(1) A probability model respecting those covariance observations: Gaussian Maximum entropy probability distribution for a given covariance observation (shown zero mean for notational convenience) : 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 Image pixels Inverse covariance matrix

Power spectra of typical images Experimentally, the power spectrum as a function of Fourier frequency is observed to follow a power law.

Random draw from Gaussian spectral model

Noise removal (in frequency domain), under Gaussian assumption Variance of white, Gaussian additive noise Observed Fourier component Estimated Fourier component Power law prior probability on estimated Fourier component 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): Posterior probability for X

Noise removal, under Gaussian assumption original With Gaussian noise of std. dev added, giving PSNR=22.06 (1) Denoised with Gaussian model, PSNR= (try to ignore JPEG compression artifacts from the PDF file)

(2) The wavelet marginal model Histogram of wavelet coefficients, c, for various images. Parameter determining width of distribution Parameter determining peakiness of distribution Wavelet coefficient value

Random draw from the wavelet marginal model

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

P(x, y) = P(x|y) P(y) so P(x|y) P(y) = P(y|x) P(x) P(x, y) = P(x|y) P(y) so P(x|y) P(y) = P(y|x) P(x) and P(x|y) = P(y|x) P(x) / P(y) Bayes theorem The parameters you want to estimate What you observe Prior probability Likelihood function Constant w.r.t. parameters x. P(x, y) = P(x|y) P(y) By definition of conditional probability Using that twice

P(x) Bayesian MAP estimator for clean bandpass coefficient values Let x = bandpassed image value before adding noise. Let y = noise-corrupted observation. By Bayes theorem P(x|y) = k P(y|x) P(x) P(y|x) P(x|y) P(y|x) y y = 25

Bayesian MAP estimator Let x = bandpassed image value before adding noise. Let y = noise-corrupted observation. By Bayes theorem P(x|y) = k P(y|x) P(x) y P(y|x) P(x|y) y = 50

Bayesian MAP estimator Let x = bandpassed image value before adding noise. Let y = noise-corrupted observation. By Bayes theorem P(x|y) = k P(y|x) P(x) y P(y|x) P(x|y) y = 115

P(x) P(y|x) y y = 25 P(x|y) y P(y|x) P(x|y) y = 115 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,, as function of observed coefficient value, y Simoncelli and Adelson, Noise Removal via Bayesian Wavelet Coring

original With Gaussian noise of std. dev added, giving PSNR=22.06 (1) Denoised with Gaussian model, PSNR=27.87 (2) Denoised with wavelet marginal model, PSNR=

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.

Statistics of pairs of wavelet coefficients Contour plots of the joint histogram of various wavelet coefficient pairs Conditional distributions of the corresponding wavelet pairs

(3) Gaussian scale mixtures Wavelet coefficient probability A mixture of Gaussians of scaled covariances observed Gaussian scale mixture model simulation 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.

original With Gaussian noise of std. dev added, giving PSNR=22.06 (1) Denoised with Gaussian model, PSNR= (3) Denoised with Gaussian scale mixture model, PSNR=30.86 (2) Denoised with wavelet marginal model, PSNR=29.24