Wavelet Regularised Maximum Entropy

Reconstructing maps of fluctuations in the Cosmic Microwave Background (CMB) radiation from observed data requires a number of steps in the data analysis. Among these are the removal of receiver noise, the deconvolution of the image and the separation and subtraction of foreground contaminants. The latter is usually achieved by the use of multifrequency observations, which exploit the different spectral behaviour of the physical foreground sources. Here we will consider the case of a single image and develop a maximum entropy method (MEM) for the reconstruction that involves a regularisation in wavelet space. In a simplified model we will assume that the data D observed by an experiment are given by a convolution of the true sky signal or image h with the point spread function P of the instrument, plus some Gaussian random noise :