Recovering an image from its Fourier magnitude only
Phase retrieval recovers a signal from its Fourier intensity |F(u)|² alone — information lost in X-ray crystallography, astronomy, and coherent imaging. The Hybrid Input-Output (HIO) algorithm (Fienup 1982) alternates Fourier magnitude projection with a real-space support constraint: outside the support, x_{n+1} = x_n − β·P_F⁻¹[x_n]. With oversampling ratio σ > 2 the problem is generically solvable.