Joseph Salmon

Non-local Methods with Shape-Adaptive Patches (NLM-SAP)

We propose in this paper an extension of the Non-Local Means (NL-Means) denoising algorithm. The idea is to replace the usual square patches used to compare pixel neighborhoods with various shapes that can take advantage of the local geometry of the image. We provide a fast algorithm to compute the NL-Means with arbitrary shapes thanks to the Fast Fourier Transform. We then consider local combinations of the estimators associated with various shapes by using Stein’s Unbiased Risk Estimate (SURE). Experimental results show that this algorithm improve the standard NL-Means performance and is close to state-of-the-art methods, both in terms of visual quality and numerical results. Moreover, common visual artifacts usually observed by denoising with NL-Means are reduced or suppressed thanks to our approach.

Image denoising with patch-based PCA: local versus global

In recent years, overcomplete dictionaries combined with sparse learning techniques became extremely popular in computer vision. While their usefulness is undeniable, the improvement they provide in specific tasks of computer vision is still poorly understood. The aim of the present work is to demonstrate that for the task of image denoising, nearly state-of-the-art results can be achieved using orthogonal dictionaries only, provided that they are learned directly from the noisy image. To this end, we introduce three patchbased denoising algorithms which perform hard thresholding on the coefficients of the patches in image-specific orthogonal dictionaries. The algorithms differ by the methodology of learning the dictionary: local PCA, hierarchical PCA and global PCA.We carry out a comprehensive empirical evaluation of the performance of these algorithms in terms of accuracy and running times. The results reveal that, despite its simplicity, PCA-based denoising appears to be competitive with the state-of-the-art denoising algorithms, especially for large images and moderate signal-to-noise ratios.

Patch reprojections for Non-Local methods

Since their introduction in image denoising, the family of non-local methods, whose Non-Local Means (NL-Means) is the most famous member, has proved its ability to challenge other powerful methods such as wavelet based approaches or variational techniques. Though simple to implement and efficient in practice, the classical NL-Means algorithm suffers from several limitations: noise artifacts are created around edges and regions with few repetitions in the image are not treated at all. In this paper, we present an easy to implement and time efficient modification of the NL-Means based on a better reprojection from the patches space to the original pixel space, specially designed to reduce the artifacts due to the rare patch effect. We compare the performance of several reprojection schemes on a toy example and on some classical natural images.

Sharp oracle inequalities for aggregation of affine estimators

We consider the problem of combining a (possibly uncountably infinite) set of affine estimators in non-parametric regression model with heteroscedastic Gaussian noise. Focusing on the exponentially weighted aggregate, we prove a PAC-Bayesian type inequality that leads to sharp oracle inequalities in discrete but also in continuous settings. The framework is general enough to cover the combinations of various procedures such as least square regression, kernel ridge regression, shrinking estimators and many other estimators used in the literature on statistical inverse problems. As a consequence, we show that the proposed aggregate provides an adaptive estimator in the exact minimax sense without neither discretizing the range of tuning parameters nor splitting the set of observations. We also illustrate numerically the good performance achieved by the exponentially weighted aggregate.

From patches to pixels in Non-Local methods: Weighted-average reprojection

Since their introduction in denoising, the family of non local methods, whose Non-Local Means (NL-Means) is the most famous member, has proved its ability to challenge other powerful methods such as wavelet based approaches or variational techniques. Though simple to implement and efficient in practice, the classical NL-Means suffers from ringing artifacts around edges. In this paper, we present an easy to implement and time efficient modification of the NL-means based on a better reprojection from the patches space to the original (image) pixel space. We illustrate the performance of our method on a toy example and on some classical images.

Poisson noise reduction with non-local PCA

Photon limitations arise in spectral imaging, nuclear medicine, astronomy and night vision. The Poisson distribution used to model this noise has variance equal to its mean so blind application of standard noise removals methods yields significant artifacts. Recently, overcomplete dictionaries combined with sparse learning techniques have become extremely popular in image reconstruction. The aim of the present work is to demonstrate that for the task of image denoising, nearly state-of-the-art results can be achieved using small dictionaries only, provided that they are learned directly from the noisy image. To this end, we introduce patch-based denoising algorithms which perform an adaptation of PCA (Principal Component Analysis) for Poisson noise. We carry out a comprehensive empirical evaluation of the performance of our algorithms in terms of accuracy when the photon count is really low. The results reveal that, despite its simplicity, PCA-flavored denoising appears to be competitive with other state-of-the-art denoising algorithms.

NL-Means and aggregation procedures

Patch based denoising methods, such as the NL-Means, have emerged recently as simple and efficient denoising methods. This paper provides a new insight on those methods by showing their connection with recent statistical aggregation techniques. Within this aggregation framework, we propose some novel patch based denoising methods. We provide some theoretical justification and then explain how to implement them with a Monte Carlo based algorithm.

Oracle Inequalities and Minimax Rates for Nonlocal Means and Related Adaptive Kernel-Based Methods

This paper describes a novel theoretical characterization of the performance of non-local means (NLM) for noise removal. NLM has proven effective in a variety of empirical studies, but little is understood fundamentally about how it performs relative to classical methods based on wavelets or how various parameters (e.g., patch size) should be chosen. For cartoon images and images which may contain thin features and regular textures, the error decay rates of NLM are derived and compared with those of linear filtering, oracle estimators, variable-bandwidth kernel methods, Yaroslavskys filter and wavelet thresholding estimators. The trade-off between global and local search for matching patches is examined, and the bias reduction associated with the local polynomial regression version of NLM is analyzed. The theoretical results are validated via simulations for 2D images corrupted by additive white Gaussian noise.

A two-stage denoising filter: The preprocessed Yaroslavsky filter

This paper describes a simple image noise removal method which combines a preprocessing step with the Yaroslavsky filter for strong numerical, visual, and theoretical performance on a broad class of images. The framework developed is a two-stage approach. In the first stage the image is denoised by a classical denoising method (e.g., wavelet or curvelet thresholding). In the second step a modification of the Yaroslavsky filter is performed on the original noisy image, where the weights of the filters are governed by pixel similarities in the denoised image from the first stage. The procedure is supported by theoretical guarantees, achieves very good performance for cartoon images, and can be computed much more quickly than current patch-based denoising algorithms.

On debiasing restoration algorithms: applications to total-variation and nonlocal-means

Bias in image restoration algorithms can hamper further analysis, typically when the intensities have a physical meaning of interest, e.g., in medical imaging. We propose to suppress a part of the bias – the method bias – while leaving unchanged the other unavoidable part – the model bias. Our debiasing technique can be used for any locally affine estimator including \(\ell _1\) regularization, anisotropic total-variation and some nonlocal filters.