideal) and suitable penalties reflecting the desired image decomposition. The fact that algorithms derived from such minimization problems can be used to construct (deep) learning architectures has spurred the development of algorithms that can be trained for a specifically desired image decomposition, e.g., into cartoon and texture. While many such methods are very successful, theoretical guarantees are only scarcely available. To this end, in this contribution, we formalize a general class of intersection point problems encompassing a wide range of (learned) image decomposition models, and we give an existence result for a large subclass of such problems, i.e., giving the existence of a fixed point of the corresponding algorithm. This class generalizes classical model-based variational problems, such as the TV-$\ell^2$-model or the more general TV-Hilbert model. To illustrate the potential for learned algorithms, novel (nonlearned) choices within our class show comparable results in denoising and texture removal.

中文翻译：

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用于图像分解学习的具有不动点的广义相交算法

SIAM 成像科学杂志，第 14 卷，第 3 期，第 1273-1305 页，2021 年 1 月。
在图像处理中，经典方法最小化一个合适的函数，该函数在计算可行性（函数的凸性是理想的）和反映所需图像分解的合适惩罚之间取得平衡。源自此类最小化问题的算法可用于构建（深度）学习架构这一事实刺激了算法的发展，这些算法可针对特定所需的图像分解进行训练，例如分解为卡通和纹理。虽然许多这样的方法非常成功，但几乎没有理论保证。为此，在这个贡献中，我们形式化了一个一般类别的交点问题，包括广泛的（学习的）图像分解模型，并且我们给出了此类问题的一个大子类的存在结果，即，给出相应算法的不动点的存在。本课程概括了经典的基于模型的变分问题，例如 TV-$\ell^2$-模型或更一般的 TV-Hilbert 模型。为了说明学习算法的潜力，我们班级中的新颖（非学习）选择在去噪和纹理去除方面显示了可比的结果。