Wiener filtering

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. [38

]). This has the same affect as chosing a Gaussian prior so that Equation 16

becomes

where $C$ is the covariance matrix of the image vector $H$ given by

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

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

where the Wiener filter, $W$ , is given by

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 ]).