(1) A probability model respecting those covariance observations: Gaussian • Maximum entropy probability distribution for a given covariance observation (shown zero mean for notational convenience): Inverse covariance matrix P( x ) exp( 12 x T Cx1 x ) Image pixels • If we rotate coordinates to the Fourier basis, the covariance matrix in that basis will be diagonal. So in that model, each Fourier transform coefficient is an independent Gaussian random variable of covariance D( ) E (| F ( ) |2 ) Power spectra of typical images Experimentally, the power spectrum as a function of Fourier frequency is observed to follow a power law. E (| F ( ) | ) 2 http://www.cns.nyu.edu/pub/eero/simoncelli05a-preprint.pdf A Random draw from Gaussian spectral model http://www.cns.nyu.edu/pub/eero/simoncelli05a-preprint.pdf Noise removal (in frequency domain), under Gaussian assumption Posterior Observed Fourier component Estimated Fourier probability for X component Power law prior probability on estimated Fourier component P( X | Y ) exp( || Y X || 2 ) exp( X ( 2 2 n A 1 ) X 2) Variance of white, Gaussian additive noise Setting to zero the derivative of the the log probability of X gives an analytic form for the optimal estimate of X (or just complete the square): Xˆ ( ) A Y ( ) 2 A n Noise removal, under Gaussian assumption original With Gaussian noise of std. dev. 21.4 added, giving PSNR=22.06 (try to ignore JPEG compression artifacts from the PDF file) http://www.cns.nyu.edu/pub/eero/simoncelli05a-preprint.pdf (1) Denoised with Gaussian model, PSNR=27.87 (2) The wavelet marginal model Histogram of wavelet coefficients, c, for various images. http://www.cns.nyu.edu/pub/eero/simoncelli05a-preprint.pdf P(c) exp( | c s | ) p Wavelet coefficient value Parameter determining peakiness of distribution Parameter determining width of distribution Random draw from the wavelet marginal model http://www.cns.nyu.edu/pub/eero/simoncelli05a-preprint.pdf And again something that is reminiscent of operations found in V1… An application of image pyramids: noise removal Image statistics (or, mathematically, how can you tell image from noise?) Noisy image Clean image Pixel representation, image histogram Pixel representation, noisy image histogram bandpassed representation image histogram Pixel domain noise image and histogram Bandpass domain noise image and histogram Noise-corrupted full-freq and bandpass images But want the bandpass image histogram to look like this Bayes theorem By definition of conditional probability P(x, y) = P(x|y) P(y) so Using that twice P(x|y) P(y) = P(y|x) P(x) and P(x|y) = P(y|x) P(x) / P(y) The parameters you want to estimate Constant w.r.t. Likelihood parameters x. function Prior probability What you observe Bayesian MAP estimator for clean bandpass coefficient values Let x = bandpassed image value before adding noise. Let y = noise-corrupted observation. By Bayes theorem y = 25 P(x|y) = k P(y|x) P(x) y P(x) P(y|x) P(y|x) P(x|y) P(x|y) Bayesian MAP estimator Let x = bandpassed image value before adding noise. Let y = noise-corrupted observation. By Bayes theorem y = 50 P(x|y) = k P(y|x) P(x) y P(y|x) P(x|y) Bayesian MAP estimator Let x = bandpassed image value before adding noise. Let y = noise-corrupted observation. By Bayes theorem y = 115 P(x|y) = k P(y|x) P(x) y P(y|x) P(x|y) y = 25 y = 115 y P(x) P(y|x) P(y|x) P(x|y) y P(x|y) For small y: probably it is due to noise and y should be set to 0 For large y: probably it is due to an image edge and it should be kept untouched MAP estimate, x̂ , as function of observed coefficient value, y x̂ y http://www-bcs.mit.edu/people/adelson/pub_pdfs/simoncelli_noise.pdf Simoncelli and Adelson, Noise Removal via Bayesian Wavelet Coring original With Gaussian noise of std. dev. 21.4 added, giving PSNR=22.06 (2) Denoised with wavelet marginal model, PSNR=29.24 http://www.cns.nyu.edu/pub/eero/simoncelli05a-preprint.pdf (1) Denoised with Gaussian model, PSNR=27.87 M. F. Tappen, B. C. Russell, and W. T. Freeman, Efficient graphical models for processing images IEEE Conf. on Computer Vision and Pattern Recognition (CVPR) Washington, DC, 2004. Motivation for wavelet joint models Note correlations between the amplitudes of each wavelet subband. http://www.cns.nyu.edu/pub/eero/simoncelli05a-preprint.pdf Statistics of pairs of wavelet coefficients Contour plots of the joint histogram of various wavelet coefficient pairs Conditional distributions of the corresponding wavelet pairs http://www.cns.nyu.edu/pub/eero/simoncelli05a-preprint.pdf (3) Gaussian scale mixtures P( x ) 1 exp( x ( z) x ) 1 T 2 N 2 (2 ) | z | 12 Pz ( z ) dz observed Wavelet coefficient probability A mixture of Gaussians of scaled covariances z is a spatially varying hidden variable that can be used to (a) Create the non-gaussian histograms from a mixture of Gaussian densities, and (b) model correlations between the neighboring wavelet coefficients. Gaussian scale mixture model simulation original With Gaussian noise of std. dev. 21.4 added, giving PSNR=22.06 (1) Denoised with Gaussian model, PSNR=27.87 (3) Denoised with Gaussian scale mixture model, PSNR=30.86 http://www.cns.nyu.edu/pub/eero/simoncelli05a-preprint.pdf (2) Denoised with wavelet marginal model, PSNR=29.24