The calibration method based on polynomial fitting can obtain the camera response function (CRF) curve and image exposure ratio under the lack of camera exposure time, and has wide applicability. However, the method has the problems of iterative dispersion and low calibration accuracy, thus affecting its practical applications. We analyze the flow of the traditional polynomial fitting calibration methods and find that the calibration data set contains a large amount of invalid data under the global error function, which not only reduces the quantity of effective calibration data but also causes inaccurate iterative image exposure ratio parameters. To this end, we propose an improved joint local error function calibration method, which can select the calibration data between two images with similar exposures to avoid the introduction of invalid terms and make the data for calculating the polynomial coefficients and exposure ratios consistent. The calibration results of the public data set and an industrial camera show that the improved method has better convergence, the color three-channel CRF curves are more compactly distributed than that of the traditional methods, and the average deviation of the exposure ratio between channels is reduced by 49.83% and 42.25% respectively. The code of the improved calibration method can be downloaded at https://github.com/GuanBanglei/CRF_Calibration.

We improve the traditional CRF polynomial fitting calibration method to make the calibration results more accurate. Firstly, by analyzing the flow of the traditional calibration method, the reason for the dispersion of the calibration process and the inaccuracy of the results is the large number of invalid terms in the calibration data set. This results in inconsistencies in the set employed to calculate the polynomial coefficients and exposure ratios. Secondly, we rewrite the global error function as a local error function and select the calibration data by dividing two images with adjacent exposure levels into a group to avoid invalid terms in the calibration set. In this case, the set of calculated polynomial coefficients is the same as that of data adopted to compute the exposure ratio. During the iterative computation, the equations for all multiple exposure combinations are united to ensure global optimization. Thirdly, the improved method is tested on the publicly available data set office and an industrial camera respectively. Compared with the traditional method, the improved method outputs more compact CRF curves for the three color channels with better consistency of exposure ratio data.

Firstly, our method has better calculation accuracy. From the exposure ratio values among images of different exposure levels in Table 3, we find that the maximum exposure ratio difference between different color channels is 0.1506 and the average difference is 0.0603, while the corresponding values are 0.0664 and 0.0333 respectively in our method. The maximum difference and the average difference have a 59.96% reduction and a 49.83% reduction respectively. For the industrial camera (Table 4), the maximum deviation is reduced by 63.35% and the average deviation by 42.25%. Secondly, a reasonable explanation is given for the distribution of CRF curves for the three color channels. In Fig. 2, the B-channel curve is at the top, the G-channel curve is in the middle, and the R-channel curve is at the bottom. This is because the three color channels have different quantum absorption efficiencies for the spectrum. As shown in Fig. 5, in the absorption spectrum of silicon from 400 to 950 nm, the envelope of the B channel is the smallest, the R channel is the largest, and the G channel is the middle. For the uniform ambient spectrum, the B channel has the smallest pixel value, the R channel has the largest, and the G channel has the middle. It means that for the same pixel value, the B channel represents the largest irradiance, the G channel is the second largest, and the R channel is the smallest. As for the industrial camera, the G channel is slightly smaller than the R channel due to the working wavelength of the ordinary lens, with the working wavelength of ordinary lenses being about 360-780 nm. However, the B channel still indicates the highest radiation, demonstrating the distribution reasonableness of the calibration curves in Fig. 4. Thirdly, polynomials with an odd maximum order are more suitable for convergence during iterations. For the adopted data set, the iterative process is dispersed when the maximum order is 4 and 6, and overfitting occurs in the B and R channels when the maximum order is 5. The optimal result of the Office data set is obtained when the maximum order is 3.

The proposed improved polynomial fitting CRF calibration method can address the inconsistency between the coefficients of the solved iterative polynomials and the set of exposure ratio data, which exists in the traditional calibration method, and enhance the accuracy of the CRF calibration and exposure ratio calculation of the images. The calibration results on the public data set and an industrial camera show that the maximum deviation of the exposure ratios between different color channels is reduced by 59.96% and 63.35% respectively, and the average deviation is reduced by 49.83% and 42.25% respectively. The distribution reasonableness of CRF curves is demonstrated by analyzing the spectral quantum absorption efficiency of the three channels of the color camera. Finally, the relationship between the highest order of the fitting polynomial and the convergence of the CRF calibration curves is discussed to provide guidance for the practical applications of the proposed method.