biostudies-literatureRambo RPDepartment of EnergyFoundation for the National Institutes of HealthBasic Energy SciencesNational Institutes of HealthCancer Prevention and Research Institute of Texas2511-2522https://www.ebi.ac.uk/biostudies/studies/S-EPMC12414664biostudies-literatureUnknown124(15)Small-angle X-ray scattering (SAXS) of particles in solution informs on the conformational states and assemblies of biological macromolecules (bioSAXS) outside of cryo- and solid-state conditions. In bioSAXS, the SAXS measurement under dilute conditions is resolution limited, and through an inverse Fourier transform, the measured SAXS intensities directly relate to the physical space occupied by the particles via the P(r)-distribution. Yet, this inverse transform of SAXS data has been historically cast as an ill-posed, ill-conditioned problem requiring an indirect approach. Here, we show that through the applications of matrix and information theories, the inverse transform of SAXS intensity data is a well-conditioned problem. The so-called ill-conditioning of the inverse problem is directBiophysical journalInformation theory optimization of signals from small-angle scattering measurements.PMC12414664P30 GM124169R35 CA220430Tainer JARambo RPfalseInformation theory optimization of signals from small-angle scattering measurements.Small-angle X-ray scattering (SAXS) of particles in solution informs on the conformational states and assemblies of biological macromolecules (bioSAXS) outside of cryo- and solid-state conditions. In bioSAXS, the SAXS measurement under dilute conditions is resolution limited, and through an inverse Fourier transform, the measured SAXS intensities directly relate to the physical space occupied by the particles via the P(r)-distribution. Yet, this inverse transform of SAXS data has been historically cast as an ill-posed, ill-conditioned problem requiring an indirect approach. Here, we show that through the applications of matrix and information theories, the inverse transform of SAXS intensity data is a well-conditioned problem. The so-called ill-conditioning of the inverse problem is direct2025-01-01T00:00:00Z2025 Aug2026-06-02T02:26:37.795Z2026-04-13T03:11:51.066ZS-EPMC124146644058181710.1016/j.bpj.2025.06.031