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Section: New Results

Statistical aggregation methods for image denoising and estimation

Participants : Charles Kervrann, Frédéric Lavancier.

In the line of the Non-Local means [43] and ND-SAFIR [9], [10], [5] denoising algorithms, we have proposed a novel adaptive estimator based on the weighted average of observations taken in a neighborhood with weights depending on image data. The idea is to compute adaptive weights that best minimize an upper bound of the pointwise $L_2$ risk. In the framework of adaptive estimation, we show that the “oracle” weights depend on the unknown image and are optimal if we consider triangular kernels instead of the commonly-used Gaussian kernel. Furthermore, we propose a way to automatically choose the spatially varying smoothing parameter for adaptive denoising. Under conventional minimal regularity conditions, the obtained estimator converges at the usual optimal rate. The implementation of the proposed algorithm is also straightforward. The simulations show that our algorithm improves significantly the classical NL-means, and is competitive when compared to the more sophisticated NL-means filters both in terms of PSNR values and visual quality.

Previously, we investigated statistical aggregation methods which optimally combine several estimators to produce a boosted solution [11]. In this range of work, we also introduced in [24] a general method to combine estimators in order to produce a better estimate. From a theoretical point of view, we proved that this method is optimal in some sense. It is illustrated on standard statistical problems in parametric and semi-parametric models where the averaging estimator outperforms the initial estimators in most cases. This method has been subsequently adapted in [39] to models in spatial statistics. As part of an on-going work, we are applying this method to improve patch-based image denoising algorithms.

References:  [24] [39]

Collaborators: Qiyu Jin (School of Mathematical Science, Inner Mongolia University, China),

                          Ion Grama and Quansheng Liu (University of Bretagne-Sud, Vannes),

                          Paul Rochet (Laboratoire de Mathématiques Jean Leray (LMJL), University of Nantes).