Unwrapped Phase Estimation Research Articles

Local-polynomial-approximation-based phase unwrapping using state space analysis

A new method of phase unwrapping is proposed for the absolute phase estimation based on the first-order polynomial phase approximation within a symmetric window generated around each pixel. The accurate estimation of polynomial coefficients is performed by formulating them as the elements of a state vector in a state space model. The extended Kalman filter offers a robust approach for the state estimation with the capability of handling high noise power. The polynomial coefficients estimates obtained at a given pixel are used as the initial conditions of the Kalman filter for its neighboring pixel, which results in the unwrapped phase estimation. The simulation and experimental results validate the performance of the proposed phase unwrapping method along with its ability of handling the masked phase fringe patterns with the help of a predefined binary mask and a pixel selection strategy.

Direct unwrapped phase estimation in phase shifting interferometry using Levenberg–Marquardt algorithm

An algorithm for the direct unwrapped phase estimation from the linearly phase shifted interferograms is presented. The temporal fringe intensity along each pixel is represented as a function of fringe amplitude, phase step and the searched phase. These parameters are estimated in the nonlinear least squares sense using the Levenberg–Marquardt algorithm. The proposed method allows the masked interferograms to be handled using a pixel selection approach to provide the appropriate initial conditions at a given pixel utilizing the estimated parameters at one of its neighboring pixels, which results in direct unwrapped phase estimation. Simulation results are provided to evaluate the performance of the proposed method as a function of noise power, spatially varying phase step, number of interferograms and phase step detuning error. The experimental results are also provided in the case of a holographic interferometry setup.

Simultaneous estimation of unwrapped phase and phase derivative from a closed fringe pattern

We propose a new approach for the direct estimation of the unwrapped phase from a single closed fringe pattern. The fringe analysis is performed along a given row/column at a time by approximating the phase with a weighted linear combination of linearly independent basis functions. Gaussian radial basis functions with equally distributed centers and a fixed variance are considered for the phase approximation. A state space model is defined with the weights of the basis functions as the state vector elements. Extended Kalman filter is effectively utilized for the accurate state estimation. A fringe density estimation based criteria is established to select whether the phase estimation is performed in a row by row or column by column manner. In the seed row/column decided based on this criteria, the optimal basis dimension is computed. The proposed method effectively renders itself in the simultaneous estimation of the phase and the phase derivative. The proposed phase modeling approach also allows us to successfully demodulate the low density fringe patterns. Simulation and experimental results validate the practical applicability of the proposed method.

Estimation of Phase and Its Higher Order Derivatives from a Single Complex Interferogram

This paper reports a method for the simultaneous estimation of unwrapped phase and higher order phase derivatives from a single phase fringe pattern recorded in an optical interferometric setup, thereby overcoming substantial barriers to achieving such measurements. The proposed method considers the interference phase as a weighted linear combination of Gaussian radial basis functions defined along a given row or column at a time. The Gaussian radial basis functions are defined with a constant standard deviation and equally spaced centers. Unscented Kalman filter is employed for the accurate estimation of the weights of the basis functions using the state space representation of the spatial evolution of the interferogram. The estimated weights along with the numerically computed gradients of the basis functions also provide the estimations of phase derivatives of arbitrary order. The proposed representation of interference phase along with the unscented Kalman filter provides high robustness against the speckle noise. Simulation study is preformed to evaluate the dependence of the phase and phase derivative estimation accuracy on the selection of basis dimension and the noise level. Experimental results demonstrate the practical applicability of the proposed method.

Phase estimation from noisy phase fringe patterns using linearly independent basis functions

A novel technique is proposed for obtaining unwrapped phase estimation from a highly noisy exponential phase field. In this technique, the interference phase is represented as a linear combination of linearly independent and pre-defined basis functions along each row/column of the phase field at a time. Consequently, the problem of phase estimation is converted into the problem of the estimation of the weights of the basis functions. The extended Kalman filter formulation allows for the accurate estimation of these weights. The simulation results indicate that the formulation offers a strong noise robustness in the phase estimation. Experimental results obtained using digital holographic interferometry and digital speckle pattern interferometry set-ups are provided to demonstrate the practical applicability of the proposed method.