Abstract. We propose a unified view of nonlocal methods for single-image denoising, for which BM3D is the most popular representative, that operate by gathering noisy patches together according to their similarities in order to process them collaboratively. Our general estimation framework is based on the minimization of the quadratic risk, which is approximated in two steps, and adapts to photon and electronic noises. Relying on an unbiased risk estimate (URE) for the first step and on “internal adaptation,” a concept borrowed from deep learning theory, for the second, we show that our approach enables one to reinterpret and reconcile previous state-of-the-art nonlocal methods. Within this framework, we propose a novel denoiser called NL-Ridge that exploits linear combinations of patches. While conceptually simpler, we show that NL-Ridge can outperform well-established state-of-the-art single-image denoisers.

In the past decade, deep neural networks have revolutionized image denoising in achieving significant accuracy improvements by learning on datasets composed of noisy/clean image pairs. However, this strategy is extremely dependent on training data quality, which is a well-established weakness. To alleviate the requirement to learn image priors externally, single-image (a.k.a., self-supervised or zero-shot) methods perform denoising solely based on the analysis of the input noisy image without external dictionary or training dataset. This work investigates the effectiveness of linear combinations of patches for denoising under this constraint. Although conceptually very simple, we show that linear combinations of patches are enough to achieve state-of-the-art performance. The proposed parametric approach relies on quadratic risk approximation via multiple pilot images to guide the estimation of the combination weights. Experiments on images corrupted artificially with Gaussian noise as well as on real-world noisy images demonstrate that our method is on par with the very best single-image denoisers, outperforming the recent neural network-based techniques, while being much faster and fully interpretable.

In many information processing systems, it may be desirable to ensure that any change of the input, whether by shifting or scaling, results in a corresponding change in the system response. While deep neural networks are gradually replacing all traditional automatic processing methods, they surprisingly do not guarantee such normalization-equivariance (scale + shift) property, which can be detrimental in many applications. To address this issue, we propose a methodology for adapting existing neural networks so that normalization-equivariance holds by design. Our main claim is that not only ordinary convolutional layers, but also all activation functions, including the ReLU (rectified linear unit), which are applied element-wise to the pre-activated neurons, should be completely removed from neural networks and replaced by better conditioned alternatives. To this end, we introduce affine-constrained convolutions and channel-wise sort pooling layers as surrogates and show that these two architectural modifications do preserve normalization-equivariance without loss of performance. Experimental results in image denoising show that normalization-equivariant neural networks, in addition to their better conditioning, also provide much better generalization across noise levels.

We propose a unified view of unsupervised non-local methods for image denoising that linearly combine noisy image patches. The best methods, established in different modeling and estimation frameworks, are two-step algorithms. Leveraging Stein’s unbiased risk estimate (SURE) for the first step and the “internal adaptation”, a concept borrowed from deep learning theory, for the second one, we show that our NL-Ridge approach enables to reconcile several patch aggregation methods for image denoising. In the second step, our closed-form aggregation weights are computed through multivariate Ridge regressions. Experiments on artificially noisy images demonstrate that NL-Ridge may outperform well established state-of-the-art unsupervised denoisers such as BM3D and NL-Bayes, as well as recent unsupervised deep learning methods, while being simpler conceptually.

This work tackles the issue of noise removal from images, focusing on the well-known DCT image denoising algorithm. The latter, stemming from signal processing, has been well studied over the years. Though very simple, it is still used in crucial parts of state-of-the-art "traditional" denoising algorithms such as BM3D. For a few years however, deep convolutional neural networks (CNN), especially DnCNN, have outperformed their traditional counterparts, making signal processing methods less attractive. In this paper, we demonstrate that a DCT denoiser can be seen as a shallow CNN and thereby its original linear transform can be tuned through gradient descent in a supervised manner, improving considerably its performance. This gives birth to a fully interpretable CNN called DCT2net. To deal with remaining artifacts induced by DCT2net, an original hybrid solution between DCT and DCT2net is proposed combining the best that these two methods can offer; DCT2net is selected to process non-stationary image patches while DCT is optimal for piecewise smooth patches. Experiments on artificially noisy images demonstrate that two-layer DCT2net provides comparable results to BM3D and is as fast as DnCNN algorithm.