(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 ) exp( ) ( 1 2 1 x C x x P x T Image pixels Inverse covariance matri ) | ) ( (| ) ( 2 F E D
29
Embed
(1) A probability model respecting those covariance observations: Gaussian Maximum entropy probability distribution for a given covariance observation.
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
(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
)exp()( 12
1 xCxxP xT Image pixels
Inverse covariance matrix
)|)((|)( 2 FED
Power spectra of typical images
Experimentally, the power spectrum as a function of Fourier frequency is observed to follow a power law.
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.
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. 21.4 added, giving PSNR=22.06