Fringe projection profilometry has been widely used for 3-D measurement, but the gamma nonlinearity severely affects the accuracy. Ideally, the histogram of the undistorted phase is uniform, but that of the distorted phase becomes nonuniform. In practice, adequate sampling pixels are required to calculate the phase histogram from modulated fringe patterns. Therefore, this article proposes an effective shifted-phase histogram equalization (SHE) method for nonlinear correction. Based on the finding that the distorted histogram contains three symmetric valleys in theory, if we shift the wrapped phase by $\pm 2\pi $ /3, the shifted phases will share the same histogram as the original phase. Without additional patterns, we can easily obtain two shifted phases. Combining the original and two shifted phases, the number of sampling pixels will increase by three times, and we can obtain more accurate histogram than the original phase. To make the histogram of the distorted phase be uniform as that of the undistorted phase, we directly apply histogram equalization on the distorted phase to estimate the undistorted phase. Moreover, this article proposes a spline interpolation optimization (SIO) method to eliminate the discretization error caused by histogram equalization. Simulations and experiments have been carried out, and their results indicate that the proposed SHE method performs better than the conventional phase histogram equalization (PHE) method, and the proposed SIO method can further reduce the discretization error.

For conventional camera calibration methods, well-focused images are necessary to detect features accurately. However, this requirement causes practical inconveniences to image acquisition for long- and short-distance photogrammetry. In this study, three active phase-shift circular grating (PCG) arrays are used as calibration patterns. The PCGs’ centers are regarded as feature points that can be accurately extracted by ellipse fitting of $2\pi$-phase points even though patterns are substantially blurred. In the experiments, Gaussian filters are utilized to blur pattern images, and different standard deviations are set for different fuzzy degrees. Pattern images with different defocusing degrees are also captured. The period and number of PCGs and noise are considered. Experimental results indicate that our method is accurate, reliable, and insensitive to image defocusing.

Three-dimensional (3D) measurement of metal surfaces is one of fundamental tasks for product life-cycle management of complex equipment, which is meaningful but challenging due to its optical characteristics of highly reflectivity. To reliably reconstruct 3D metal surfaces, commonly-used techniques heavily rely on multiple exposures for optimal fusion but do not fit to high-efficient monitoring. To alleviate this reliance, we propose a novel single-exposure method called deep sinusoidal prior (DSP) for damaged phase recovery of highly reflective surfaces. Specifically, the sinusoidal hypothesis is instilled into an untrained deep neural network (DNN) as two-stream information, in order to bypass the problem of brightness enhancement. Utilizing elaborately designed loss functions, this approach enables us to restore the accurate phase encoding by fitting the DNN to two-stream sinusoidal priors. Experimental results demonstrate the proposed DSP method has superior performances on damaged phase recovery requiring no training samples. For instance, measuring a standard workpiece, absolute errors of the DSP method have been decreased substantially (81.69% and 59.49%) compared with the direct measurement and achieved similar accuracy (0.0754 mm vs. 0.0744 mm) compared with the reference. Most strikingly, the proposed method, for the first time, demonstrates a new perspective of recovering the reliable phase from a degraded one itself, contributing to the superior generalization capability insensitive to fringe frequencies, imaging settings and variant scenes.

Stereo cameras have been widely used for three-dimensional (3D) photogrammetry, and stereo calibration is a crucial process to estimate the intrinsic and extrinsic parameters. This paper proposes a stereo calibration method with absolute phase target by using horizontal and vertical phase-shifting fringes. The one-to-one mapping from the world points to the image points that can be recovered by referring to the absolute phase and then used to calibrate the stereo cameras. Compared with traditional methods that only use feature points within the overlapping field-of-view (FOV), the proposed method can use all feature points within the overlapping and non-overlapping FOVs. Besides, since phase is more robust against camera defocusing than intensity, the target images can be captured regardless of the depth-of-field (DOF). With the advantages of whole-field capability and defocusing tolerability, the target placement becomes very flexible. Both simulations and experiment results demonstrate the robustness and accuracy of the proposed method.

Binary coding methods have been widely used for phase unwrapping. However, traditional temporal binary coding methods require a sequence of binary patterns to encode the fringe order information. This paper presents a spatial binary coding (SBC) method that encodes the fringe order into only one binary pattern. Each stripe of the sinusoidal phase-shifting patterns is corresponding to an N -bit codeword of the binary pattern. A robust stripe-wise decoding scheme is also developed to extract the N -bit codeword, then fringe order can be determined, and stripe-wise phase unwrapping can be performed. Experiment results confirm that the SBC method can correctly recover the absolute phase of measured objects with only one additional binary pattern.