During the past decade, the dictionary learning has been a hot topic in sparse representation. With theoretical guarantees, a low-coherence dictionary is demonstrated to optimize the sparsity and improve the accuracy of the performance of signal reconstruction. Two strategies have been investigated to learn incoherent dictionaries: (i) by adding a decorrelation step after the dictionary updating (e.g. INK-SVD), or $(ii)$ by introducing an additive penalty term of the mutual coherence to the general dictionary learning problem. In this paper, we propose a third method, which learns an incoherent dictionary by solving a constrained quadratic programming problem. Therefore, we can learn a dictionary with a prior fixed coherence value, which cannot be realized by the second strategy. Moreover, it updates the dictionary by considering simultaneously the reconstruction error and the incoherence, and thus does not suffer from the performance reduction of the first strategy.

The constrained quadratic programming problem is difficult problem due to its non-smoothness and non-convexity. To deal with the problem, a two-step alternating method is used: sparse coding by solving a problem of mixed-integer programming and dictionary updating by the hybrid method of augmented Lagrangian and alternating proximal linearized minimization. Finally, extensive experiments conducted in image denoising demonstrate the relevance of the proposed method, and illustrate the relation between coherence of dictionary and reconstruction quality.

在过去的十年中，字典学习一直是稀疏表示的热门话题。在理论上得到保证，演示了一种低相干字典，可优化稀疏性并提高信号重建性能的准确性。已经研究了两种策略来学习不连贯的词典：（i）通过在字典更新后添加去相关步骤（例如INK-SVD），或$（\mathrm{一世}\mathrm{一世}）$通过引入一般字典学习问题的相互连贯性的加法惩罚项。在本文中，我们提出了第三种方法，该方法通过解决约束二次规划问题来学习不连贯的字典。因此，我们可以学习具有先验固定相干值的字典，而这是第二种策略无法实现的。此外，它通过同时考虑重构误差和不连贯性来更新字典，因此不会遭受第一策略的性能降低的困扰。

约束二次规划问题由于其非光滑性和非凸性而成为难题。为了解决该问题，使用了两步交替方法：通过解决混合整数编程和字典的问题来进行稀疏编码，该问题是通过增强拉格朗日方法和交替近端线性最小化的混合方法进行的。最后，在图像去噪中进行的大量实验证明了该方法的相关性，并说明了字典一致性与重建质量之间的关系。