(1) A probability model respecting those covariance observations: Gaussian

1
(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

Inverse covariance matrix
Image pixels
2
Power spectra of typical images
Experimentally, the power spectrum as a function
of Fourier frequency is observed to follow a
power law.
http//www.cns.nyu.edu/pub/eero/simoncelli05a-prep
rint.pdf
3
Random draw from Gaussian spectral model
http//www.cns.nyu.edu/pub/eero/simoncelli05a-prep
rint.pdf
4
Noise removal (in frequency domain), under
Gaussian assumption
Observed Fourier component
Power law prior probability on estimated Fourier
component
Estimated Fourier component
Posterior probability for X
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)
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Noise removal, under Gaussian assumption
(1) Denoised with Gaussian model, PSNR27.87
With Gaussian noise of std. dev. 21.4 added,
giving PSNR22.06
original
(try to ignore JPEG compression artifacts from
the PDF file)
http//www.cns.nyu.edu/pub/eero/simoncelli05a-prep
rint.pdf
6
(2) The wavelet marginal model

• Histogram of wavelet coefficients, c, for various
  images.

http//www.cns.nyu.edu/pub/eero/simoncelli05a-prep
rint.pdf
Parameter determining peakiness of distribution
Parameter determining width of distribution
Wavelet coefficient value
7
Random draw from the wavelet marginal model
http//www.cns.nyu.edu/pub/eero/simoncelli05a-prep
rint.pdf
8
And again something that is reminiscent of
operations found in V1
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An application of image pyramidsnoise removal
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Image statistics (or, mathematically, how can you
tell image from noise?)
Noisy image
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Clean image
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Pixel representation, image histogram
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Pixel representation, noisy image histogram
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bandpassed representation image histogram
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Pixel domain noise image and histogram
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Bandpass domain noise image and histogram
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Noise-corrupted full-freq and bandpass images
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Bayes theorem
P(x, y) P(xy) P(y) so P(xy) P(y) P(yx)
P(x)
P(x, y) P(xy) P(y) so P(xy) P(y) P(yx)
P(x) and P(xy) P(yx) P(x) / P(y)
Using that twice
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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(xy) k P(yx) P(x)
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Bayesian MAP estimator
Let x bandpassed image value before adding
noise. Let y noise-corrupted observation. By
Bayes theorem
y 50
P(xy) k P(yx) P(x)
y
P(yx)
P(xy)
21
Bayesian MAP estimator
Let x bandpassed image value before adding
noise. Let y noise-corrupted observation. By
Bayes theorem
y 115
P(xy) k P(yx) P(x)
y
P(yx)
P(xy)
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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
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MAP estimate, , as function of observed
coefficient value, y
Simoncelli and Adelson, Noise Removal via
Bayesian Wavelet Coring
http//www-bcs.mit.edu/people/adelson/pub_pdfs/sim
oncelli_noise.pdf
24
(1) Denoised with Gaussian model, PSNR27.87
With Gaussian noise of std. dev. 21.4 added,
giving PSNR22.06
original
(2) Denoised with wavelet marginal model,
PSNR29.24
http//www.cns.nyu.edu/pub/eero/simoncelli05a-prep
rint.pdf
25
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.
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Motivation for wavelet joint models
Note correlations between the amplitudes of each
wavelet subband.
http//www.cns.nyu.edu/pub/eero/simoncelli05a-prep
rint.pdf
27
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-prep
rint.pdf
28
(3) Gaussian scale mixtures
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.
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(1) Denoised with Gaussian model, PSNR27.87
With Gaussian noise of std. dev. 21.4 added,
giving PSNR22.06
original
(3) Denoised with Gaussian scale mixture model,
PSNR30.86
http//www.cns.nyu.edu/pub/eero/simoncelli05a-prep
rint.pdf