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The Essential Guide to Image Processing- P8:We are in the middle of an exciting period of time in the field of image processing. Indeed, scarcely a week passes where we do not hear an announcement of some new technological breakthrough in the areas of digital computation and telecommunication. | 210 CHAPTER 9 Capturing Visual Image Properties with Probabilistic Models by the conditional density of the observed noisy image y given the original clean image x P y x exp y -x 2 2a-2 where is the variance of the noise. Using Bayes rule we can reverse the conditioning by multiplying by the prior probability density on x P x y exp y - x 2 202 P x . An estimate x for x may now be obtained from this posterior density. One can for example choose the x that maximizes the probability the maximum a posteriori or MAP estimate or the mean of the density the minimum mean squared error MMSE or Bayes Least Squares BLS estimate . If we assume that the prior density is Gaussian then the posterior density will also be Gaussian and the maximum and the mean will then be identical x y Cx Cx Io-2 1y where I is an identity matrix. Note that this solution is linear in the observed noisy image y. This linear estimator is particularly simple when both the noise and signal covariance matrices are diagonalized. As mentioned previously under the spectral model the signal covariance matrix may be diagonlized by transforming to the Fourier domain where the estimator maybe written as A W F -----2 A y 7 n G S where F a and G a are the Fourier transforms of x y and y respectively. Thus the estimate may be computed by linearly rescaling each Fourier coefficient individually. In order to apply this denoising method one must be given or must estimate the parameters A y and an see Chapter 11 for further examples and development of the denoising problem . Despite the simplicity and tractability of the Gaussian model it is easy to see that the model provides a rather weak description of images. In particular while the model strongly constrains the amplitudes of the Fourier coefficients it places no constraint on their phases. When one randomizes the phases of an image the appearance is completely destroyed 13 . As a direct test one can draw sample images from the distribution by simply generating .
