American Institute of Mathematical Sciences

Advanced
August  2013, 7(3): 813-838. doi: 10.3934/ipi.2013.7.813

How to explore the patch space

Jose-Luis Lisani 1, , Antoni Buades 2, and  Jean-Michel Morel 3,

1. 

Universitat de les Illes Balears, Crta. de Valldemossa, km 7.5, 07122 Palma de Mallorca
2. 

Universitat de les Illes Balears, Ctra Valldemossa km 7.5, Palma de Mallorca, 07122
3. 

CMLA, ENS Cachan, 61 avenue du Président Wilson, 94235 Cachan Cedex

Received  January 2013 Revised  April 2013 Published  September 2013

Patches are small images (typically $8\times 8$ to $12\times 12$) extracted from natural images. The ``patch space'' is the set of all observable patches extracted from digital images in the world. This observable space is huge and should permit a sophisticated statistical analysis. In the past ten years, statistical inquiries and applications involving the ``patch space'' have tried to explore its structure on several levels. The first attempts have invalidated models based on PCA or Fourier analysis. Redundant bases (or patch dictionaries) obtained by independent component analysis (ICA) or related processes have tried to find a reduced set of patches on which every other patch obtains a sparse decomposition. Optimization algorithms such as EM have been used to explore the patch space as a Gaussian mixture. The goal of the present paper is to review this literature, and to extend its methodology to gain more insight on the independent components of the patch space.
    The conclusion of our analysis is that the sophisticated ICA tools introduced to analyze the patch space require a previous geometric normalization step to yield non trivial results. Indeed, we demonstrate by a simple experimental setup and by the analysis of the literature that, without this normalization, the patch space structure is actually hidden by the rotations, translations, and contrast changes. Thus, ICA models applied on a random set of patches boil down to segmenting the patch space depending on insignificant dimensions such as the patch orientation or the position of its gradient barycenter. When, instead of exploring the raw patches, one decides to explore the quotient of the set of patches by these action groups, a geometrically interpretable patch structure is revealed.
Keywords: Principal component analysis (PCA), sparse representation, image patch models., independent component analysis (ICA), digital image representation.
Mathematics Subject Classification: 6.
Citation: Jose-Luis Lisani, Antoni Buades, Jean-Michel Morel. How to explore the patch space. Inverse Problems & Imaging, 2013, 7 (3) : 813-838. doi: 10.3934/ipi.2013.7.813
References:
[1]

M. Aharon, Michael Elad and A. Bruckstein, K-SVD: Design of dictionaries for sparse representation,, IEEE Transactions on Image Processing, (2005), 9.  doi: 10.1109/TSP.2006.881199.  Google Scholar

[2]

Michal Aharon, Michael Elad and Alfred Bruckstein, K-SVD: An algorithm for designing overcomplete dictionaries for sparse representation,, IEEE Transactions on Signal Processing, 54 (2006), 4311.  doi: 10.1109/TSP.2006.881199.  Google Scholar

[3]

C. V. Angelino, E. Debreuve and M. Barlaud, et al, Confidence-based denoising relying on a transformation-invariant, robust patch similarity exploring ways to improve patch synchronous summation,, In, (2011).   Google Scholar

[4]

A. J. Bell and T. J. Sejnowski, The independent components of natural scenes are edge filters,, Vision Research, 37 (1997), 3327.  doi: 10.1016/S0042-6989(97)00121-1.  Google Scholar

[5]

A. Buades, B. Coll and J. M. Morel, A review of image denoising algorithms, with a new one,, Multiscale Modeling Simulation, 4 (2005), 490.  doi: 10.1137/040616024.  Google Scholar

[6]

A. Buades, M. Lebrun and J. M. Morel, Implementation of the "non-local bayes'' image denoising algorithm,, Image Processing On Line (http:www.ipol.im), (2012), 1.   Google Scholar

[7]

J. Canny, A computational approach to edge detection,, IEEE Trans. Pattern Analysis and Machine Intelligence, 8 (1986), 679.  doi: 10.1109/TPAMI.1986.4767851.  Google Scholar

[8]

P. Chatterjee and P. Milanfar, Patch-based near-optimal image denoising,, IEEE Transactions on Image Processing: A Publication of the IEEE Signal Processing Society, 21 (2011), 1635.  doi: 10.1109/TIP.2011.2172799.  Google Scholar

[9]

S. F. Cotter, R. Adler, R. D. Rao and K. Kreutz-Delgado, Forward sequential algorithms for best basis selection,, In, 146 (1999), 235.  doi: 10.1049/ip-vis:19990445.  Google Scholar

[10]

K. Dabov, A. Foi, V. Katkovnik and K. Egiazarian, Image denoising by sparse 3-D transform-domain collaborative filtering,, IEEE Transactions on Image Processing, 16 (2007), 2080.  doi: 10.1109/TIP.2007.901238.  Google Scholar

[11]

A. Delorme and Makeig S, Eeglab: An open source toolbox for analysis of single-trial eeg dynamics,, Journal of Neuroscience Methods, 134 (2004), 9.  doi: 10.1016/j.jneumeth.2003.10.009.  Google Scholar

[12]

A. Efros and T. Leung, Texture synthesis by non parametric sampling,, In, 2 (1999), 1033.  doi: 10.1109/ICCV.1999.790383.  Google Scholar

[13]

M. Elad and M. Aharon, Image denoising via sparse and redundant representations over learned dictionaries,, Image Processing, 15 (2006), 3736.  doi: 10.1109/TIP.2006.881969.  Google Scholar

[14]

A. Foi and G. Boracchi, Foveated self-similarity in nonlocal image filtering,, In, (8291), 829110.  doi: 10.1117/12.912217.  Google Scholar

[15]

S. Geman and D. Geman, Stochastic relaxation, gibbs distributions and the bayesian restoration of images,, IEEE Pat. Anal. Mach. Intell., 6 (1984), 721.   Google Scholar

[16]

S. Grewenig, S. Zimmer and J. Weickert, Rotationally invariant similarity measures for nonlocal image denoising,, Journal of Visual Communication and Image Representation, 22 (2011), 117.  doi: 10.1016/j.jvcir.2010.11.001.  Google Scholar

[17]

Guillermo Sapiro Guoshen Yu, DCT image denoising: A simple and effective image denoising algorithm,, Image Processing On Line, (2011).   Google Scholar

[18]

David H Hubel, "Eye, Brain, and Vision,", Scientific American Library New York, (1988).   Google Scholar

[19]

A. Hyvarinen, Fast and robust fixed-point algorithms for independent component analysis,, IEEE Transactions on Neural Networks, 10 (1999), 626.  doi: 10.1109/72.761722.  Google Scholar

[20]

A. Hyvarinen, The fixed-point algorithm and maximum likelihood estimation for independent component analysis,, Neural Processing Letters, 10 (1999), 1.   Google Scholar

[21]

A. Hyvarinen and E. Oja, Independent component analysis: Algorithms and applications,, Neural Networks, 13 (2000), 411.  doi: 10.1016/S0893-6080(00)00026-5.  Google Scholar

[22]

Z. Ji, Q. Chen, Q. S. Sun and D. S. Xia, A moment-based nonlocal-means algorithm for image denoising,, Information Processing Letters, 109 (2009), 1238.  doi: 10.1016/j.ipl.2009.09.007.  Google Scholar

[23]

I. T. Jolliffe, N. T. Trendafilov and M. Uddin, A modified principal component technique based on the Lasso,, Journal of Computational and Graphical Statistics, 12 (2003), 531.  doi: 10.1198/1061860032148.  Google Scholar

[24]

M. Lebrun, M. Colom, A. Buades and JM Morel, Secrets of image denoising cuisine,, Acta Numerica, 21 (2012), 475.  doi: 10.1017/S0962492912000062.  Google Scholar

[25]

A. B. Lee, K. S. Pedersen and D. Mumford, The nonlinear statistics of high-contrast patches in natural images,, International Journal of Computer Vision, 54 (2003), 83.   Google Scholar

[26]

M. S. Lewicki and T. J. Sejnowski, Learning overcomplete representations,, Neural computation, 12 (2000), 337.  doi: 10.1162/089976600300015826.  Google Scholar

[27]

Y. Lou, P. Favaro, S. Soatto and A. Bertozzi, Nonlocal similarity image filtering,, Image Analysis and Processing-ICIAP 2009, 5716 (2009), 62.  doi: 10.1007/978-3-642-04146-4_9.  Google Scholar

[28]

J. Mairal, F. Bach, J. Ponce and G. Sapiro, Online learning for matrix factorization and sparse coding,, The Journal of Machine Learning Research, 11 (2010), 19.   Google Scholar

[29]

J. Mairal, F. Bach, J. Ponce, G. Sapiro and A. Zisserman, Non-local sparse models for image restoration,, In, (2009), 2272.  doi: 10.1109/ICCV.2009.5459452.  Google Scholar

[30]

D. Martin, C. Fowlkes, D. Tal and J. Malik, A database of human segmented natural images and its application to evaluating segmentation algorithms and measuring ecological statistics,, In, 2 (2001), 416.  doi: 10.1109/ICCV.2001.937655.  Google Scholar

[31]

Y. Meyer, "Wavelets-Algorithms and Applications,", Wavelets-Algorithms and applications Society for Industrial and Applied Mathematics (SIAM), (1993).   Google Scholar

[32]

B. A. Olshausen, D. J. Field, et al, Sparse coding with an overcomplete basis set: A strategy employed by V1,, Vision research, 37 (1997), 3311.  doi: 10.1016/S0042-6989(97)00169-7.  Google Scholar

[33]

L. U. Perrinet, Role of homeostasis in learning sparse representations,, Neural computation, 22 (2010), 1812.  doi: 10.1162/neco.2010.05-08-795.  Google Scholar

[34]

Javier Portilla, Vasily Strela, Martin J. Wainwright and Eero P. Simoncelli, Image denoising using scale mixtures of gaussians in the wavelet domain,, IEEE Trans. Image Process, 12 (2003), 1338.  doi: 10.1109/TIP.2003.818640.  Google Scholar

[35]

W. H. Richardson, Bayesian-based iterative method of image restoration,, JOSA, 62 (1972), 55.  doi: 10.1364/JOSA.62.000055.  Google Scholar

[36]

WF Sun, YH Peng and WL Hwang, Modified similarity metric for non-local means algorithm,, Electronics Letters, 45 (2009), 1307.  doi: 10.1049/el.2009.2406.  Google Scholar

[37]

L. Yaroslavsky and M. Eden, "Fundamentals of Digital Optics,", Birkhäuser, (2003).  doi: 10.1007/978-1-4612-0845-7.  Google Scholar

[38]

G. Yu, G. Sapiro and S. Mallat, Image modeling and enhancement via structured sparse model selection,, In, (2010), 1641.  doi: 10.1109/ICIP.2010.5653853.  Google Scholar

[39]

G. Yu, G. Sapiro and S. Mallat, Solving inverse problems with piecewise linear estimators: From gaussian mixture models to structured sparsity,, IEEE Trans. Image Process, 21 (2012), 2481.  doi: 10.1109/TIP.2011.2176743.  Google Scholar

[40]

S. Zimmer, S. Didas and J. Weickert, A rotationally invariant block matching strategy improving image denoising with non-local means,, In, (2008).   Google Scholar

[41]

T. Zito, N. Wilbert, L. Wiskott and P. Berkes, Modular toolkit for data processing (MDP): A python data processing frame work,, Front. Neuroinform., 2 (2008).   Google Scholar

[42]

D. Zoran and Y. Weiss, From learning models of natural image patches to whole image restoration,, In, (2011), 479.  doi: 10.1109/ICCV.2011.6126278.  Google Scholar

[43]

H. Zou, T. Hastie and R. Tibshirani, Sparse principal component analysis,, Journal of Computational and Graphical Statistics, 15 (2006), 265.  doi: 10.1198/106186006X113430.  Google Scholar

show all references

References:
[1]

M. Aharon, Michael Elad and A. Bruckstein, K-SVD: Design of dictionaries for sparse representation,, IEEE Transactions on Image Processing, (2005), 9.  doi: 10.1109/TSP.2006.881199.  Google Scholar

[2]

Michal Aharon, Michael Elad and Alfred Bruckstein, K-SVD: An algorithm for designing overcomplete dictionaries for sparse representation,, IEEE Transactions on Signal Processing, 54 (2006), 4311.  doi: 10.1109/TSP.2006.881199.  Google Scholar

[3]

C. V. Angelino, E. Debreuve and M. Barlaud, et al, Confidence-based denoising relying on a transformation-invariant, robust patch similarity exploring ways to improve patch synchronous summation,, In, (2011).   Google Scholar

[4]

A. J. Bell and T. J. Sejnowski, The independent components of natural scenes are edge filters,, Vision Research, 37 (1997), 3327.  doi: 10.1016/S0042-6989(97)00121-1.  Google Scholar

[5]

A. Buades, B. Coll and J. M. Morel, A review of image denoising algorithms, with a new one,, Multiscale Modeling Simulation, 4 (2005), 490.  doi: 10.1137/040616024.  Google Scholar

[6]

A. Buades, M. Lebrun and J. M. Morel, Implementation of the "non-local bayes'' image denoising algorithm,, Image Processing On Line (http:www.ipol.im), (2012), 1.   Google Scholar

[7]

J. Canny, A computational approach to edge detection,, IEEE Trans. Pattern Analysis and Machine Intelligence, 8 (1986), 679.  doi: 10.1109/TPAMI.1986.4767851.  Google Scholar

[8]

P. Chatterjee and P. Milanfar, Patch-based near-optimal image denoising,, IEEE Transactions on Image Processing: A Publication of the IEEE Signal Processing Society, 21 (2011), 1635.  doi: 10.1109/TIP.2011.2172799.  Google Scholar

[9]

S. F. Cotter, R. Adler, R. D. Rao and K. Kreutz-Delgado, Forward sequential algorithms for best basis selection,, In, 146 (1999), 235.  doi: 10.1049/ip-vis:19990445.  Google Scholar

[10]

K. Dabov, A. Foi, V. Katkovnik and K. Egiazarian, Image denoising by sparse 3-D transform-domain collaborative filtering,, IEEE Transactions on Image Processing, 16 (2007), 2080.  doi: 10.1109/TIP.2007.901238.  Google Scholar

[11]

A. Delorme and Makeig S, Eeglab: An open source toolbox for analysis of single-trial eeg dynamics,, Journal of Neuroscience Methods, 134 (2004), 9.  doi: 10.1016/j.jneumeth.2003.10.009.  Google Scholar

[12]

A. Efros and T. Leung, Texture synthesis by non parametric sampling,, In, 2 (1999), 1033.  doi: 10.1109/ICCV.1999.790383.  Google Scholar

[13]

M. Elad and M. Aharon, Image denoising via sparse and redundant representations over learned dictionaries,, Image Processing, 15 (2006), 3736.  doi: 10.1109/TIP.2006.881969.  Google Scholar

[14]

A. Foi and G. Boracchi, Foveated self-similarity in nonlocal image filtering,, In, (8291), 829110.  doi: 10.1117/12.912217.  Google Scholar

[15]

S. Geman and D. Geman, Stochastic relaxation, gibbs distributions and the bayesian restoration of images,, IEEE Pat. Anal. Mach. Intell., 6 (1984), 721.   Google Scholar

[16]

S. Grewenig, S. Zimmer and J. Weickert, Rotationally invariant similarity measures for nonlocal image denoising,, Journal of Visual Communication and Image Representation, 22 (2011), 117.  doi: 10.1016/j.jvcir.2010.11.001.  Google Scholar

[17]

Guillermo Sapiro Guoshen Yu, DCT image denoising: A simple and effective image denoising algorithm,, Image Processing On Line, (2011).   Google Scholar

[18]

David H Hubel, "Eye, Brain, and Vision,", Scientific American Library New York, (1988).   Google Scholar

[19]

A. Hyvarinen, Fast and robust fixed-point algorithms for independent component analysis,, IEEE Transactions on Neural Networks, 10 (1999), 626.  doi: 10.1109/72.761722.  Google Scholar

[20]

A. Hyvarinen, The fixed-point algorithm and maximum likelihood estimation for independent component analysis,, Neural Processing Letters, 10 (1999), 1.   Google Scholar

[21]

A. Hyvarinen and E. Oja, Independent component analysis: Algorithms and applications,, Neural Networks, 13 (2000), 411.  doi: 10.1016/S0893-6080(00)00026-5.  Google Scholar

[22]

Z. Ji, Q. Chen, Q. S. Sun and D. S. Xia, A moment-based nonlocal-means algorithm for image denoising,, Information Processing Letters, 109 (2009), 1238.  doi: 10.1016/j.ipl.2009.09.007.  Google Scholar

[23]

I. T. Jolliffe, N. T. Trendafilov and M. Uddin, A modified principal component technique based on the Lasso,, Journal of Computational and Graphical Statistics, 12 (2003), 531.  doi: 10.1198/1061860032148.  Google Scholar

[24]

M. Lebrun, M. Colom, A. Buades and JM Morel, Secrets of image denoising cuisine,, Acta Numerica, 21 (2012), 475.  doi: 10.1017/S0962492912000062.  Google Scholar

[25]

A. B. Lee, K. S. Pedersen and D. Mumford, The nonlinear statistics of high-contrast patches in natural images,, International Journal of Computer Vision, 54 (2003), 83.   Google Scholar

[26]

M. S. Lewicki and T. J. Sejnowski, Learning overcomplete representations,, Neural computation, 12 (2000), 337.  doi: 10.1162/089976600300015826.  Google Scholar

[27]

Y. Lou, P. Favaro, S. Soatto and A. Bertozzi, Nonlocal similarity image filtering,, Image Analysis and Processing-ICIAP 2009, 5716 (2009), 62.  doi: 10.1007/978-3-642-04146-4_9.  Google Scholar

[28]

J. Mairal, F. Bach, J. Ponce and G. Sapiro, Online learning for matrix factorization and sparse coding,, The Journal of Machine Learning Research, 11 (2010), 19.   Google Scholar

[29]

J. Mairal, F. Bach, J. Ponce, G. Sapiro and A. Zisserman, Non-local sparse models for image restoration,, In, (2009), 2272.  doi: 10.1109/ICCV.2009.5459452.  Google Scholar

[30]

D. Martin, C. Fowlkes, D. Tal and J. Malik, A database of human segmented natural images and its application to evaluating segmentation algorithms and measuring ecological statistics,, In, 2 (2001), 416.  doi: 10.1109/ICCV.2001.937655.  Google Scholar

[31]

Y. Meyer, "Wavelets-Algorithms and Applications,", Wavelets-Algorithms and applications Society for Industrial and Applied Mathematics (SIAM), (1993).   Google Scholar

[32]

B. A. Olshausen, D. J. Field, et al, Sparse coding with an overcomplete basis set: A strategy employed by V1,, Vision research, 37 (1997), 3311.  doi: 10.1016/S0042-6989(97)00169-7.  Google Scholar

[33]

L. U. Perrinet, Role of homeostasis in learning sparse representations,, Neural computation, 22 (2010), 1812.  doi: 10.1162/neco.2010.05-08-795.  Google Scholar

[34]

Javier Portilla, Vasily Strela, Martin J. Wainwright and Eero P. Simoncelli, Image denoising using scale mixtures of gaussians in the wavelet domain,, IEEE Trans. Image Process, 12 (2003), 1338.  doi: 10.1109/TIP.2003.818640.  Google Scholar

[35]

W. H. Richardson, Bayesian-based iterative method of image restoration,, JOSA, 62 (1972), 55.  doi: 10.1364/JOSA.62.000055.  Google Scholar

[36]

WF Sun, YH Peng and WL Hwang, Modified similarity metric for non-local means algorithm,, Electronics Letters, 45 (2009), 1307.  doi: 10.1049/el.2009.2406.  Google Scholar

[37]

L. Yaroslavsky and M. Eden, "Fundamentals of Digital Optics,", Birkhäuser, (2003).  doi: 10.1007/978-1-4612-0845-7.  Google Scholar

[38]

G. Yu, G. Sapiro and S. Mallat, Image modeling and enhancement via structured sparse model selection,, In, (2010), 1641.  doi: 10.1109/ICIP.2010.5653853.  Google Scholar

[39]

G. Yu, G. Sapiro and S. Mallat, Solving inverse problems with piecewise linear estimators: From gaussian mixture models to structured sparsity,, IEEE Trans. Image Process, 21 (2012), 2481.  doi: 10.1109/TIP.2011.2176743.  Google Scholar

[40]

S. Zimmer, S. Didas and J. Weickert, A rotationally invariant block matching strategy improving image denoising with non-local means,, In, (2008).   Google Scholar

[41]

T. Zito, N. Wilbert, L. Wiskott and P. Berkes, Modular toolkit for data processing (MDP): A python data processing frame work,, Front. Neuroinform., 2 (2008).   Google Scholar

[42]

D. Zoran and Y. Weiss, From learning models of natural image patches to whole image restoration,, In, (2011), 479.  doi: 10.1109/ICCV.2011.6126278.  Google Scholar

[43]

H. Zou, T. Hastie and R. Tibshirani, Sparse principal component analysis,, Journal of Computational and Graphical Statistics, 15 (2006), 265.  doi: 10.1198/106186006X113430.  Google Scholar

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