Maximum Entropy in Small-Angle Scattering

Steen Hansen
Department of Mathematics and Physics
Royal Veterinary and Agricultural University
Thorvaldsensvej 40, 1871 FRB C, Denmark


Among its many possible applicatations, small-angle scattering is used for obtaining structural information about molecules in solution. The interest for performing solution experiments appears for example in biophysics, where it may be crucial to preserve the exact and functionally active structure of the biomolecule. The loss of information owing to the random orientation of the molecules in the solution makes it important to extract the maximum information from the measured scattering profile.

For interpretation of the experimental results it is often relevant to represent the scattering data in direct space, which requires a Fourier transform of the experimental data. A direct Fourier transform is of limited use, owing to noise, smearing and truncation. By indirect Fourier transformation some of the problems can be overcome, but the indirect transformation is an underdetermined problem requiring regularisation to be used. MaxEnt is ideally suited for such problems as it stabilizes the solution by allowing prior information to be included in a logical and transparent way. Calculating the number of good directions measured in a small-angle scattering experiment shows the need for additional knowledge to be used in the analysis of the data.

For small-angle scattering prior information about the structure of the molecule usually will be available from other techniques as e.g. electron microscopy.

Examples are given demonstrating the performance of MaxEnt compared to some of the various other methods used for indirect Fourier transformation in small-angle scattering (ad hoc smoothness constraints (the method most frequently used) decomposition in exotic functional systems and Tikhonov regularisation). It is shown how the choice of prior influences the solution, and given the choice between several models how the ``best'' can be chosen by maximization of the entropy (in particular, any unknown parameter in the prior (e.g. the overall size of the molecule) can be found by simultaneous maximization of the entropy of the distribution to be estimated and this parameter).

MaxEnt 94 Abstracts /