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Rank constrained nuclear norm minimization with application to image denoising

机译：秩受约束的核规范最小化在图像去噪中的应用

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In the low rank matrix approximation problem, the well known nuclear norm minimization (NNM) problem plays a crucial role and attracts significant interests in recent years. In NNM the regularization parameter λ plays a decisive part, λ controls both the rank of the solution and the extent of the thresholding. However, it is hard for a single λ to balance the two issuses, and meanwhile the solving method calls singular value decomposition (SVD), of which the computational complexity is impracticable when the scale of the problem becomes large. This paper presents a rank constrained nuclear norm minimization (RNNM) method, in which the rank and the extent of thresholding are controlled separately by an added parameter k. More importantly, by proving its equivalence with an unconstrained bi-convex optimization problem RNNM can be solved in SVD free manner. In this paper, a SOR (Successive Over Relaxation) algorithm is designed for the equivalent bi-convex problem and its convergence is proved. We show that RNNM has a unique global optimal solution although being non-convex. We explicitly analyse the structure of the solution for the bi-convex problem and show some interesting properties. Finally, we verify the effectiveness of RNNM in image denoising. Experimental results show that the proposed solving method works faster than SVD based method. Thanks to the well balance of rank and thresholding, RNNM achieves superior results than the state-of-the-art methods in image denoising such as BM3D, SAIST in terms of both quantity measure and visual quality.
机译：在低秩矩阵逼近问题中，众所周知的核规范最小化（NNM）问题起着至关重要的作用，并且近年来引起了广泛的关注。在NNM中，正则化参数λ起着决定性的作用，λ既控制解的秩，又控制阈值的程度。但是，单个λ很难平衡这两个问题，同时求解方法称为奇异值分解（SVD），当问题的规模变大时，其计算复杂性将变得不切实际。本文提出了一种秩约束核准则最小化（RNNM）方法，其中秩和阈值范围由附加参数k分别控制。更重要的是，通过用无约束双凸优化问题证明其等效性，可以用无SVD的方式解决RNNM。本文针对等效双凸问题设计了一种SOR算法（Successive Over Relaxation），并证明了其收敛性。我们证明了RNNM是非凸的，但它具有唯一的全局最优解。我们显式分析双凸问题的解决方案的结构，并显示一些有趣的性质。最后，我们验证了RNNM在图像去噪中的有效性。实验结果表明，所提出的求解方法比基于SVD的方法更快。得益于等级和阈值的良好平衡，RNNM在图像去噪方面比BM3D，SAIST等最新技术在数量测量和视觉质量方面均取得了优异的结果。

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