Scattering-based structural reconstruction by dimensional elevation

This study outlines a conceptually new approach for reconstructing the neutron scattering length density profile, Δρ(r), directly from small-angle neutron scattering (SANS) intensity profiles, I(Q). The method is built upon a universal operator A, fundamental to scattering processes, which relates I(Q) to Δρ(r) through the covariance matrix X ≡ Δρ(r)Δρ(r)^†. In contrast to conventional SANS data analysis techniques, this approach eliminates the need to predefine a model of Δρ(r) in the regression process. This capability inherently addresses challenges often encountered in existing spectral inversion analysis, such as convergence to local minima due to incomplete analytical models, insufficient orthogonal basis vectors, or non-orthogonality among basis functions in model-free approaches. By extending spectral regression analysis from the vector space of I(Q) to the higher-dimensional space of AXA^†, the PhaseLift framework imposes convexity on the regression process. This ensures the stable and computationally efficient reconstruction of the universal minimum Δρ(r) from I(Q). Numerical benchmarks and experimental validations confirm the reliability of this approach in tackling neutron scattering inverse problems. The method establishes a robust and flexible framework for advancing neutron scattering data analysis, with the potential to significantly enhance both the precision and efficiency of experiments across various scientific domains. It provides a solid foundation for further research into the interpretation and application of scattering data.