Zeng Gengsheng L
Department of Computer Science, Utah Valley University, Orem, UT 84058, USA.
Axioms. 2025 Aug;14(8). doi: 10.3390/axioms14080605. Epub 2025 Aug 4.
In compressed sensing, it is believed that the norm minimization is the best way to enforce a sparse solution. However, the norm is difficult to implement in a gradient-based iterative image reconstruction algorithm. The total variation (TV) norm minimization is considered a proper substitute for the norm minimization. This paper points out that the TV norm is not powerful enough to enforce a piecewise-constant image. This paper uses the limited-angle tomography to illustrate the possibility of using the norm to encourage a piecewise-constant image. However, one of the drawbacks of the norm is that its derivative is zero almost everywhere, making a gradient-based algorithm useless. Our novel idea is to replace the zero value of the norm derivative with a zero-mean random variable. Computer simulations show that the proposed norm minimization outperforms the TV minimization. The novelty of this paper is the introduction of some randomness in the gradient of the objective function when the gradient is zero. The quantitative evaluations indicate the improvements of the proposed method in terms of the structural similarity (SSIM) and the peak signal-to-noise ratio (PSNR).
在压缩感知中，人们认为 范数最小化是实现稀疏解的最佳方法。然而， 范数在基于梯度的迭代图像重建算法中难以实现。总变分（TV）范数最小化被认为是 范数最小化的合适替代方法。本文指出，TV 范数在强制生成分段常数图像方面不够强大。本文使用有限角度断层扫描来说明使用 范数来促进分段常数图像的可能性。然而， 范数的一个缺点是其导数几乎处处为零，这使得基于梯度的算法无用。我们的新想法是用零均值随机变量代替 范数导数的零值。计算机模拟表明，所提出的 范数最小化优于 TV 最小化。本文的新颖之处在于，当梯度为零时，在目标函数的梯度中引入了一些随机性。定量评估表明，所提出的方法在结构相似性（SSIM）和峰值信噪比（PSNR）方面有所改进。
