基于非凸的全变分和低秩混合正则化的图像去模糊模型和算法

非盲图像去模糊问题是从已知核的带噪声的线性卷积变换中恢复原始图像.如果噪声是满足高斯分布的,则可以直接使用最小二乘求解.然而在大多数情况下,去模糊问题都是高度病态的,直接求解无法做到.因此,通常的做法是通过抽取原始图像的已知统计先验信息进行正则化来帮助求解问题.两种常用的正则化是低秩和全变分.早期的相关工作单独使用这两种正则化.直到几年前,人们才考虑将这两种正则化结合起来.已有的结果表明,混合正则化模型比单一模型具有更好的性能.然而,目前的混合正则化方法只是采用凸方法,非凸的工作仍然是空白的.考虑到非凸正则化在很多种情况下都比凸正则化的效果要好,因此本文使用L1/2范数和Schatten-1/2范数提出了一种新的非凸混合模型.我们使用这两个非凸函数,因为它们的近端算子很容易计算.这种非凸混合正则化模型本质上是一个非凸线性约束问题,可以通过交替方向乘子法求解.然而,非凸性使得交替方向乘子法收敛十分困难.因此,我们转向求解原问题的惩罚问题.将交替最小化方法应用于惩罚问题就可以得到提出的算法,其中每个子步骤只涉及非常简单的计算.由于惩罚参数很大时,交替极小化算法速度会很慢,为了加速算法,针对惩罚参数我们使用了预热技术,即选取很小的初值但是在迭代过程中不断将参数增大.我们证明了该算法的收敛性.数值实验验证了本文提出的模型和算法的有效性.在非常温和的假设下,我们证明了算法的收敛性.数值实验验证了本文提出的模型和算法的有效性.

Nonconvex Low Rank and Total Variation Regularized Model and Algorithm for Image Deblurring

Non-blind image deblurring aims to reconstruct the original image for noised linear convolution transform with some known kernel.If the noise is Gaussian,one might use the least square minimization with the observed image and the kernel for this task.However,in most cases,the convolution operator makes the problem very ill-posed and hard to solve directly.To this end,regularizations,which are recruited to characterize known statistical priors about the original images,are then developed to help.Among them,two frequently used ones are the low-rank and totalvariation regularizations.The earlier related works employed them separately,that is,just using one of them,not both.Until several years ago,people have considered combing these two regularizations together.Existing results show that the hybrid regularized model performs much better than the single one.However,the current composite regularization just uses the convex methodology.The nonconvex implementations are still missing.Considering nonconvex regularizations can beat convex ones in various cases,in this paper,we propose a novel nonconvex hybrid model in which,the L1/2 and Schatten-1/2 norms are used.We use these two nonconvex functions due to that their proximal maps are easy to calculate even without convexity.Both L1/2 and Schatten-1/2 norms enjoy closedform proximal maps.The proposed nonconvex hybrid regularized model is naturally a nonconvex linearly constrained problem which can be solved by the alternating directions of multipliers.However,the nonconvexity breaks the theoretical guarantees.Thus,we turn to solve the penalty problem rather than the original form.The alternating minimization method applied to the penalty then yields the proposed algorithm,in which,each substep just involves very simple computations.In our algorithm,an important parameter is the penalty parameter.If it is infinity,the penalty is then identical to the original problem.But the large penalty parameter will make the algorithm iterate slowly.Thus,to improve the speed and narrow the penalty problem and the original one,for the penalty parameter,we use a warm-up technique,that is,increasing the penalty parameter in the iterations.The convergence of the algorithm is proved under very mild assumptions,which can be easily satisfied in applications.The numerical experiments are conducted on six natural test images.The performance of the proposed algorithm verifies the convergence theory.Comparisons with other algorithms demonstrate the efficiency of our algorithm.