Gradient reconstruction for the phase recovery from a single interferogram with closed fringes

A new technique for the phase gradient estimation encoded in a single interferogram is proposed. The gradient is calculated numerically solving a differential equation obtained from the interferogram's derivatives in orthogonal directions. The phase gradient is assumed to vary almost linearly among adjacent pixels in a small window. A regularized term is aggregated to the differential equation which enables us to find the solution for the phase derivatives adjusting a plane in a minimization process. Both phase derivatives terms are obtained simultaneously from a set of linear equations that results from the minimization process. The algorithm requires a small initial region with the phase, the phase derivatives, and the sine of the phase already estimated. The calculated values of the sine of the phase from the initial region are used as a regularized term to solve the differential equation. The phase derivatives solution is then propagated from the initial region until the whole interferogram field is processed. Each value of the sine of the phase found is aggregated in the regularized term which makes the solution stable. The initial region may be easily found applying a band pass filter in the frequency domain as done with the Fourier method. The phase of the interferogram is calculated with a least square method using the information of the phase derivatives found with the proposed technique. The feasibility of the described approach for phase gradient reconstruction is tested in simulated and experimental data.