This paper focuses on the development of a space-variant regularization model
for solving an under-determined linear inverse problem. The case study is a
medical image reconstruction from few-view tomographic noisy data. The primary
objective of the proposed optimization model is to achieve a good balance
between denoising and the preservation of fine details and edges, overcoming
the performance of the popular and largely used Total Variation (TV)
regularization through the application of appropriate pixel-dependent weights.
The proposed strategy leverages the role of gradient approximations for the
computation of the space-variant TV weights. For this reason, a convolutional
neural network is designed, to approximate both the ground truth image and its
gradient using an elastic loss function in its training. Additionally, the
paper provides a theoretical analysis of the proposed model, showing the
uniqueness of its solution, and illustrates a Chambolle-Pock algorithm tailored
to address the specific problem at hand. This comprehensive framework
integrates innovative regularization techniques with advanced neural network
capabilities, demonstrating promising results in achieving high-quality
reconstructions from low-sampled tomographic data.

该研究聚焦于开发一种用于解决欠定线性反问题的空间变异正则化模型。研究案例为从少视角层析噪声数据中重建医学图像。通过应用适当的像素相关权重，该优化模型的主要目标是在去噪和保留细节和边缘之间取得良好平衡，克服了广泛使用的总变差（TV）正则化方法的性能。提出的策略利用梯度逼近来计算空间变异 TV 权重。为此，设计了一个卷积神经网络，使用其训练中的弹性损失函数来逼近真值图像及其梯度。此外，本文对所提出的模型进行了理论分析，展示了其解的唯一性，并展示了针对特定问题的 Chambolle-Pock 算法。这一综合框架将创新的正则化技术与先进的神经网络能力相结合，展示出在从低采样层析数据中实现高质量重建方面的有 promising 结果。

少视角层析成像中由学习技术增强的空间变差全变差

Space-Variant Total Variation boosted by learning techniques in few-view  tomographic imaging

The Chambolle-Pock algorithm (CPA), also known as the primal-dual hybrid
gradient method (PDHG), has surged in popularity in the last decade due to its
success in solving convex/monotone structured problems. This work provides
convergence results for problems with varying degrees of (non)monotonicity,
quantified through a so-called oblique weak Minty condition on the associated
primal-dual operator. Our results reveal novel stepsize and relaxation
parameter ranges which do not only depend on the norm of the linear mapping,
but also on its other singular values. In particular, in nonmonotone settings,
in addition to the classical stepsize conditions for CPA, extra bounds on the
stepsizes and relaxation parameters are required. On the other hand, in the
strongly monotone setting, the relaxation parameter is allowed to exceed the
classical upper bound of two. Moreover, sufficient convergence conditions are
obtained when the individual operators belong to the recently introduced class
of semimonotone operators. Since this class of operators encompasses many
traditional operator classes including (hypo)- and co(hypo)monotone operators,
this analysis recovers and extends existing results for CPA. Several examples
are provided for the aforementioned problem classes to demonstrate and
establish tightness of the proposed stepsize ranges.

透过所谓的斜弱 Minty 条件，提供用于具有不同程度的（非）单调性问题的收敛结果。我们的研究结果表明，步长和松弛参数范围不仅取决于线性映射的范数，还取决于其其他奇异值。此外，在非单调设置中，除了 CPA 的经典步长条件外，还需要额外的步长和松弛参数界限。然而，在强单调设置下，允许松弛参数超过 2 的经典上界。此外，当个别算子属于最近引入的半单调算子类时，我们得到了足够的收敛条件。由于此类算子包括许多传统算子类，如（假设）- 和共（假设）单调算子，因此此分析恢复并扩展了 CPA 的现有结果。我们提供了几个示例，以演示和建立所提步长范围的紧致性。

Chambolle-Pock 算法在无单调性情况下的收敛性

Convergence of the Chambolle-Pock Algorithm in the Absence of  Monotonicity

In this paper, we propose a new primal-dual algorithm for minimizing $f(x) +
g(x) + h(Ax)$, where $f$, $g$, and $h$ are proper lower semi-continuous convex
functions, $f$ is differentiable with a Lipschitz continuous gradient, and $A$
is a bounded linear operator. The proposed algorithm has some famous
primal-dual algorithms for minimizing the sum of two functions as special
cases. E.g., it reduces to the Chambolle-Pock algorithm when $f = 0$ and the
proximal alternating predictor-corrector when $g = 0$. For the general convex
case, we prove the convergence of this new algorithm in terms of the distance
to a fixed point by showing that the iteration is a nonexpansive operator. In
addition, we prove the $O(1/k)$ ergodic convergence rate in the primal-dual
gap. With additional assumptions, we derive the linear convergence rate in
terms of the distance to the fixed point. Comparing to other primal-dual
algorithms for solving the same problem, this algorithm extends the range of
acceptable parameters to ensure its convergence and has a smaller per-iteration
cost. The numerical experiments show the efficiency of this algorithm.

在这篇文章中，我们提出了一种新的原始 - 对偶算法，用于最小化具有 Lipschitz 连续梯度的可微下凸函数 $f$，下半连续的凸函数 $g$ 和线性算子 $A$ 的 $h (Ax)$ 的总和，证明了此算法的收敛性，并比较了其与其他多种算法在参数可接受范围和迭代代价方面的优点。