where y = (y 1 , y 2 , . . . , y N ) is the N-size vector of the measured scattering data, x = (x 1 , x 2 , . . . , x M ) is the M-size vector of the real-space amplitudes, and S is the N × M matrix with elements S ik = sin(s i r k ). Equation 10 provides a simple expression for the amplitudes as