Sparse models and their variants have been extensively investigated, and have achieved great success in image denoising. Compared with recently proposed deep-learning-based methods, sparse models have several advantages: (1) Sparse models do not require a large number of pairs of noisy images and the corresponding clean images for training. (2) The performance of sparse models is less reliant on the training data, and the learned model can be easily generalized to natural images across different noise domains. In sparse models, ${\ell }_0$ norm penalty makes the problem highly non-convex, which is difficult to be solved. Instead, ${\ell }_1$ norm penalty is commonly adopted for convex relaxation, which is considered as the Laplacian prior from the Bayesian perspective. However, many previous works have revealed that ${\ell }_1$ norm regularization causes a biased estimation for the sparse code, especially for high-dimensional data, e.g., images. In this paper, instead of using the ${\ell }_1$ norm penalty, we employ an improper prior in the sparse model and formulate a hierarchical sparse model for image denoising. Compared with other competitive methods, experiment results show that our proposed method achieves a better generalization for images with different characteristics across various domains, and achieves state-of-the-art performance for image denoising on several benchmark datasets.

稀疏模型及其变体已被广泛研究，并在图像去噪中取得了巨大的成功。与最近提出的基于深度学习的方法相比，稀疏模型具有以下优点：（1）稀疏模型不需要大量的噪声图像对和相应的干净图像来进行训练。（2）稀疏模型的性能较少依赖训练数据，并且学习的模型可以轻松地推广到跨不同噪声域的自然图像。在稀疏模型中${\ell }_0$规范罚则使问题高度不凸，这是很难解决的。反而，${\ell }_{\mathrm{1个}}$凸松弛通常采用范数罚则，从贝叶斯角度看，它被认为是拉普拉斯先验。但是，以前的许多作品都揭示了${\ell }_{\mathrm{1个}}$规范正则化会导致稀疏代码（尤其是高维数据，例如图像）的偏差估计。在本文中，不要使用${\ell }_{\mathrm{1个}}$规范惩罚，我们在稀疏模型中采用了不当先验，并为图像去噪制定了一个层次化的稀疏模型。与其他竞争方法相比，实验结果表明，我们提出的方法可以更好地概括具有不同特征的图像，并且在多个基准数据集上都能达到最新的图像去噪性能。