The interatomic distance distribution,P(r), is a valuable tool for evaluating the structure of a molecule in solution and represents the maximum structural information that can be derived from solution scattering data without further assumptions. Most current instrumentation for scattering experiments (typically CCD detectors) generates a finely pixelated two-dimensional image. In continuation of the standard practice with earlier one-dimensional detectors, these images are typically reduced to a one-dimensional profile of scattering intensities,I(q), by circular averaging of the two-dimensional image. Indirect Fourier transformation methods are then used to reconstructP(r) fromI(q). Substantial advantages in data analysis, however, could be achieved by directly estimating theP(r) curve from the two-dimensional images. This article describes a Bayesian framework, using a Markov chain Monte Carlo method, for estimating the parameters of the indirect transform, and thusP(r), directly from the two-dimensional images. Using simulated detector images, it is demonstrated that this method yieldsP(r) curves nearly identical to the referenceP(r). Furthermore, an approach for evaluating spatially correlated errors (such as those that arise from a detector point spread function) is evaluated. Accounting for these errors further improves the precision of theP(r) estimation. Experimental scattering data, where no ground truth referenceP(r) is available, are used to demonstrate that this method yields a scattering and detector model that more closely reflects the two-dimensional data, as judged by smaller residuals in cross-validation, thanP(r) obtained by indirect transformation of a one-dimensional profile. Finally, the method allows concurrent estimation of the beam center andDmax, the longest interatomic distance inP(r), as part of the Bayesian Markov chain Monte Carlo method, reducing experimental effort and providing a well defined protocol for these parameters while also allowing estimation of the covariance among all parameters. This method provides parameter estimates of greater precision from the experimental data. The observed improvement in precision for the traditionally problematicDmaxis particularly noticeable.
Determination of hard X-ray polarization from two-dimensional images
