Stable Desmearing of Slit-Collimated SAXS Patterns by Adequate Numerical Conditioning

Abstract

Small-angle X-ray scattering (SAXS) patterns from slit cameras (`Kratky cameras') require a subsequent desmearing procedure in order to obtain the pinhole scattering curve that is suitable for subsequent structure analysis. Since the corresponding integral equation contains a singularity, its solutions are usually unstable and fail if large noise is present. It is demonstrated how analytical stability can be achieved by physically reliable conditioning of the experimental data, introduction of the Moore–Penrose pseudoinverse of the equation's discretized integral operator and solving the equation by a FFT algorithm. This ensures the consistency of the solution as well as its stability, and hence its convergence. This solution can account for arbitrarily nonsymmetrical primary-beam profiles. The algorithm does not require antecedent smoothing of the scattering curve. It allows on the contrary low-pass filter smoothing during desmearing but remains stable despite large noise contributions.