Maximum entropy image reconstruction-a practical non-information-theoretic approach

Abstract

The maximum entropy method (MEM) of image reconstructtion is discussed in the context of incomplete Fourier information (as in aperture synthesis). Several current viewpoints on the conceptual foundation of the method are analysed and found to be unsatisfactory. It is concluded that the MEM is a form of model-fitting, the model being a non-linear transform of a band-limited function. A whole family of 'entropies' can be constructed to give reconstructions which (a) are individually unique, (b) have sharpened peaks and (c) have flattened baselines. The widely discussed 1nB and - B1nB forms of the entropy are particular cases and lead to Lorentzian and Gaussian shaped peaks respectively. However, they hardly exhaust the possibilities-for example, B^½ is equally good. The two essential features of peak sharpening and baseline flattening are shown to depend on a parameter which can be controlled by adding a suitable constant to the zero spacing correlation ρ₀₀. This process, called FLOATing, effectively tames much of the unphysical behaviour noted in earlier studies of the MEM. A numerical scheme for obtaining the MEM reconstruction is described. This incorporates the FLOAT feature and uses the fast Fourier transform (FFT), requiring about a hundred FFTs for convergence. Using a model brightness distribution, the MEM reconstructions obtained for different entropies and different values of the resolution parameter are compared. The results substantiate the theoretically deduced properties of the MEM. To allow for noise in the data, the least-squares approach has been widely used. It is shown that this method is biased since it leads to deterministic residuals which do not have a Gaussian distribution. It is suggested that fitting the noisy data exactly has the advantage of being unbiased even though the noise appears in the final map. A comparison of the strengths and weaknesses of the MEM and CLEAN suggests that the MEM already has a useful role to play in image reconstruction.