The accuracy of camera calibration directly determines the precision of 3D measurements, underscoring its importance in the research of 3D measurement techniques. Due to the advantages of high speed and high resolution, line-scan cameras are increasingly in demand for 3D measurement applications, necessitating high-precision calibration. Currently, there are two main methods for line-scan camera calibration: dynamic calibration and static calibration. Dynamic calibration is a process that requires the uniform movement of the line-scan camera with a displacement stage, while simultaneously scanning a calibration target, thereby generating 2D images for calibration purposes. Although this method realizes high accuracy, the calibration process is complex and often unfeasible in industrial settings where camera movement is restricted. On the other hand, static calibration depends on a parallel line calibration board, designed according to the principle of projective invariance. This method determines the calibration parameters by calculating the intersection coordinates of camera lines on the calibration target, which is performed by measuring the distance between parallel lines in the images. Static calibration methods reduce calibration costs and improve flexibility. However, they are susceptible to nonlinear mapping errors, insufficient feature information, and low calibration accuracy when the target is out of focus. To address these challenges, in this study, we propose a high-precision calibration method for line-scan cameras based on an absolute phase target. Leveraging the advantages of phase targets, such as high-precision positioning, rich feature points, and robustness against defocus, we design a phase target and calibration method appropriate for line-scan camera calibration. By combining the strengths of phase targets and assistance from a complementary area scan camera, this method establishes an accurate correspondence between line-scan camera images and spatial points via absolute phase information, thereby realizing high-precision calibration for line-scan cameras.

Initially, a phase target suitable for line-scan cameras is designed, composed of phase-shifted fringe targets and Gray codes. To circumvent issues of phase unwrapping failures and wrapped phase calculation in line-scan camera images, we introduce a non-orthogonal absolute phase target. The absolute phase values of slanted fringe targets and vertical fringe targets are employed to encode the coordinates of feature points on the target. Subsequently, the phase-shifted fringe target image of the target is displayed on a monitor, and the line-scan camera captures the fringe targets to compute the wrapped phase values on the target within the image. During phase unwrapping, Gray codes are utilized to resolve phase ambiguities in slanted fringe targets by encoding the phase levels of the first column of the slanted fringe target image. The decoded Gray code values are first multiplied by 2π,then added to the unwrapped phase of the slanted fringe target in the image, and finally the unambiguous absolute phase values are obtained. Finally, the phase target is placed in various spatial positions, and an auxiliary frame camera is utilized to determine the relative spatial positions between the targets. This process creates an accurate alignment between the line-scan camera images and spatial points. A two-step calibration process is then deployed to calculate the intrinsic parameters of the line-scan camera and the coordinate transformation between the two cameras.

To verify the feasibility and accuracy of the proposed method, we conduct tests using simulated and real data. In the simulation experiments, we investigate the impact of image noise, defocus level, and lens distortion on the calibration results. In the image noise test, the results (Fig. 5) indicate that the maximum residual values of the intrinsic parameters fy and v0 do not exceed 3.5 pixel, the absolute residual of the translation vector remains under 0.9 mm, and the root-mean-square-error (RMSE) of reprojection peaks at 0.165 pixel. In the defocus image test, the image undergoes convolution with a defocus point spread function. The results (Fig. 6) demonstrate that the residuals of fy and v0 typically stay below 1.4 pixel, the residual of the translation vector remains under 0.5 mm, and the reprojection error maintains relative stability. In the lens distortion test, we introduce a first-order radial distortion to the image and draw a comparison between the phase target and the geometric target based on projective invariance. The results (Fig. 7) show that the calibration residuals and RMSE of the phase target outperform those of the cross-ratio target. These findings highlight that the proposed method exhibits strong robustness and resistance to defocus, and adapts more efficiently to image distortion and noise compared to traditional methods.

In the practical calibration experiments, the calibration system (Fig. 8) is constructed using the devices specified in Table 2. The system is calibrated three times utilizing the proposed method and cross-ratio target. To assess the accuracy of the algorithms, the residuals and root mean square errors of all reprojection points are computed using the calibration results (Table 3). The computation results (Table 4) reveal that the maximum residual of the reprojection points for the phase target calibration is 0.468 pixel, and the maximum RMSE is 0.091 pixel. Conversely, the geometric cross-ratio target results in the maximum residual of 2.366 pixel and the maximum RMSE of 0.496 pixel. This significant improvement in accuracy suggests that the proposed method outperforms traditional methods, thereby demonstrating superior calibration precision.

In this study, a high-precision calibration method for line-scan cameras is proposed using an absolute phase target. During calibration, two sets of non-orthogonal fringe target images are captured by the line scan camera and an auxiliary frame camera. The absolute phase and target position relationship are calculated to establish a highly accurate correspondence between the spatial coordinates of feature points and the image coordinates of the line scan camera. The initial values are obtained using direct linear transform (DLT) and further refined via nonlinear optimization to obtain the calibration parameters. Experimental results demonstrate that the proposed method can realize a mean reprojection error of 0.089 pixel in practical calibration, which represents a reduction of more than 70% when compared to existing geometric targets based on parallel lines. Although the proposed method exhibits reduced flexibility, it effectively improves the calibration accuracy of line scan cameras. Furthermore, the method is capable of efficiently completing the calibration even in the presence of defocus, and thereby, satisfying the high-precision calibration requirements of line scan cameras for optical 3D measurement applications.