This is an brief outline of the method introduced by Otto Glatter.[2] For simplicity, we use in the following.
In indirect Fourier transformation, a guess on the largest distance in the particle is given, and an initial distance distribution function is expressed as a sum of cubic spline functions evenly distributed on the interval (0,):
| | (1) |
where are scalar coefficients. The relation between the scattering intensity and the is:
| | (2) |
Inserting the expression for pi(r) (1) into (2) and using that the transformation from to is linear gives:
where is given as:
The 's are unchanged under the linear Fourier transformation and can be fitted to data, thereby obtaining the coefficients . Inserting these new coefficients into the expression for gives a final . The coefficients are chosen to minimise the of the fit, given by:
where is the number of datapoints and is the standard deviations on data point . The fitting problem is ill posed and a very oscillating function would give the lowest despite being physically unrealistic. Therefore, a smoothness function is introduced:
- .
The larger the oscillations, the higher . Instead of minimizing , the Lagrangian is minimized, where the Lagrange multiplier is denoted the smoothness parameter.
The method is indirect in the sense that the FT is done in several steps: .
O. Glatter (1977). "A new method for the evaluation of small-angle scattering data". Journal of Applied Crystallography. 10 (5): 415–421. doi:10.1107/s0021889877013879. P.B. Moore (1980). "Small-angle scattering. Information content and error analysis". Journal of Applied Crystallography. 13 (2): 168–175. doi:10.1107/s002188988001179x. B. Vestergaard and S. Hansen (2006). "Application of Bayesian analysis to indirect Fourier transformation in small-angle scattering". Journal of Applied Crystallography. 39 (6): 797–804. doi:10.1107/S0021889806035291.