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We propose a novel image denoising strategy based on an enhanced sparse representation in transformdomain. The enhancement of the sparsity is achieved by grouping similar 2D image fragments (e.g. blocks) into 3D data arrays which we call "groups".
Collaborative filtering is a special procedure developed to deal with these 3D groups. We realize it using the three successive steps: 3D transformation of 3D group, shrinkage of transform spectrum, and inverse 3D transformation. The result is a 3D estimate that consists of the jointly filtered grouped image blocks. By attenuating the noise, the collaborative filtering reveals even the finest details shared by grouped blocks and at the same time it preserves the essential unique features of each individual block. The filtered blocks are then returned to their original positions.
Because these blocks are overlapping, for each pixel we obtain many different estimates which need to be combined. Aggregation is a particular averaging procedure which is exploited to take advantage of this redundancy.
A significant improvement is obtained by a specially developed collaborative Wiener filtering.
We develop algorithms based on this novel denoising strategy. The experimental results presented here demonstrate that the developed methods achieve stateoftheart denoising performance in terms of both peak signaltonoise ratio and subjective visual quality.

Application of BMxD to signaldependent noise: 

 Poisson and mixed PoissonGaussian (photonlimited imaging)  Rice distribution (MRI)  Clipped PoissonGaussian (raw data) 
σ¹/PSNR  Salesman 288x352@50 
Tennis 240x352@150 
Flower garden 240x352@150 
Miss America 288x360@150 
Costguard 144x176@300 
Foreman 288x352@300 
Bus 288x352@150 
Bicycle 576x720@30 
5 / 34.16  40.44  38.47  36.49  41.58  38.25  39.77  37.55  40.89 
10 / 28.13  37.21  34.68  32.11  39.61  34.78  36.46  33.32  37.62 
15 / 24.61  35.44  32.63  29.81  38.64  33.00  34.64  31.05  35.67 
20 / 22.11  34.04  31.20  28.24  38.85  31.71  33.30  29.57  34.18 
25 / 20.18  32.79  30.11  27.00  37.10  30.62  32.19  28.48  32.90 
30 / 18.59  31.68  29.22  25.89  36.41  29.68  31.27  27.59  31.77 
35 / 17.25  30.72  28.56  25.16  36.87  28.92  30.56  26.91  30.85 
¹ The noisy images/videos were created by adding zeromean white Gaussian noise with the following MATLAB commands:
randn('seed', 0);
noisy = original + sigma*randn(size(original));
σ  Salesman 288x352@50 
Tennis 240x352@150 
Flower garden 240x352@150 

Noisy  Denoised  Noisy  Denoised  Noisy  Denoised  
15  24.71  35.44  24.78  32.61  24.88  29.78 
20  22.32  34.07  22.33  31.18  22.46  28.21 
Matteo Maggioni
Enrique SánchezMonge
Alessandro Foi
Aram Danielyan
Kostadin Dabov
Vladimir Katkovnik
Karen Egiazarian
Compressed Sensing Image Reconstruction, Image Upsampling, and Image/Video SuperResolution via Recursive Spatially Adaptive Filtering
A preliminary version of BM3D using exclusively the DFT
ShapeAdaptive Transforms Filtering (Pointwise SADCT algorithm).
M. Maggioni, E. SánchezMonge, and A. Foi, “Joint Removal of Random and FixedPattern Noise through Spatiotemporal Video Filtering”, IEEE Trans. Image Process., vol. 23, no. 10, pp. 42824296, 2014. http://doi.org/10.1109/TIP.2014.2345261
M. Maggioni, V. Katkovnik, K. Egiazarian, and A. Foi, “A Nonlocal TransformDomain Filter for Volumetric Data Denoising and Reconstruction”, IEEE Trans. Image Process., vol. 22, no. 1, pp. 119133, January 2013. doi:10.1109/TIP.2012.2210725
M. Maggioni, G. Boracchi, A. Foi, and K. Egiazarian, “Video Denoising, Deblocking and Enhancement Through Separable 4D Nonlocal Spatiotemporal Transforms”, IEEE Trans. Image Process., vol. 21, no. 9, pp. 39523966, September 2012. doi:10.1109/TIP.2012.2199324
A. Danielyan, V. Katkovnik, and K. Egiazarian, “BM3D Frames and Variational Image Deblurring”, IEEE Trans. Image Process., vol. 21, no. 4, pp. 17151728, April 2012. doi:10.1109/TIP.2011.2176954
M. Maggioni and A. Foi, “Nonlocal transformdomain denoising of volumetric data with groupwise adaptive variance estimation”, Proc. SPIE Electronic Imaging 2012, Computational Imaging X, 829622, Burlingame (CA), USA, January 2012. doi:10.1117/12.912109
A. Danielyan, A. Foi, V. Katkovnik, and K. Egiazarian, “Spatially adaptive filtering as regularization in inverse imaging: compressive sensing, upsampling, and superresolution”, in SuperResolution Imaging (P. Milanfar, ed.), CRC Press / Taylor & Francis, ISBN: 9781439819302, September 2010 Examples of superresolution reconstruction as zipped Matlab MATfiles.
V. Katkovnik, A. Foi, K. Egiazarian, and J. Astola, “From local kernel to nonlocal multiplemodel image denoising”, Int. J. Computer Vision, vol. 86, no. 1, pp. 132, January 2010. doi:10.1007/s1126300902727
K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “BM3D Image Denoising with ShapeAdaptive Principal Component Analysis”, Proc. Workshop on Signal Processing with Adaptive Sparse Structured Representations (SPARS'09), SaintMalo, France, April 2009. Poster