Although I cannot say I completely grasp the underlying theory, the Bayesian approach to the Inverse Fourier Transformation of (isotropic) small-angle scattering patterns certainly appeals to me. The idea is that the small-angle scattering pattern can be transformed (back) into real-space, resulting in either a distance distribution function p(r), a correlation function gamma(r) (=r^2 p(r) ), or through double derivation of the result, into a chord length distribution (CLD). The Bayesian approach removes the user-defined input requirements of the standard IFT.
So what can you do with all this real-space information? Well, first of all, being in real space means that one’s intuition can once more be applied (because intuition does not work in reciprocal space). For example: a maximum probability at a certain radius really may indicate that this is a characteristic length scale in the system. Secondly, the real-space p(r) may be a lot easier to fit than the scattering pattern itself, especially for odd shapes for which no analytical scattering function exists. Lastly, it shows exactly the amount of information inherent in the SA(X)S pattern, making it easier for those new in the field to understand the limits of the number of extractable parameters.
More about this in future posts. If you’re interested, I can point towards the following references:
Hansen. Bayesian estimation of hyperparameters for indirect Fourier transformation in small-angle scattering. Journal of Applied Crystallography (2000) vol. 33 pp. 1415-1421
Hansen. Estimation of chord length distributions from small-angle scattering using indirect Fourier transformation. Journal of Applied Crystallography (2003) vol. 36 pp. 1190-1196
Hansen. Simultaneous estimation of the form factor and structure factor for globular particles in small-angle scattering. J Appl Crystallogr (2008) vol. 41 pp. 436-445
Pons et al. Modeling of chord length distributions. Chem Eng Sci (2006) vol. 61 (12) pp. 3962-3973