Because of its advantages of non-contact, flexibility, and high measurement accuracy, phase-shift profilometry is widely used to obtain three-dimensional shape information, which is particularly valuable in the fields of industrial inspection, precision manufacturing, and medical imaging. However, traditional phase-shift fringe calibration methods involve cumbersome steps, high error sensitivity, and high time consumption. To solve these problems, a polynomial calibration model is introduced to improve calibration accuracy. Theoretically, the higher the order of the calibration model, the higher the 3D reconstruction accuracy. However, as the order of the calibration model increases, the number of parameters grows rapidly. For example, the third-order polynomial calibration model needs to fit 60 parameters, which not only increases the computational complexity but also leads to computational instability, which in turn affects the calibration accuracy and computational efficiency. Therefore, there is an urgent need for an algorithm that can quickly solve the high-order polynomial calibration model to improve the accuracy and efficiency of 3D reconstruction methods based on polynomial calibration models. In this paper, we propose a fast calibration method for fringe projection based on high-order polynomial models, which has both the efficient iterative property of the stochastic sparse Kaczmarz algorithm and the selective strategy of the greedy algorithm, realizing fast fitting of high-order polynomial calibration models and thus improving the computational efficiency and accuracy of 3D reconstruction.

In this paper, a fast calibration method for high-order polynomial calibration models is proposed with the aim of addressing the cumbersome steps, time-consuming computation, and low accuracy in the calibration of traditional phase-shift profilometry (PSP) systems. First, based on the construction of a high-order polynomial calibration model, the sparse greedy random Kaczmarz algorithm (SGRK) is used to fit the model parameters and obtain the coefficient matrix of the calibration model. This algorithm combines the efficient iterative properties of the stochastic sparse Kaczmarz algorithm with the selective strategy of the greedy algorithm, which significantly improves the speed of model fitting while ensuring computational accuracy and stability. Specifically, the method establishes the relationship among object 3D coordinates, pixel coordinates, and their phases through polynomial fitting, thus realizing fast optimization of high-order polynomial models. The application of the algorithm not only accelerates the calibration process but also effectively reduces the computational complexity and avoids the computational instability and efficiency bottlenecks that may occur in traditional methods. Ultimately, by substituting the pixel coordinates of the object fringe image with the corresponding absolute phase into the polynomial model, a high-precision 3D reconstruction of the object is accomplished. The proposed method provides a more efficient and accurate calibration solution for PSP systems, which has important theoretical significance and a wide range of practical applications, particularly in the fields of industrial inspection and precision manufacturing.

Traditional polynomial-fitting methods often face problems such as computational speed degradation and unstable computational results, which limit their wide application in complex applications. Therefore, this paper proposes a high-order polynomial calibration method that incorporates the SGRK algorithm. Compared with the traditional least squares fitting method (LSM), the SGRK algorithm significantly improves the computational efficiency and stability through its efficient iterative property and greedy strategy and successfully overcomes the problems of slow computation speed and poor instability that exist in the traditional method. With this method, the fitting of high-order polynomial models can be completed in a shorter time, which significantly improves the speed (Tables 1 and 2) and accuracy (Figs. 6 and 7) of 3D reconstruction. The experimental results verify that the proposed method can effectively reduce the calibration time and significantly improve the reconstruction accuracy while ensuring the calibration accuracy (Table 3). In different scenarios, the proposed method shows obvious advantages compared with the traditional calibration method (Table 3 and Fig. 8), particularly when dealing with complex geometries, and the calibration error and reconstruction accuracy are significantly improved (Fig. 9). By optimizing the iterative process of the algorithm, the proposed method not only improves the overall computational efficiency but also effectively copes with the demand for high-precision and high-efficiency 3D reconstruction.

In conclusion, this paper proposes a fast calibration method based on the SGRK algorithm, which effectively solves a series of problems in traditional phase-shift profilometry calibration. By introducing the SGRK algorithm, the fitting efficiency and accuracy of the high-order polynomial calibration model are significantly improved, providing new ideas and methods for the development of 3D reconstruction technology. The experimental results verify the advantages of the proposed method in terms of calibration speed and 3D reconstruction accuracy, indicating that the method has important application value, especially in fields requiring efficient and accurate 3D reconstruction, such as cultural relic protection and biomedicine.