6.2 Wiener filtering

In the absence of non-Gaussian sources it is possible to simplify the Maximum Entropy method using the quadratic approximation to the entropy (Hobson et al. [38Jump To The Next Citation Point]). This has the same affect as chosing a Gaussian prior so that Equation 16View Equation becomes
1- † −1 Pr(H ) ∝ exp [− 2 H C H ] (19 )
where C is the covariance matrix of the image vector H given by
C = H †H. (20 )
The solution to the analysis with this Bayesian prior is called Wiener filtering (see, for example, [12] and [92]) and has been applied to many data sets in the past when non-Gaussianity could be ignored. The data D can be written as
D = B ⋆ H + 𝜖 (21 )
where the convolution of the image vector H is with B, the beam response of the instrument and frequency dependance of H. 𝜖 is the noise vector. In this case, the best reconstructed image vector, ˆ H, is given by
ˆ H = W D (22 )
where the Wiener filter, W, is given by
W = CB †(BCB † + N )−1 (23 )
and N is the noise covariance matrix given by † N = 𝜖 𝜖. It is important to note that not only the CMB signal but all the foregrounds are implicitly assumed to be non-Gaussian in this method ([38] [45] and [37]).
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