Gao Wenjing, Kemao Qian
School of Computer Engineering, Nanyang Technological University, Singapore, 639798.
Appl Opt. 2012 Jan 20;51(3):328-37. doi: 10.1364/AO.51.000328.
Based on the windowed Fourier transform, the windowed Fourier ridges (WFR) algorithm and the windowed Fourier filtering algorithm (WFF) have been developed and proven effective for fringe pattern analysis. The WFR algorithm is able to estimate local frequency and phase by assuming the phase distribution in a local area to be a quadratic polynomial. In this paper, a general and detailed statistical analysis is carried out for the WFR algorithm when an exponential phase field is disturbed by additive white Gaussian noise. Because of the bias introduced by the WFR algorithm for phase estimation, a phase compensation method is proposed for the WFR algorithm followed by statistical analysis. The mean squared errors are derived for both local frequency and phase estimates using a first-order perturbation technique. These mean square errors are compared with Cramer-Rao bounds, which shows that the WFR algorithm with phase compensation is a suboptimal estimator. The above theoretical analysis and comparison are verified by Monte Carlo simulations. Furthermore, the WFR algorithm is shown to be slightly better than the WFF algorithm for quadratic phase.
基于加窗傅里叶变换，已开发出加窗傅里叶脊线（WFR）算法和加窗傅里叶滤波算法（WFF），并证明它们在条纹图案分析中是有效的。WFR算法能够通过假设局部区域内的相位分布为二次多项式来估计局部频率和相位。本文针对指数相位场受到加性高斯白噪声干扰时的WFR算法进行了全面而详细的统计分析。由于WFR算法在相位估计中引入了偏差，因此针对WFR算法提出了一种相位补偿方法，并进行了统计分析。使用一阶微扰技术推导了局部频率和相位估计的均方误差。将这些均方误差与克拉美 - 罗界进行比较，结果表明具有相位补偿的WFR算法是次优估计器。上述理论分析和比较通过蒙特卡罗模拟得到了验证。此外，对于二次相位，WFR算法表现得略优于WFF算法。
