有约束最小二乘图像复原算法设计与实现

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有约束最小二乘图像复原算法设计与实现

摘要正则化图像复原方法是通过引入图像先验知识相关的正则项，将不适定问题转化为近似的适定问题，从而获得稳定的近似解的过程。有约束最小二乘法是解决图像去模糊及加性噪声去除的通用方法。本文针对常见的图像模糊和高斯噪声两类退化因素进行了图像复原算法的设计。首先，实现了基于稀疏性约束的梯度投影算法，算法通过选择不同的搜索方式分为GPSR-Basic算法和GPSR-BB算法，实验结果表明GPSR-BB算法具有更优的细节保持能力及更快的求解速度；其次，实现了一种动量梯度投影算法，该算法在梯度下降方向上增加一个动量项，提高了算法的快速性和鲁棒性；最后，设计了一种自适应参数的动量梯度投影算法（A-Momentum），采用Barzila Borwein 的近似方式来获得参数的自适应设置，从而减少迭代次数。实验结果表明在相同的截止条件下，A-Momentum算法仅需22次迭代即可到达稳定解，而GPSR-Basic算法和GPSR-BB算法则分别需要48次和47次。

关键词图像复原；梯度投影算法；动量梯度下降；自适应

ABSTRACT The regularization image restoration methods are to convert the ill-posed problem into an approximate problem by introducing the regular term related to the prior knowledge of the image, so as to obtain a stable approximate solution. Constrained least squares is a general method to solve image de-blur and additive noise removal. In this paper, we design the image restoration algorithm for the common image blur and Gaussian noise degradation factors. Firstly, the gradient projection algorithm based on sparsity constraint is realized. The algorithm is divided into GPSR-Basic algorithm and GPSR-BB algorithm by selecting different search methods. The experimental results show that the GPSR-BB algorithm has better detail retention ability and faster Secondly, a momentum gradient projection algorithm is implemented. The algorithm adds a momentum term to the descending direction of the gradient, which improves the fastness and robustness of the algorithm. Finally, a momentum gradient

projection of adaptive parameters is designed. The algorithm (A-Momentum) uses Barzila Borwein's approximation to obtain the adaptive setting of the parameters, thereby reducing the number of iterations. The experimental results show that the A-Momentum algorithm only needs 22 iterations to reach the stable solution under the same cut-off condition, while the GPSR-Basic algorithm and the GPSR-BB algorithm need 48 times and 47 times respectively.

KEY WORDS Image restoration, gradient projection algorithm, momentum gradient descent, adaptive

1 前言 (1)

1.1 研究背景及意义 (1)

1.2 图像复原方法与研究现状 (1)

1.3 本文主要内容 (3)

2 有约束最小二乘法基础 (4)

2.1 图像退化模型 (4)

2.2 有约束最小二乘法 (5)

2.2.1 图像复原的正则化模型 (5)

2.2.2 基于稀疏特性的正则化模型 (6)

2.2.3 TV正则化模型 (8)

2.2.4 正则化方法的优点 (8)

3 梯度投影算法 (10)

3.1 梯度投影（GPSR）算法 (10)

3.2 GPSR Basic算法 (11)

3.3 GPSR BB算法 (13)

3.4 实验结果分析 (15)

4 动量梯度投影法 (18)

4.1 动量梯度下降方向 (18)

4.2 动量梯度投影算法 (19)

4.3 自适应动量梯度投影算法 (21)

4.5 实验结果分析 (24)

5 结论 (28)

致谢 (29)

参考文献 (30)

附录 (31)