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Abstract
The three-channel spectral sensitivity of a trichromatic camera represents the characteristics of system color space. It is a mapping bridge from the spectral information of a scene to the response value of a camera. In this paper, we propose an estimation method for three-channel spectral sensitivity of a trichromatic camera. It includes calibration experiment by orthogonal test design and the data processing by window filtering. The calibration experiment was first designed by an orthogonal table of the 9-level and 3-factor. A rough estimation model of spectral sensitivity is established on the data pairs of the system input and output in calibration experiments. The data of rough estimation is then modulated by two window filters on frequency and spatial domain. The Luther-Ives condition and the smoothness condition are introduced to design the window, and help to achieve the optimal estimation of the system spectral sensitivity. Finally, the proposed method is verified by some comparison experiments. The results show that the estimated spectral sensitivity is basically consistent with the measured results of the monochromator experiments, the relative full-scale errors of the RGB three-channel is obviously lower than the Wiener filtering method and the Fourier band-limitedness method. The proposed method can estimate the spectral sensitivity of the trichromatic digital camera very well, which is of great significance for the colorimetric characterization and evaluation of imaging systems.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The spectral sensitivity describes the relative efficiency of the system radiation detection at different wavelengths. It is a mapping relationship between the color channel responsivity and the wavelength for a color imaging system. It is also a foundation for the colorimetric characterization of a trichromatic camera [1–3]. Spectral sensitivity is an important reference index for system applications and performance evaluation. The estimation of the spectral sensitivity is of great significance for system colorimetric characterization, evaluation and improvement of the system colorimetric accuracy, spectral reflectance reconstruction, and system simulation [4]. However, some spectral sensitivity measuring equipment are very expensive. Some factors such as noise, system response nonuniformity and system linearity all affect colorimetrical characterization. The spectral sensitivity estimation is an ill-posed problem. These factors affect the estimation of camera spectral sensitivity. The measurement of the imaging system spectral sensitivity can be classified as the direct measurement method and the indirect estimation method [5–7]. The direct method is the main method for spectral sensitivity measurement at present. It demands monochromators and other devices. Stable monochromatic light is the key factor for this method [5]. This type of method includes standard substitution, direct comparison, broadband filter and Fourier transform methods, which has more requirements for experiments and instruments [8–16]. Most of the indirect estimation methods are based on some constraints [17–21]. Other researches are based on the input spectrum [22–29]. There are two typical input spectrum-based estimation methods, which are estimation methods based on some known input spectral characteristics [26,27] and also based on fluorescent reflection targets [28]. Another special research is based on a set of collection basis function, which comes from some camera spectral sensitivity [29]. All indirect methods almost depend on the estimation model. Therefore, the accuracy of the basis function method is mainly affected by the type of basis function. The input spectrum-based estimation method often involves the operation of the pseudo-inverse of a matrix. Due to the influence of noise and the dimensionality of the matrix, this type of problem is always an ill-posed inverse problem. Therefore, the accuracy of the sensitivity estimation is limited by the solving method. Two important factors of indirect estimation methods are the mapping data pair and the estimation model. It involves two difficulties: input space sampling, inverse problem modeling and its solution. The orthogonal test design is a statistical test design method; it can help in the input space sampling. An orthogonal table can guarantee orthogonality of samples combination.
In this paper, we propose a spectral sensitivity estimation method. The sRGB standard display is adopted as the input light source and an orthogonal test is designed in the calibration experiment. It helps to determine the input color samples. A spectral irradiance colorimeter is used to measure the input spectrum information. Combining with the output of the imaging system, the pairs of input and output data can be obtained. A rough estimation of the system's spectral sensitivity can be obtained from the mapping relationship. The Luther-Ives condition and smoothness condition are further integrated to optimize the spectral sensitivity. For reduction the number of color patches, we also provide some selection of color patches. The proposed method avoids the dependence on monochromatic light, and overcomes the ill-posedness in the spectral sensitivity estimation solution.
2. Spectral sensitivity estimation model
According to the physical process of radiation transmission, the three-channel response value of a trichromatic imaging system can be described as:
When the calibration data pairs $({\textbf D},{\mathrm{\boldsymbol \Phi}} )$ is known, the spectral sensitivity matrix ${\textbf T}$ of the imaging system can be estimated by the calibration data pair, thus
where $f$ represents some mapping relation. Because the matrix of ${\mathrm{\boldsymbol \Phi}}$ has a low effective dimension usually, its condition number is very large. In order to avoid the ill-posedness of spectral sensitivity estimation, which is caused by inverse matrix solution, and if the principal component number is close to 3, we can convert the Eq. (5). Assuming ${\textbf H}\textrm{ = }{{\textbf T}^{ - 1}}$, the least squares method is used to estimate the inverse matrix of the system spectral sensitivity. The estimation model is shown here in Eq. (6)3. Estimation method of spectral sensitivity method
Aiming at the estimation of the spectral sensitivity of imaging systems, an estimation method is proposed in this paper. The method introduces a calibration method of a kind of orthogonal test and a processing of window filtering optimization. The detailed flow of the method is shown in Fig. 1.
3.1 Orthogonal test design for calibration experiment
The estimation of the spectral sensitivity function requires input and output data pairs from calibration experiments. The best situation is that these data pairs are independent and have full coverage. When the 8 bits data is adopted, the total color number of RGB color space is ${\textrm{2}^\textrm{8}} \times {\textrm{2}^\textrm{8}} \times {\textrm{2}^\textrm{8}}$=16777216. A large full-scale test is very difficult. An orthogonal test design is used to solve this problem. A 3-factor and 9-level orthogonal test is designed for the input colors in our method. The 3-factor is R, G and B respectively. The range of every channel is from 0 to 255. This range is divided into 9 levels. An orthogonal table of ${L_{81}}({{9^{10}}} )$ is taken for calibration experiments. Under the condition of orthogonal test, the color space is completely covered. The orthogonal test sampling selection from RGB digital values and All 81 colors are shown in Fig. 2.
Fig. 2. Orthogonal test sampling of input colors. (a) Orthogonal test sampling of RGB digital values. (b) All 81 colors in one picture
3.2 Optimized estimation by window filtering
In order to solve the accuracy problem, some constraint conditions are added to optimize the estimation. For optimal design of camera spectral sensitivities [31–42], the Luther-Ives condition is set as the prior condition for solutions. The accurate acquisition of the RGB value of the camera depends on the combination of its image sensor and filter, and the design basis of the filter meets the Luther-Ives condition. The degree of fit with the Luther-Ives condition determines its colorimetric measurement accuracy. It is well known that the filters need not be exact duplicates of the color-matching functions (CMFs) but only needs to be a nonsingular transformation of them. The inclusion of an illumination spectrum in the light path makes the fabrication of filters more difficult. Since the reproduction is viewed by a human observer, the measurements can be limited to those properties that permit the creation of an image that will appear the same as the original [43]. The CIE has tabulated CMFs for this purpose. These functions together with an illuminant are used to take a necessary data. The desired tristimulus values are obtained by [39]:
For these reasons, the spatial and frequency domain windows width are selected on the three spectral stimulus values of the W. S. Stiles 10° field of view. The frequency spectrum of three spectral stimulus values of the W. S. Stiles 10° and its corresponding cutoff frequency of filter design is shown in Fig. 3. The cutoff frequency is determined by the maximum of frequency spectrum of CMFs of the W. S. Stiles 10°. In our design the cut-off frequency corresponds to the frequency of a quarter of the maximum spectrum value.
Fig. 3. Frequency spectrum of CMFs of the W. S. Stiles 10° and its corresponding cutoff frequency of filter design. (a) Red channel. (b) Green channel. (c) Blue channel
According to the cutoff frequency of filter design, a Hanning window filter is added on the frequency domain [44]. When the Nyquist frequency is normalized to 0.5, the cut-off normalized frequency is designed as [0.0119, 0.0093, 0.0146] in frequency-domain window. Then a rectangular window filtering is performed on the space domain, the detail settings of these windows are shown in Fig. 4.
4. Experimental verification
4.1 Experimental setup
The Schematic of the experimental setup is shown in Fig. 5. The experiment was performed in a dark room. An LCD display of sRGB standard is used as a light source to show the designed digital color chips. A spectral irradiance colorimeter and a trichromatic camera are needed. A BOE BOD065D (15.6-inch, 344.232mm×193.536 mm) LCD display is adopted to output the digital color patches. A SPIC-200 spectral irradiance colorimeter is adopted as spectrum measuring device. A Canon EOS600D color camera is adopted by manual settings. F/3.5, ISO=100, and exposure time t=0.02s is set. The distance between the LCD display and the spectral irradiance colorimeter is 185.19 mm. The camera was 1 meter away from the display. The main light axes of the camera, the spectral irradiance colorimeter, and the LCD display are placed in a straight line. During the experiment, the radiance of the LCD display is collected by the camera and the spectral irradiance colorimeter. In order to minimize errors, only one color patch is imaged in just one shot which fills the monitor. The sample interval is 1 nm. Finally, the raw TIFF format data with 16-bit linearity and JPEG format data are both collected. It means that there are two types of sensitivities. These two types of sensitivities are derived from their corresponding data.
4.2 Spectral sensitivity estimation
As mentioned above, the 81 color patches were measured by the spectral irradiance colorimeter. According to the positional relation, the irradiance on the image plane of the camera is computed from the measurement of the spectral irradiance colorimeter. There are also 81 groups of data for the three-channel response value of the camera. The camera image plane irradiance and its output value constitute input-output data pairs. Sensitivity varies depending on the type of data collected. Therefore the spectral pass band also depends on the original data. In order to illustrate the results more clearly, some settings and processing are described as follows: 1) to avoid spectrum leakage and spectrum aliasing, the prolongation is performed in the spatial domain in our method. The total number is 8001 after prolongation for each channel; 2) the Frequency window is set as Fig. 4.; 3) combining the results of rough estimation and CMFs, the spatial window is set; 4) the smoothness is guaranteed by cubic smoothing spline technique; 5) and non-negativity restriction is performed at last.
4.2.1 Spectral sensitivity under raw TIFF format
The rough estimation of spectral sensitivity curve under raw TIFF format with 16-bit linearity is shown in Fig. 6(a). The optimized estimation curve of spectral irradiance sensitivity is shown in Fig. 6(b).
Fig. 6. Spectral sensitivity estimation results under raw TIFF format with 16-bit linearity, (a) Results of rough estimation, (b) Results of optimized estimation
Combining the character of the rough estimation and CMFs, the spatial window is set and shown in Fig. 7. The spatial window is first set by the CMFs. The three spatial windows are [400 nm, 700 nm], [400 nm, 645 nm] and [395 nm, 526 nm] respectively. For the green channel, the data between 420 nm to 500 nm fluctuates. Here, the marginal fluctuation section is cut off. This fluctuation section is different from others. There are two fluctuations and they may come from the sampling effect.
Fig. 7. The spatial window setting combined the rough estimation data and CMFs under raw TIFF format with 16-bit linearity. (a) R channel, (b) G channel. (c) B channel
Some detailed data in process under raw TIFF format with 16-bit linearity are shown in Fig. 8. As seen here, the frequency window and spatial window play an important role in estimation.
Fig. 8. The experimental data in process under raw format with 16-bit linearity. (a) After adding frequency window of R channel. (b) After adding space window of R channel. (c) After cubic smoothing spline of R channel. (d) After adding frequency window of G channel. (e) After adding space window of G channel. (f) After cubic smoothing spline of G channel. (g) After adding frequency window of B channel. (h) After adding space window of B channel. (i) After cubic smoothing spline of B channel
4.2.2 Spectral sensitivity under JPEG format
The JPEG RGB data is nonlinear. A uniform flat field is used to calibrate nonlinearity of our camera. The nonlinearity coefficient of the experimental camera is shown in Fig. 9. Here, the nonlinearity coefficient represents the ratio of two kinds of data. When the same energy is gathered, one corresponds to the output of the system we linearized and the other to the output of the nonlinearity system we adopted.
Because nonsingular linear transformations only change the form of the mathematical model of the system, the spatial window and frequency window both are not affected by nonsingular linear transformations. Combining the character of the rough estimation and CMFs, the spatial window is set and shown in Fig. 10.
Fig. 10. The spatial window setting combined the rough estimation data and CMFs under JPEG format with linearity calibration. (a) R channel. (b) G channel. (c) B channel
As seen here, the spatial windows have changed compared with the raw format data. It is because some transformations are performed using the raw format data. Some detailed data in process are shown in Fig. 11.
Fig. 11. The experimental data in process under JPEG format with linearity calibration. (a) After adding frequency window of R channel. (b) After adding space window of R channel. (c) After cubic smoothing spline of R channel. (d) After adding frequency window of G channel. (e) After adding space window of G channel. (f) After cubic smoothing spline of G channel. (g) After adding frequency window of B channel. (h) After adding space window of B channel. (i) After cubic smoothing spline of B channel
The rough estimation of spectral sensitivity under JPEG format with linearity calibration is shown in Fig. 12(a). The optimized estimation of spectral irradiance sensitivity is shown in Fig. 12(b). As seen here, the spectral sensitivities under linearization output RGB is different from that of raw format TIFF data with 16-bit linearity.
Fig. 12. Spectral sensitivity estimation results under JPEG format with linearity calibration, (a) Results of rough estimation, (b) Results of optimized estimation
4.3 Verification of spectral sensitivity estimation
In order to verify the proposed method, another experiment is performed. The 71SW151 monochromator of OFN instrument was used in the experiment, and the 71LX150A spherical xenon lamp of OFN instrument was used as the light source. The monochromatic light has an output at an interval of 10 nm at the range of 380nm-780 nm. The PR-715 spectrophotometer of Photo Research Company was used to measure the radiance of monochromatic light; the sensitivity of the camera is measured by these instruments. The spot of the monochromator images is shown in Fig. 13.
The accuracy of the estimated spectral sensitivity is evaluated by the relative full-scale error between two experimental results. It is described as:
where ${S_0}$ is the measured spectral sensitivity value based on the monochromator; $\max ({{S_0}} )$ represents the maximum measured sensitivity value of a channel; $MSE$ is the average of absolute error, which can be calculated by the following equation:The comparison between the measured spectral irradiance sensitivity curves and the estimated value of our method under raw format with 16-bit linearity are shown in Fig. 14 (a). The comparison between the measured spectral irradiance sensitivity curves and the estimated value of our method under JPEG format with linearity calibration are shown in Fig. 14 (b).
Fig. 14. Comparison of two experimental results. (a) Raw TIFF format with 16-bit linearity. (b) JPEG format with linearity calibration
There are some popular methods, which can be used to compare with our method. These works are the Wiener estimation [19] and Fourier band-limitedness method [20,21]. The comparison data is shown in Fig. 15. Here, the coefficient of correlation is set as 0.9 in Wiener method and the 5 Fourier basis is set in Fourier band-limitedness method.
Fig. 15. The estimation of other methods. (a) Wiener filtering method under raw format with 16-bit linearity. (b) Fourier band-limitedness method under raw format with 16-bit linearity. (c) Wiener filtering method under JPEG format with linearity calibration. (d) Fourier band-limitedness method under JPEG format with linearity calibration.
As seen here, the proposed method in this paper is closer to the data from monochromator experiments. The relative full-scale errors of the three channels of the above different methods are shown as Table 1.
Table 1. The relative full-scale errors of the three channels
According to the experimental data, the indices of the relative full-scale error of the three channels are 9.44%, 8.26% and 7.94% under raw TIFF format with 16-bit linearity data. As seen here, the average relative full-scale error of the three channels of our method is lower than that of Wiener filtering method by 7.03%, and lower than that of Fourier band-limitedness method by 8.68%. The indices of the relative full-scale error of the three channels are 14.75%, 8.04% and 7.69% under JPEG format with linearity calibration. As seen here, the average relative full-scale error of the three channels of our method is lower than that of Wiener filtering method by 11.98%, and lower than that of Fourier band-limitedness method by 15.78%.
4.4 Selection of color patches
The generation of 81 color chips using a monitor is a tedious task and takes a long time. This is a drawback of the presented method. According to Ref. [30], radiative monitor samples are the worst due to the fact that all of them generated by a monitor are in fact a linear combination of R, G and B channels spectral emmitances. This means that the condition number of the matrix of spectral emmitance of the samples is high and severely rank deficient. It also means that only three color patches are enough. We need to do some selection for our method. Therefore, a kind of Monte Carlo method is used to solve this problem. We build a uniform sampling model for selecting the 81 color patches and their spectral radiance data. 3 color patches and their spectral radiance data are selected at random. The results of 50,000 times under raw format with 16-bit linearity data are shown as Fig. 16.
Fig. 16. Sensitivity rough estimation under raw format with 16-bit linearity using Monte Carlo method. (a) Rough estimations of 50,000 groups data by 3 color patches. (b) Mean absolute error of 50,000 groups data.
At the same time, the results of 50,000 times under JPEG format data with linearity calibration are shown as Fig. 17.
Fig. 17. Sensitivity rough estimation under JPEG format with linearity calibration using Monte Carlo method, (a) Rough estimations of 50,000 groups data by 3 color patches, (b) Mean absolute error of 50,000 groups data
Figure 16(a) and Fig. 17(a) show all data of rough estimation under two types of format. As seen here, some estimations have very bad effect. The mean absolute error between two results of 3 color patches and 81 color patches is also computed. We sorted all data corresponding to 3 color patches. The minimum mean absolute error is the best combination of 3 color patches. The rough estimation results corresponding to the minimum mean absolute error is shown as Fig. 18. The minimum mean absolute error is 2.5980% under raw TIFF format with 16-bit linearity, and it is 8.0466% under JPEG format with linearity calibration. The combination of RGB values for optimal 3 color patches is [191, 223, 127], [159, 31, 191], and [0, 223, 223] under raw TIFF format with 16-bit linearity. The combination of RGB values for optimal 3 color patches is [31, 191, 223], [159, 223, 95], and [31, 255, 0] under JPEG format with linearity calibration.
Fig. 18. The rough estimation results corresponding to the minimum mean absolute error, (a) Raw TIFF format data with 16-bit linearity. (b) JPEG format data with linearity calibration
As seen here, the rough estimation can still be conducted by 3 color patches. The results will exit some errors. The optimization sensitivity estimation results corresponding to the optimal 3 color patches is also performed. The relative full-scale errors of the three channels by optimal 3 color patches are shown in Table 2. We can see that the selection of 3 color patches produce greater errors than 81 color patches. However, the error is not that significant. For raw TIFF format data with 16-bit linearity, the average of the relative full-scale errors of the three channels is greater than that of 81 color patches by 0.05%, and for JPEG format data, it is greater by 1.12%.
Table 2. The relative full-scale errors of the three channels by optimal 3 color patches
There are also some other color patches, which can be used to estimate the spectral sensitivity using monitors. According to our experiments, for raw TIFF format data with 16-bit linearity, they are {[159, 63, 223], [255, 95, 63], [0, 223, 223]}, {[127, 127, 255], [63, 255, 31], [0, 223, 223]}, {[127, 127, 255], [191, 223, 127], [0, 223, 223]} and so on. For JPEG format data with linearity calibration, they are {[159, 223, 95], [127, 127, 255], [31, 255, 0]}, {[159, 255, 127], [191, 191, 95], [31, 255, 0]}, {[31, 255, 0], [191 63 255], [31, 255, 0]} and so on. Although these color patches will produce greater error than the optimal selection. These color patches still can be used to estimate the spectral sensitivity using our method.
5. Conclusions
This paper proposes an optimized estimation method of spectral sensitivity of trichromatic digital camera. In this method, an LCD display was used as a light source, a 9-level and 3-factor orthogonal test was used to determine the input sample points, and a spectral irradiance colorimeter were used to measure the input radiation. A rough estimation model of spectral sensitivity was established based on the calibration experimental data. The Luther-Ives condition and the smoothness and non-negativity restriction condition are integrated to constrain the solution; the spectral sensitivity of the trichromatic imaging system can be estimated by these optimized operations. The experimental results show that the spectral sensitivity obtained by this method is basically consistent with the measured results of the monochromator experiment, and the relative full-scale errors of the RGB three-channel is obviously lower than Wiener filtering method and Fourier band-limitedness method. However, generation of the 81 color patches using a monitor is a tedious task and takes a long time. This is a drawback of the presented method. For reduction the number of color patches, we also provide some selection of 81 color patches. The optimal 3 color patches is given by Monte Carlo method. For raw TIFF format data with 16-bit linearity, the average of the relative full-scale errors of the three channels using optimal selection is greater than that of 81 color patches by 0.05%, and for JPEG format data with linearity calibration, it is greater by 1.12%. This research is of great significance to the colorimetric characterization and system evaluation of the imaging system.
Funding
Natural Science Foundation of Liaoning Province (2019-ZD-0292); National Natural Science Foundation of China (61975012).
Acknowledgements
The authors would like to thank Mr. Rolin for polishing the English text of this paper.
Disclosures
The authors declare no conflicts of interest.
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