HLSVD SNR improving in SAXS imaging

The approach aims at enhancing the signal-to-background ratio by extracting the relevant features of the folded two-dimensional pattern onto the one-dimensional profile, in order to reconstruct a two-dimensional synthetic background pattern to fluctuate according to Poisson statistics. The sample features are extracted by the one-dimensional profile by using a subspace-based parameter estimation method, also known as Hankel Lanczos singular value decomposition (HLSVD) technique. The guideline of our two-step approach is as follows: in the first one the multiparametric set is chosen according to a reduced SVD spectrum (large bottom thresholded singular value differences), in order to reproduce the background profile; the fine-tuned (signal) multiparameters are then obtained by applying the SVD to the (fluctuated) background subctracted data set. While applying this two-step procedure to a noiseless data set does not add more insight, due to the linearity of the SVD (model order K equals the sum of the two submodel orders), it dramatically enhances the signal over the background once the (poissonian) noise is accounted for. In that respect, although the method has been originally developed for a gaussian (additive) noise, it strightfowardly applies to poissonian (multiplicative) one as long as the counting statistics is sufficiently high (for lower statistics it breaks the SVD linearity as well).