We propose the fractional-order total variation (TV) algorithm with nonlocal self-similarity for image reconstruction in compressed sensing to alleviate texture details deterioration and eliminate staircase artifacts, which results from the TV algorithms. The Grünwald–Letnikov fractional-order differential operators, which consider more neighboring image pixels and use four different directions to handle fractional-order gradients, are used to replace the integer-order differential operators. To suppress the staircase artifacts, modified nonlocal means operators are introduced into our method, which can contain prior image structural information and update the Lagrangian multipliers. An efficient augmented Lagrangian algorithm is used to solve the TV problem. Numerical results show that the algorithm achieves remarkable performance improvements at various sampling ratios. Compared with fractional-order TV-based projections onto convex sets, the maximum gains of peak signal-to-noise ratio and structural similarity index with all images are up to 2.52 dB and 0.0178, respectively, and the algorithm performs the better for preserving details and eliminating the staircase effect at the cost of taking more time.

中文翻译：

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具有非局部自相似性的分数阶总变分算法用于图像重建

我们提出了具有非局部自相似性的分数阶总变分（TV）算法，用于压缩传感中的图像重建，以减轻纹理细节恶化并消除阶梯伪像，这是由TV算法产生的。使用Grünwald–Letnikov分数阶微分算子来代替整数阶微分算子，该算子考虑了更多的相邻图像像素并使用四个不同的方向来处理分数阶梯度。为了抑制阶梯伪像，将改进的非局部均值算子引入到我们的方法中，该算子可以包含先前的图像结构信息并更新拉格朗日乘数。一种有效的增强拉格朗日算法用于解决电视问题。数值结果表明，该算法在各种采样率下均实现了显着的性能提升。与凸集上基于分数阶电视的投影相比，所有图像的峰值信噪比和结构相似性指标的最大增益分别高达2.52 dB和0.0178，并且该算法在保留细节方面表现更好并消除楼梯效应，但要花费更多时间。