SVR image denoising [SVR-ID]
Code and results for the paper
 "Image Denoising with Kernels Based on Natural Image Relations"
V. Laparra, J. Gutierrez, G. Camps and J. Malo
Journal of Machine Learning Research, submitted, 2009

Abstract
 A successful class of image denoising methods is based on Bayesian approaches working in wavelet representations. The performance of these improves when explicitly including relations among the local frequency coefficients. However, in these techniques, image estimates can be obtained only for particular combinations of analytical models of signal and noise, thus precluding its straightforward extension to deal with other arbitrary noise sources.
 
In this paper,  we propose an alternative non-explicit way to take into account the relations among natural image wavelet coefficients for denoising: we use Support Vector Regression (SVR) in the wavelet domain to enforce these relations in the estimated signal. Since relations among the coefficients are specific to the signal, the regularization property of SVR is exploited to remove the noise, which does not share this feature. The specific signal relations, are encoded in an anisotropic kernel obtained from mutual information measures computed on a representative image database. In the proposed scheme, training considers minimizing the Kullback-Leibler divergence (KLD) between the estimated and actual probability functions (or histograms) of signal and noise in order to enforce similarity up to the higher (computationally estimable) order. Due to its non-parametric nature, the method can eventually cope with different noise sources without the need of an explicit re-formulation, as it is strictly necessary under parametric Bayesian formalisms.
 
Results under several noise levels and noise sources show that: (1) the proposed method outperforms conventional wavelet methods that assume coefficient independence, (2) it is similar to state-of-the-art methods that do explicitly include these relations when the noise source is Gaussian, and (3) it gives numerical and visual better performance when more complex, realistic noise sources are considered. Therefore, the proposed machine learning approach can be seen as a more flexible (model-free) alternative to the explicit description of wavelet coefficient relations.

Download

Results: SVR_denoising_results.rar

Software: denoising_SVR_2.zip

Help on Results

The *.rar file contains *.eps figures of all the denoising experiments in the tables of the above mentioned paper.

  • The experiments were done on the standard images Lena, Barbara and Boats.
  • The considered algorithms include (see text for details on the algorithms):
  • Hard thresholding (HT) (Donoho 95)
  • Soft thresholding (ST) (Donoho 95)
  • Bayesian Gaussian wavelet denoising (BG) (Figueiredo 01)
  • Bayesian Laplacian wavelet denoising (BL) (Simoncelli 99)
  • Bayesian GSM wavelet denoising (GSM) (Portilla 03)
  • Kernel based regresion in the wavelet domain (SVR) (Laparra 09)
  • Optimal Kernel based regresion in the wavelet domain (SVRopt) (Laparra 09)
  • The considered noise sources include (see text for details on the noise sources):
  • White additive Gaussian noise (variances var_200 and var_400)
  • JPEG encoding noise (quality factors jpeg_Q_7, jpeg_Q_9)
  • JPEG2000 (coarser quantization -jpeg2000_b_2- and finer quantization -jpeg2000_b_1-)
  • Vertical striping (vst)
  • IRIS noise (IRIS)

The results are organized as follows:

  • Each folder is named as, "image_noise"
  • Inside each folder the corresponding image files are named as, "image_noise_method"

Help on Software
  • The provided software is an implementation of the proposed SVR based approach.
  • See the readme file included in the *.zip file for details

References [Donoho 95]
David L. Donoho and Iain M. Johnstone.  Adapting to unknown smoothness via wavelet shrinkage. Journal of the American Statistical Association, 90:1200–1224, 1995.

[Figueiredo 01]
M. Figueiredo and R. Nowak. Wavelet-based image estimation: an empirical Bayes approach using Jeffrey’s noninformative prior. IEEE Transactions on Image Processing, 10(9):1322–1331, 2001.

[Simoncelli 99]
E. Simoncelli. Bayesian denoising of visual images in the wavelet domain.  In Bayesian Inference in Wavelet Based Models, pages 291–308.  Springer-Verlag, New York, 1999.

[Portilla 03]
J. Portilla, V. Strela, M. Wainwright, and E. Simoncelli. Image denoising using a scale mixture of Gaussians in the wavelet domain.  IEEE Transactions on Image Processing, 12(11):1338–1351, 2003.

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