Relaxation procedure for phase retrieval of nonnegative signals

The paper presents an iterative procedure called the relaxation of autocorrelation equations (RAE) for solving the phase retrieval problem for nonnegative signals. First, the phase retrieval problem is formulated in the spatial domain as a set of polynomial equations with autocorrelations as known data and signal values as unknowns. Then, the RAE procedure solves these equations by recognizing one unknown at a time. While other unknowns are held constant at previously estimated values, a single unknown is varied inside the nonnegative region to globally minimize the sum of squared residuals of the equations with respect to the unknown. In every iteration, this procedure is repeated for each signal value. Since the sum of squared residuals is nonincreasing, the algorithm will either converge to a solution or stagnate; ways to overcome stagnation are suggested. The key feature of the RAE procedure is that unlike iterative transform algorithms, it allows direct control over bounding values of the signal at all times. Several numerical examples illustrate the RAE procedure