The methods for calibrating a linear camera are limited

and most of them require special and high-accuracy equipment.The calibration of a linear camera is performed to obtain focus length

principal point

and distortion parameters.A linear model cannot be solved without additional constraints because the imaging model for a linear camera differs from that for a planar camera.This study proposed a new method for calibrating a linear camera based on a two-axis pan-tilt mechanism using angle measurement and a black-and-white striped board.We reduced the number of internal

external

and distortion parameters

and then made the model solvable by simplifying the planar camera model and providing considerable constraint information.We mainly used the least squares method and iterative optimization to obtain stable results.Results showed that the accuracy of this study could reach the same micron level of other research that used expensive equipment. The most popular method for planar camera calibration is the method of Zhang

which uses a chess board to gain world and image coordinate points and the congruent relationship among these points to calculate homography matrix.The internal parameters can be determined from the homography matrix via singular value decomposition

and the distortion parameters are obtained using Brown's method.However

a linear camera model differs from a planar camera model.Therefore

the method of Zhang cannot be used to calibrate a linear camera

and constraints are insufficient to solve the model.We established a linear camera model by limiting the dimensions of a planar camera model to only one based on the planar camera model and method of Zhang.Only two internal parameters should be solved

namely

focus length and principal point.However

many external parameters were required to be calculated

but the constraints were inadequate

thereby causing the model to become unsolvable.We designed a calibration board to create additional constraints to reduce the unknown external parameters.We then proposed a procedure for calibrating a linear camera

which adjusted the pose of the two-axis pan-tilt mechanism and the black-and-white striped board.Finally

the external parameters could be reduced to only two.However

the linear camera model remained unsolvable.We used a two-axis pan-tilt mechanism to obtain angle information to solve the problem.We calculated the internal and external parameters of the linear camera by using the edge points of the black-and-white striped board obtained via edge detection

the angles measured using the two-axis pan-tilt mechanism

the geometric model

and the least squares method.We acquired the initial value of the internal parameters and integrated them into the distortion formula

which was simplified from the distortion model for the planar camera model

to determine the initial value of the distortion parameters.Subsequently

iterative optimization was performed to optimize the internal and distortion parameters to gain stable results.The applicability and precision of this calibration method depend on three factors:the precision of the two-axis pan-tilt mechanism

the lens angle

and the resolution of the CCD camera. In the experiment

we repeatedly measured the internal and distortion parameters of three lenses (6 mm

8 mm

and 10 mm) based on the proposed method and analyzed the iterative optimization of the internal and distortion parameters

which could achieved a stable result within four iterations.We then used relative standard deviation (RSD) to calculate calibration accuracy.The RSD of the focus length of the three lenses was less than 0.05%

whereas the RSD of the principal point was less than 0.1%.Therefore

the proposed method is feasible and stable.Focal length calibration accuracy was better than 5 μm

whereas principal calibration accuracy was better than 3 μm.Compared with other studies that used expensive equipment for calibration

we could achieve the same micron level accuracy with a low cost.We simply used a two-axis pan-tilt mechanism and a printed calibration board

and calibration could be performed anywhere using this simple equipment.This approach can save significant money

time

and work.Nevertheless

the applicability of this method depends on the precision of the two-axis pan-tilt mechanism

the lens angle

and the resolution of the CCD camera.If the step size of the two-axis pan-tilt mechanism is 0.012 9° and the resolution of the CCD camera is 1 436

then this calibration method can work the best on a 16 mm lens.For a lens with longer focus length

the precision of the two-axis pan-tilt mechanism should be increased. This study proposed a new method for calibrating a linear camera

which established the model for a linear camera

derived the formulas

and provided the calibration process.The feasibility of this method was also analyzed.The linear camera model was built by simplifying the planar camera model.The number of parameters to be solved was reduced to a minimum by adjusting the pose of the two-axis pan-tilt mechanism and the black-and-white striped board.We used angle information from the two-axis pan-tilt mechanism to make the model solvable

and iterative optimization was performed to obtain reliable and stable internal and distortion parameters.Three lenses with different focus lengths were tested based on our method to calculate the internal and distortion parameters.The result showed that the accuracy of this study could reach the same micron level as that of other research that used expensive equipment.We can test only lenses with a focus length that does not exceed 16 mm because of the limit in the precision of the two-axis pan-tilt mechanism and the resolution of the CCD camera.If we intend to calibrate lenses with longer focus lengths

then we will require a two-axis pan-tilt mechanism with higher precision.Further study will be conducted using better equipment to test more lens under different situations.