PURPOSE:The adaptive steepest descent projection onto convex set (ASD-POCS) algorithm is a promising algorithm for constrained total variation (TV) type norm minimization models in computed tomography (CT) image reconstruction using sparse and/or noisy data. However, in ASD-POCS algorithm, the existing gradient expression of the TV-type norm appears too complicated in the implementation code and reduces image reconstruction speed. To address this issue, this work aims to develop and test a simple and fast ASD-POCS algorithm. METHODS:Since the original algorithm is not derived thoroughly, we first obtain a simple matrix-form expression by thorough derivation via matrix representations. Next, we derive the simple matrix expressions of the gradients of TV, adaptive weighted TV (awTV), total p-variation (TpV), high order TV (HOTV) norms by term combinations and matrix representations. The deep analysis is then performed to identify the hidden relations of these terms. RESULTS:The TV reconstruction experiments by use of sparse-view projections via the Shepp-Logan, FORBILD and a real CT image phantoms show that the simplified ASD-POCS (S-ASD-POCS) using the simple matrix-form expression of TV gradient achieve the same reconstruction accuracy relative to ASD-POCS, whereas it enables to speed up the whole ASD process 1.8–2.7 time fast. CONCLUSIONS:The derived simple matrix expressions of the gradients of these TV-type norms may simplify the implementation of the ASD-POCS algorithm and speed up the ASD process. Additionally, a general gradient expression suitable to all the sparse transform-based optimization models is demonstrated so that the ASD-POCS algorithm may be tailored to extended image reconstruction fields with accelerated computational speed.

中文翻译：

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一种用于图像重建的简单快速的 ASD-POCS 算法

目的：自适应最陡下降投影到凸集 (ASD-POCS) 算法是一种很有前途的算法，用于使用稀疏和/或噪声数据进行计算机断层扫描 (CT) 图像重建中的约束总变异 (TV) 类型范数最小化模型。然而，在 ASD-POCS 算法中，现有的 TV-type 范数的梯度表达在实现代码中显得过于复杂，降低了图像重建速度。为了解决这个问题，这项工作旨在开发和测试一种简单快速的 ASD-POCS 算法。方法：由于原始算法没有彻底推导，我们首先通过矩阵表示彻底推导得到一个简单的矩阵形式表达式。接下来，我们推导出 TV 梯度的简单矩阵表达式，自适应加权 TV (awTV)，总 p-variation (TpV)，通过术语组合和矩阵表示的高阶电视（HOTV）规范。然后进行深入分析以识别这些术语的隐藏关系。结果：通过 Shepp-Logan、FORBILD 和真实 CT 图像体模使用稀疏视图投影的 TV 重建实验表明，简化的 ASD-POCS (S-ASD-POCS) 使用 TV 梯度的简单矩阵形式表达相对于 ASD-POCS 实现相同的重建精度，同时它能够将整个 ASD 过程加快 1.8-2.7 倍。结论：导出的这些 TV 类型范数的梯度的简单矩阵表达式可以简化 ASD-POCS 算法的实现并加速 ASD 过程。此外，