2D phase-unwrapping algorithms (PUAs) are commonly used to obtain a continuous phase map from the sawtooth-shaped phase map. However, implementing PUAs can be time consuming, and the accuracy of those algorithms may be low if there is heavy noise. In this paper, we develop a simple and robust PUA based on the transport of intensity equation (TIE). In our method, the TIE was solved using the fast cosine transform, and a phase correction operation was introduced after the TIE was solved. Because of the phase correction operation, the proposed method can obtain a satisfactory unwrapping result even in a notably hash noise condition. The simulation and experimental results are presented to validate the effectiveness of the proposed method. The detailed software package can be found at https://ww2.mathworks.cn/matlabcentral/fileexchange/68493-robust-2d-phase-unwrapping-algorithm .

Deep learning has been widely used in phase unwrapping. However, owing to the noise of the wrapped phase, errors in wrap count prediction and phase calculation can occur, making it challenging to achieve high measurement accuracy under high-noise conditions. To address this issue, a three-stage multi-task phase unwrapping method was proposed. The phase retrieval was divided into three training stages: wrapped phase denoising, wrap count prediction, and unwrapped phase error compensation. In the first stage, a noise preprocessing module was trained to reduce noise interference, thereby improving the accuracy of the wrap count prediction and phase calculation. The second stage involved training the wrap count prediction module. A residual compensation module was added to correct the errors from the denoising results generated in the first stage. Finally, in the third stage, the phase error compensation module was trained to correct errors in the unwrapped phase calculated in the second stage. Additionally, a convolution-based multi-scale spatial attention module was proposed, which effectively reduces the interference of spatially inconsistent noise and can be applied to a convolutional neural network. The principles of the multi-task phase unwrapping method based on a three-stage training strategy were first introduced. Subsequently, the framework and training strategies for each stage were presented. Finally, the method was tested using simulated data with varying noise levels. It was compared with TIE, iterative TIE, the least squares phase unwrapping method, UNet, phaseNet2.0, and DeepLabV3 + with a phase correction operation, demonstrating the noise robustness and phase retrieval accuracy of the proposed method.

Phase unwrapping is a crucial process to obtain the absolute phase profile in many optical phase measurement techniques such as interferometry, holography, profilometry, etc. In this paper, we have studied several phase unwrapping algorithms based on solving the discrete Poisson equation. The differences among those algorithms lie in two aspects: one is the calculation of the input for the Poisson equation using the wrapped phase data and the other is the way to compute the output (unwrapped phase data) using the corresponding input. Firstly, the method to compute the input for the Poisson equation was investigated using the finite difference and fast Fourier transform (FFT) methods. Then different methods, based on FFT or discrete cosine transform, were employed to calculate the unwrapped phase, and their performances were compared in terms of accuracy and efficiency. To enhance the precision of those algorithms, an iteration strategy was introduced and its performance was investigated under different noise conditions. Finally, several pieces of real phase data was tested by using the direct and iterative methods. The detailed software package can be found online ( www.mathworks.com/matlabcentral/fileexchange/71810-phase-unwrapping-algorithms-by-solving-the-poisson-equation ).

Camera calibration plays a crucial role in 3D measurement tasks of machine vision. In typical calibration processes, camera parameters are iteratively optimized in the forward imaging process (FIP). However, the results can only guarantee the minimum of 2D projection errors on the image plane, but not the minimum of 3D reconstruction errors. In this paper, we propose a universal method for camera calibration, which uses the back projection process (BPP). In our method, a forward projection model is used to obtain initial intrinsic and extrinsic parameters with a popular planar checkerboard pattern. Then, the extracted image points are projected back into 3D space and compared with the ideal point coordinates. Finally, the estimation of the camera parameters is refined by a non-linear function minimization process. The proposed method can obtain a more accurate calibration result, which is more physically useful. Simulation and practical data are given to demonstrate the accuracy of the proposed method.

In this Letter, a novel three-dimensional (3D) measurement method, called the circular fringe projection profilometry (CFPP), is proposed. Similar to the conventional fringe projection profilometry, CFPP also requires fringe pattern projection and capture, phase demodulation, and phase unwrapping. However, it works with a totally different mechanism. CFPP recovers the height of a point by calculating its distance to the optical center of a projector along the optical axis. This distance is calculated with the aid of the divergence angle of a projected light ray and the distance between the measured point and the optical axis. The distance between the measured point and the optical axis is detected by a camera with telecentric lenses, while the divergence angle can be calculated from the phase of a captured circular fringe pattern. The validity of CFPP is confirmed by a set of experiments.

A multifrequency fringe projection phase unwrapping algorithm (MFPPUA) is important to fringe projection profilometry, especially when a discontinuous object is measured. However, a fringe order error (FOE) may occur when MFPPUA is adopted. An FOE will result in error to the unwrapped phase. Although this kind of phase error does not spread, it brings error to the eventual 3D measurement results. Therefore, an FOE or its adverse influence should be obviated. In this paper, reasons for the occurrence of an FOE are theoretically analyzed and experimentally explored. Methods to correct the phase error caused by an FOE are proposed. Experimental results demonstrate that the proposed methods are valid in eliminating the adverse influence of an FOE.