Information Theory, Inference, and Learning ... - Inference Group

Copyright Cambridge University Press 2003. On-screen viewing permitted. Printing not permitted. http://www.cambridge.org/0521642981You can buy this book for 30 pounds or $50. See http://www.inference.phy.cam.ac.uk/mackay/itila/ for links.46Deconvolution46.1 Traditional image reconstruction methodsOptimal linear filtersIn many imaging problems, the data measurements {d n } are linearly relatedto the underlying image f:d n = ∑ kR nk f k + n n . (46.1)The vector n denotes the inevitable noise that corrupts real data. In the caseof a camera which produces a blurred picture, the vector f denotes the trueimage, d denotes the blurred and noisy picture, and the linear operator Ris a convolution defined by the point spread function of the camera. In thisspecial case, the true image and the data vector reside in the same space;but it is important to maintain a distinction between them. We will use thesubscript n = 1, . . . , N to run over data measurements, and the subscriptsk, k ′ = 1, . . . , K to run over image pixels.One might speculate that since the blur was created by a linear operation,then perhaps it might be deblurred by another linear operation. We can derivethe optimal linear filter in two ways.Bayesian derivationWe assume that the linear operator R is known, and that the noise n isGaussian and independent, with a known standard deviation σ ν .(1P (d | f, σ ν , H) =(2πσν 2 exp − ∑ (d n − ∑ ))N/2 k R nkf k ) 2/ (2σν 2 ) . (46.2)nWe assume that the prior probability of the image is also Gaussian, with ascale parameter σ f .⎛⎞P (f | σ f , H) =det− 1 2 C(2πσf 2 exp ⎝− ∑ f k C kk ′f k/ ′ (2σ2f ) ⎠ . (46.3))K/2 k,k ′If we assume no correlations among the pixels then the symmetric, full rankmatrix C is equal to the identity matrix I. The more sophisticated ‘intrinsiccorrelation function’ model uses C = [GG T ] −1 , where G is a convolution thattakes us from an imaginary ‘hidden’ image, which is uncorrelated, to the realcorrelated image. The intrinsic correlation function should not be confusedwith the point spread function R which defines the image-to-data mapping.549