If we start with Bayes’ theorem which states, given a hypothesis and some data the posterior probability is the product of the likelihood and the prior probability , normalised by the evidence ,

If the instrumental noise on each frequency channel is Gaussian-distributed, then the probability distribution of the noise is a multivariate Gaussian. Assuming the expectation value of the noise to be zero at each observing frequency, the likelihood is therefore given by where the misfit statistic has been introduced.We now have to decide the form of the prior probability. Let us consider a discretised image consisting of cells, so that ; we may consider the as the components of an image vector . If we base the derivation of the prior on purely information theoretic considerations (subset independence, coordinate invariance and system independence) we are naturally led to the Maximum Entropy Method (MEM). It may be shown [85] the prior probability takes the form

where the dimensional constant depends on the scaling of the problem and may be considered as a regularising parameter, and is a model vector to which defaults in the absence of any data.In standard applications of the maximum entropy method, the image is taken to be a positive additive distribution (PAD). Nevertheless, the MEM approach can be extended to images that take both positive and negative values by considering them to be the difference of two PADS, so that

where and are the positive and negative parts of respectively. In this case, the cross entropy is given by where and and are separate models for each PAD. The global maximum of the cross entropy occurs at . The most probable image is then just the result from finding the maximum probability or, equivalently, the minimum of . This can be done by using any known minimising routine.It can be shown [38] that the Lagrange multiplier is completely defined in a Bayesian way and any prior correlation information can also be incorporated into the analysis. Also, the assignment of errors is straightforward in the Fourier domain where all the pixels in the discretised image will be independent.

Hobson et al. [38] simulated data taken by the Planck Surveyor satellite and used MEM to reconstruct the underlying CMB and foregrounds. They used six input maps (the CMB, thermal and kinetic SZ, dust emission, free-free emission and synchrotron emission) to make up the data and then added Gaussian noise to each frequency. After using MEM with the Bayesian value for and giving the algorithm the average power spectra of each channel, it was found that features in all six maps were recovered. Without any prior power spectrum information it was found that only the kinetic SZ was not recovered and all others were recovered to some degree (the CMB and dust were almost indistinguishable from the input maps with residual errors of and per pixel respectively). Figure 20 shows the results from MEM as compared to the input maps for the case with assumed average power spectrum. It is easily seen that MEM reconstructs both the Gaussian CMB and the non-Gaussian thermal SZ effect very well.

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