Book excerpt: This is the first book to present the direct method for solving the inverse problems in the X-ray multicomponent spectroscopy and small-angle X-ray scattering, X-ray diffraction tomography, grazing-incidence small-angle X-ray reflectometry of multilayer structures, and electron multibeam diffraction imaging. It considers the theory of numerical analysis of multivariate additive spectra of non-separable mixtures, and decodes data obtained using the X-ray diffraction tomography technique. The book also discusses the theory of high-resolution X-ray reflectometry (HRXR) of multilayer structures (MLS) based on the modified Parratt relationships for reflection and transmission coefficients and the phase problem in electron structural crystallography.