Phase Unwrapping Algorithm Based on Gradient Consistency

The purpose of this study is to develop a robust and accurate Phase Unwrapping （PU） algorithm that can reliably recover continuous phase information from noisy wrapped phase data while preserving structural details. In many practical measurements， noise and local phase disturbances degrade the accuracy of gradient estimation， which leads to error amplification and structural distortion during the unwrapping process. To address this challenge， this work proposes a new PU framework based on gradient consistency detection and adaptive noise correction. The main objective is to accurately identify noise-corrupted pixels in the wrapped phase map and selectively suppress their influence without altering reliable phase information. By combining localized complex-domain filtering with efficient phase reconstruction， the proposed method aims to improve both reconstruction accuracy and robustness while maintaining computational efficiency suitable for practical imaging systems.The proposed method consists of three main stages　dual wrapped phase generation， gradient consistency-based noise detection， and integral phase reconstruction. First， two wrapped phase maps are obtained from the same group of fringe patterns through forward and reverse phase-shifting demodulation. Specifically， phase extraction is performed using five fringe images in the sequences I1 →I5 and I5→I1， respectively. These two wrapped phase maps represent the same underlying phase distribution but contain opposite phase-shift directions introduced during demodulation. Under ideal noise-free conditions， the spatial gradients of the two wrapped phase maps should be identical at each pixel location. Based on this property， the proposed algorithm detects noise by evaluating the gradient consistency between the primary wrapped phase and the auxiliary wrapped phase. The gradients of the two maps are computed and compared pixel-wise， and pixels whose gradient difference exceeds a predefined threshold are classified as noisy pixels. This detection strategy enables accurate localization of noise while avoiding unnecessary processing of reliable regions. Once the noisy pixels are identified， localized filtering is applied only to these pixels in the complex domain. The wrapped phase is first transformed into a complex representation to eliminate discontinuity issues caused by the 2π periodicity. A local mean filtering operation is then iteratively performed to suppress noise while maintaining phase continuity. To prevent excessive smoothing and loss of structural details， a gradient-based stopping criterion is introduced. The filtering process for each pixel terminates once the corrected gradient becomes sufficiently consistent with the reference gradient or when the maximum number of iterations is reached. This selective filtering mechanism effectively reduces noise while preserving reliable phase structures. After the noise suppression stage， the continuous phase is reconstructed through gradient integration. Instead of solving large optimization problems， the proposed method employs a fast integral reconstruction strategy. Starting from a reference point， the unwrapped phase is obtained by accumulating the denoised gradient values across the image. This approach avoids nonlinear optimization and large matrix operations， thereby significantly improving computational efficiency.Extensive experiments were conducted using both simulated datasets and real measurement data to evaluate the performance of the proposed method. In the simulation experiments， wrapped phase maps with different noise levels were generated to analyze the robustness of the algorithm under controlled conditions. The proposed method was compared with several representative phase unwrapping techniques， including the Total Variation （TV） based method， the Least-Squares （LS） method， and the Transport-of-Intensity Equation （TIE） method. Quantitative evaluation was performed using multiple performance metrics， including Root Mean Square Error （RMSE）， Structural Similarity Index （SSIM）， and computational time. The results show that the proposed method consistently achieves lower RMSE values and higher SSIM scores across all noise levels. When the noise level is low， the proposed algorithm produces highly accurate phase reconstructions comparable to existing approaches. As the noise level increases， the advantages of the proposed method become more evident. Traditional LS and TIE methods exhibit significant error amplification due to inaccurate gradient estimation， while TV-based methods tend to oversmooth phase structures and blur sharp edges. In contrast， the proposed method effectively suppresses noise while preserving phase discontinuities and structural details. To further evaluate the stability of the algorithm， repeated experiments were performed under identical experimental conditions. Ten independent runs were conducted with different random noise realizations. The results show that the RMSE values obtained by the proposed method remain consistently low with minimal variation， indicating strong robustness and stability. In contrast， the TV method exhibits noticeable fluctuations in reconstruction error across different runs. Real-world experiments were also carried out using fringe images captured from physical samples. The test object consisted of stacked wafer surfaces forming a discontinuous phase structure. The proposed method successfully reconstructed the phase distribution and preserved sharp structural boundaries. Two-dimensional profile comparisons further demonstrate that the reconstructed phase closely matches the reference phase and maintains accurate edge information. Additionally， the computational time of the proposed method is significantly shorter than that of the TV-based method， demonstrating its efficiency advantage.The experimental results demonstrate that the proposed phase unwrapping algorithm achieves high reconstruction accuracy， strong robustness， and efficient computational performance under various noise conditions. The method effectively suppresses noise while preserving structural details， providing a reliable solution for phase reconstruction in practical measurement environments.