Practical method to derive non-linear response functions of cameras

Preprint: The final version was published as Applied Optics, 50, 2401-2407, 2011 DOI: 10.1364/AO.50.002401 Contact: Yoichiro Hanaoka [email protected]

Practical method to derive non-linear response functions of cameras for scientific imaging Yoichiro Hanaoka∗ , Isao Suzuki, and Takashi Sakurai National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan ∗

Corresponding author: [email protected]

We developed a practical method to derive response functions which convert the amount of incident light to the A/D counts of cameras for scientific imaging. In this method, we need a mechanism to accurately control the amount of incident light into cameras just within a limited dynamic range and at a limited number of steps. A variable brightness light source, which supplies the incident light into cameras, is also necessary, but we do not need to know its accurate brightness. Thus this method enables us to derive the non-linear rec 2011 Optical Society sponse functions accurately with such a simple setup.  of America OCIS codes: 040.1490, 120.5630.

1.

Introduction

Cameras for scientific imaging are often used to take data for photometry, and therefore, the output signal from the cameras is expected to have a linear relationship with the amount of light input. However, in some cameras the relationship is nonlinear. We use a XEVA-CL-640 near-infrared camera, fabricated by Xenics (Leuven, Belgium), for polarimetric observations of the Sun. This camera has a non-linear amplifier before the A/D converter, and shows complex, significantly non-linear behavior. The non-linearity makes it problematic to use the data obtained with this camera for quantitative analysis. The response functions can be derived on the basis of the cameras’ output A/D counts measured at some known levels of light input; this is the ‘direct’ method. In most cases where cameras show non-linear responses, their response functions are not much different from the linear one. In such cases, the derivation of the non-linearity corresponds to the derivation of the small, systematic deviation from the linear function. For that purpose, the direct method 1

is useful. Even if the number of steps where the amount of light input can be set is limited, the response function thus derived is usually accurate enough to meet the requirement. However, in the case of such a camera as XEVA-CL-640, it is necessary to measure the A/D counts with fine step control of the input light over the whole dynamic range. However, it is practically difficult to do such control accurately. For instance, to change exposure time is one of the easiest ways to control the amount of light input, because most of the cameras have a function to control the exposure time. However, it is pointed out that the nominal exposure time is not necessarily correct, because it may contain a constant offset [1]. To avoid the error of the exposure time setting, a method with the same exposure repeated before the CCD read-out to control the input light amount was used [1, 2]. However, with such a method it is difficult to change the exposure time by a couple of orders of magnitude. To use neutral density filters is another way to control the amount of input light. In this case, it is easy to change the amount of light by a couple of orders of magnitude, but fine control is difficult. To know accurate transmission coefficients of the filters might be another problem. Controlling an aperture with a device like an iris changes the light amount accurately to some extent, but it is particularly difficult to control the area of the aperture when it is close to zero. On the other hand, there is an ‘indirect’ method based on the comparison between the ratio of the amounts of two kinds of input light and that of the output A/D counts [1, 3]. However, this method also assumes that the response function is approximately linear and the non-linearity is small. In our case, the camera shows significant non-linearity, and it is difficult to apply this method to our case. For accurate measurements of the non-linearity, some special instruments have been proposed [4, 5], and many measurements have been done particularly by the National Physical Laboratory (UK) linearity measurement facility (e.g. [6, 7]). However, it is not easy to carry out such measurements for a non-specialized group for optical instruments, and therefore, it is desirable to carry out such measurements with a simple instrumental setup. Then we developed a practical method to derive non-linearity functions. In this method, only the following are needed. - A mechanism to control the amount of incident light from a light source accurately just within a limited dynamic range and at a limited number of steps. - A variable brightness light source, the accurate brightness of which we do not need to know. A measuring instrument including the above two enables us to carry out the measurements, on the basis of which analysis gives a non-linear response function as a form of a differential equation. The non-linear response function itself can be calculated via numerical integration. In the following sections, the principle for obtaining response functions is described in Section 2. In Section 3, an example of the derivation of a non-linear response function is

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presented. Section 4 is devoted to the summary. 2.

Principle of the Method

A sample drawing of a non-linear relationship between the amount of input light, namely the number of photons entering into cameras within exposure time, I, and the output A/D count, S, is shown in Fig. 1(a). Let us write this relation as S = f (I).

(1)

As mentioned in Section 1, if we could measure the output count for any known input light amount, we would be able to obtain this relationship directly from the measurement results, but it is practically difficult. Usually the input light amount is controlled only at a limited number of levels and only in a limited range. In such cases, to cover the whole dynamic range of the output A/D count, S, with sufficiently fine steps, a light source with variable brightness is needed in addition. The brightness of such a light source is controlled by, for instance, adjustment of the electric current, but the accurate brightness is often unknown. The measurements which we can actually do under these restricted (but realistic) conditions are shown in Fig. 1(b). Parameter T can be controlled accurately; the exposure time, the transmission of filters, the aperture of an iris are some of the possibilities of parameter T . On the other hand, parameter B is the unknown brightness of the light source. The amount of light input, I in Eq. (1) corresponds to a product, BT . With parameters B and T , the output A/D count is expressed as S = f (BT ). (2) The result of the measurements for a B with setting T at a limited number of points, Ti (i = 1, 2, 3, ...), is shown by each curve in Fig. 1(b). Therefore, each single curve in Fig. 1(b) shows the response curve only at limited number of points. On the other hand, a series of measurements carried out with various brightnesses of the light source, Bj (j = 1, 2, 3, ...), produces a set of the curves in Fig. 1(b). In this case, the measured values, Si,j (i = 1, 2, 3, ..., j = 1, 2, 3, ...), as a whole cover the entire range of S, when the brightness, B, is set at various levels appropriately. These B, T , and S will give the functional form of f in Eq. (2), but actually we cannot plot the measured points in the BT − S space, because we do not know the values of Bj ’s. Our method gives a way to combine the curves in Fig. 1(b) into a single curve in the BT − S space, namely the non-linear response function shown in Fig. 1(a). Figure 1(c) picks up two points, Si and Si+1 measured at Ti and Ti+1 under a B. If the value Ti+1 − Ti is sufficiently small, the first derivative, dS/dT , can be written as dS/dT  (Si+1 − Si )/(Ti+1 − Ti ). 3

(3)

Therefore, the values known from the measurements are not only T and S but also dS/dT . The relation I = BT means that dI = BdT , and therefore, we can write dS/dT = BdS/dI.

(4)

If we multiply both sides by T , we obtain T dS/dT = T BdS/dI = IdS/dI.

(5)

The term IdS/dI is a function of S, then we write IdS/dI = g(S).

(6)

If we know the functional form of Eq. (1), that of Eq. (6) can be also known, and vice versa. From Eq. (5), we can write function g as T dS/dT = g(S).

(7)

All of the variables in Eq. (7), namely S, T , and dS/dT , are known from the measurements, and the equation does not include B. This means that all the measured points in Fig. 1(b), which are distributed in two dimensions, are on a unique function g regardless the values of B, after the conversion from the relation between T and S to that between S and T dS/dT . Therefore, we can derive the functional form of g from the measurements without knowing accurate values of B, and eventually we can derive the functional form of the non-linear response, f . 3.

An Example of the Application of our Method

In this Section, we describe the measurements and analysis actually carried out to derive the non-linear response function of a camera, and show how our method works. 3.A. Instrumental Setup The tested camera, a XEVA-CL-640, has a 640×512 pixels InGaAs detector, which has a sensitivity range of 0.9–1.7 μm. The size of each pixel is 20μ square and the depth of the digital output is 14 bits. The detector can be cooled down by about 40 K lower than the ambient temperature. The analogue gain can be set at four different levels. The nominal exposure time can be set between 1 μs and about 100 s. It takes about 11 ms to read out a full frame, and therefore, the frame rate is 1/(exposure time + 11ms) s−1 . This type of camera is not necessarily suitable for night-time astronomical observations, which require long exposure times. However, even in astronomy, there are some suitable applications for this camera, which require short exposure times and rather high frame rates, such as solar 4

observations. Note that our method is not specialized for this camera, and the non-linearity of any camera can be evaluated with our method. The setup for the measurements is shown in Fig. 2. It consists of a light source, a light modulator, and the camera. The light source is a halogen lamp unit. The voltage of the electric power supply for the lamp can be controlled manually to change the brightness of the lamp. An interference filter, of which the transmission wavelength is centered at 1.1 μ with a FWHM of 50 nm, is placed at the exit of the lamp unit, so that the spectral characteristics of the incident light into the camera do not change with the brightness control of the lamp. We need to accurately control the relative amount of the light detected by the camera, while the brightness of the lamp is fixed to be constant. As mentioned in Section 2, generally the exposure time control of cameras is appropriate for this purpose. However, the output A/D counts of the XEVA-CL-640 camera show complicated behavior to the exposure time. Therefore we measured the non-linear response at a fixed exposure time, and we needed to prepare a device which can modulate the light amount under a fixed exposure time. Shown in Fig. 2, it consists of a fixed linear polarizer and a rotatable linear polarizer. The synthesized transmission of the two polarizers is determined by the angle between the axes of the two polarizers, θ. The transmission is proportional to (1 + cos 2θ)/2, and therefore, parameter T in Fig. 1(b) is determined by (1 + cos 2θ)/2. Accurate control of the angle between the axes of the two polarizers realizes accurate control of the amount of the incident light into the camera. 3.B.

Measurements of the A/D counts

We carried out the measurements under some different conditions, but here we show the results obtained with the gain setting at the lowest level, because in this case the non-linearity is most remarkable. The exposure time is fixed at 20 ms, and the detector temperature is set at 260 K. Under these conditions, a series of measurements is done as follows. Firstly, the brightness of the lamp is tuned so that the A/D count is well saturated when the axes of the polarizers are set to be parallel to each other to maximize the transmission of the polarizer pair. The brightness of the light source is now B1 . Then we take 5 images each at the angles between the polarizer axes of ±0◦ , ±4.5◦ , ±9◦ , ..., ±90◦ . This series of measurements gives A/D count values for 21 different amounts of the incident light. Next, the brightness of the lamp is somewhat decreased to B2 , and the same sequence of the measurements is repeated. Such sequences are carried out at various brightness settings of the light source, B3 , B4 , ..., B19 , and finally we turned off the light source, and carry out a set of measurements without light input (B20 ). An example of the images taken with the camera in these measurements is shown in Fig. 3(a). The average A/D count of this image is calculated as follows. First, we picked up

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128×128 pixels at the center of the 640×512 pixel detector as shown in Fig. 3(a), where the intensity of the incident light is approximately constant. The image in this sub-region is shown in Fig. 3(b). As seen in this figure, there are pixels showing obviously different A/D counts from the average level. These abnormal pixels are shown as black pixels in Fig. 3(c), and there are 350 such pixels. Excluding these pixels, we calculate the average A/D value based on the remaining ‘normal’ pixels. There are ten images taken at an angle θ (both at +θ and −θ) at one of Bj ’s. The average of the A/D counts of these ten images is defined as S in Fig. 1(b). 3.C. Derivation of the S − IdS/dI Relation The relation between transmission T of the polarizers and average A/D count S measured as described in Section 3.B is shown in Fig. 4(a). In this figure, each curve corresponds to a set of measurements carried out under a brightness level of the light source, B1 , B2 , ..., B20 . As described in Section 3.A, in principle the transmission, T , is expressed as (1 + cos 2θ)/2, but actually it is necessary to add some corrections. Firstly, the transmission with the 90◦ -crossed axes of the polarizers is not zero but about 0.2 %, because the polarizers are not perfect. Secondly, we found systematic errors in the transmission, up to about 0.6 %, probably due to errors in the setting of the angle between the two polarizers. These empirical corrections have been applied to transmission T in Fig. 4(a). We can find that the relation between the input light amount and the output A/D count is non-linear, particularly in the range that S < 5000, in Fig. 4(a). None of the curves alone in Fig. 4(a) show the non-linear response function with sufficiently dense sampling. If we know the accurate value of Bj for each curve, we can plot all the points in Fig. 4(a) in the BT − S space. They are densely distributed on a single curve, namely the non-linear response function, in the BT − S space. In reality, we do not know Bj for each curve. Nevertheless, we can re-organize the measured values in Fig. 4(a) with the method described in Section 2 to obtain the response function. For that purpose, we need to derive the relation between S and T dS/dT from the raw measured values. Figure 4(b) shows three points in the T − S space, Ti−1 , Ti , Ti+1 and Si−1 , Si , Si+1 , measured at the same Bj . In such a case, the value (Sj+1 − Sj−1 )/(Tj+1 − Tj−1 ) is a good approximation for the derivative, dS/dT , at [Ti , Si ]. Furthermore, the upper and lower limits of dS/dT at [Ti , Si ] can be considered to be (Si − Si−1 )/(Ti − Ti−1 ) and (Si+1 − Si )/(Ti+1 − Ti ), respectively (in the case that the curve seems to be concave). Then the value of T dS/dT at [Ti , Si ] is approximately expressed as Ti (Si+1 − Si−1 )/(Ti+1 − Ti−1 ), and it is between Ti (Si − Si−1 )/(Ti − Ti−1 ) and Ti (Si+1 − Si )/(Ti+1 − Ti ). Figure 5(a) shows all the measured points in Fig. 4(a) plotted in the S − T dS/dT space. For each point, the upper and lower limits of T dS/dT described above are connected with a

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straight line. Now, except for some points with large ambiguity, we can confirm that the data on the 20 curves in Fig. 4(a) are on a single curve, even though we do not know the values of Bj ’s. The S −T dS/dT relation in Fig. 5(a) also expresses the relation between S and IdS/dI, because IdS/dI is equivalent to T dS/dT as described in Section 2. In Eq. (6), this relation is denoted as function g. Our ultimate goal is deriving the relation between I and S, namely function f in Eq. (1), and it can be derived from the integration of function g. To carry out the integration, we need to interpolate or fit the relation to know the T dS/dT value for any S. For that purpose, we excluded the data with large ambiguity, whose difference between the upper and lower limits exceed 10 % of Ti (Si+1 − Si−1 )/(Ti+1 − Ti−1 ). The remaining 239 data out of 380, which are shown in Fig. 5(b), are used to derive function g. Figure 5(c) shows an enlargement of the low S range between 0 and 5000. Figures 5(b) and 5(c) show the complex non-linear behavior of the response function of the XEVA camera in the low S range and power-law-like response in the high S range. Note that a straight line in the S − IdS/dI space corresponds to a power-law function in the I − S space, namely S = αI β + γ, but does not necessarily correspond to a linear relation. Interpolation by smoothing Spline functions is one possible way to draw a curve on the data shown in Figs. 5(b) and 5(c), and the least-square fitting by analytical functions is another possibility. We tried the latter. The curve derived from the least-square fitting is also shown in Figs. 5(b) and 5(c) with solid grey curves. Since the measured data show complex behavior particularly in the low S range, we fitted the data with three analytical functions, which cover different ranges of S as shown in Fig. 5(c). They are sigmoid functions and expressed as range I :S = 0–1600 IdS/dI = 0.5949S − 284.2 + 670.9/[1. + exp{−0.003444(S − 309.9)}]

(8)

range II :S = 1400–3250 IdS/dI = 4.8318S + 7725.1 − 31976.6/[1. + exp{−0.000555(S − 1990.7)}]

(9)

range III :S = 2750–16383 IdS/dI = 0.9428S − 1232.8 + 73686675/[1. + exp{−0.000853(S − 11617.1)}]

(10)

In Fig. 5(c), the above three curves are shown with a dashed curve, a dotted curve, and a dash-dot curve, respectively. There are overlapping ranges of the curves; for instance, between 1400–1600, firstly at 1400, the fitted result comes 100 % from Eq. (8), and with increasing S, the contribution in the fitted result from Eq. (9) gradually increases, and finally at 1600, the fitted result comes 100 % from Eq. (9). The grey curves in Figs. 5(b) and 5(c) are the combined result of the three curves, and these curves well reproduce the measured results. 7

The residual errors of the above fitting are shown in Fig. 6. In this figure, the plus signs correspond to the measured points shown in Fig. 5(b), and each curve connecting the plus signs corresponds to one of the curves shown in Fig. 4(a). The overall distribution of the plus signs does not show any significant tendency. On the other hand, some of the curves seem to run from top-left to bottom-right systematically, and this means that there still remain systematic errors. Presumably these errors are caused by the remaining errors in T . In spite of there being room to improve the fitting by removing the systematic errors, the RMS of the errors is only 9.4, and we concluded that the fitting is already sufficiently good. 3.D. Derivation of the Non-Linear Response Function The non-linear response function, f in Eq. (1), can be derived from function g in Eq. (6), namely the fitted curve of S and IdS/dI shown in Fig. 5, with numerical integration. We derived the non-linear response function, and show it in Fig. 7(a) with a solid grey curve. For the integration of Eq. (6), the scale of I can be adopted arbitrarily. Then we define the value I for the value S of 10000 to be 10000. In Fig. 7(a), we also plotted the measured points in Fig. 4(a) with plus signs, adopting appropriate values for Bj . This figure shows that the calculated non-linear curve well reproduces the measurement results. Fig. 7(b) shows the enlargement of the low light-level range in Fig. 7(a). Note that in the range where the A/D count is less than 1000 the derived curve is convex. It is difficult to find such a thing with coarse sampling throughout the whole A/D range, which is often done in rough estimation of non-linear response functions. Our method virtually realized dense sampling in the I − S space and succeeded to show such a complicated shape of the response curve. The fitting errors between measured and calculated I values are shown in Fig. 8. In this figure, the plus signs correspond to the measured points shown in Fig. 7(a), and each curve connecting the plus signs corresponds to one of the curves shown in Fig. 4(a). The points showing large errors correspond to saturated images. The overall distribution of the plus signs except for those of the saturated data does not show any significant tendency as in Fig. 6. On the other hand, some of the curves seem to run from bottom-left to top-right systematically. This is considered to reflect the systematic errors seen in Fig. 6. However, the RMS error calculated without the saturated data is only 2.8. In most of the range of I (0–18000), this value is much smaller than 0.1 % of the I values. Therefore, we concluded that the obtained non-linear response curve reproduces the measured results with sufficiently high accuracy. Note that this error level does not show the limitation of the fitting accuracy in our method. As seen in Fig. 8, there are still systematic errors. In the case that higher accuracy is needed, more accurate estimation of the T values will improve the fitting and realize smaller error. Finally, we show some examples of the conversion from S to I using the response function

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shown in Fig. 7. Figures 9(a) and 9(b) show sample images used in the above analysis taken at the same B but at different T ’s (0.27 and 0.79, respectively). Figure 9(c) shows the ratio between the two sample images. The ratio at pixel i, j is calculated with [Sb (i, j) − Sd (i, j)]/[Sa (i, j) − Sd (i, j)], where Sa , Sb , and Sd are the A/D count in image (a), that in image (b), and that in the dark image. Since the dark image was subtracted, the amount of incident light has the same ratio in any pixels in the two images, except the bad pixels. However, the ratio in Fig. 9(c) does not distribute uniformly but shows a dark ring structure. This fact means that the calculated ratio, [Sb (i, j) − Sd (i, j)]/[Sa (i, j) − Sd (i, j)], depends on Sa and Sb . This is due to the non-linearity response of the camera. On the other hand, Fig. 9(d) shows the distribution of [Ib (i, j) − Id (i, j)]/[Ia (i, j) − Id (i, j)], where I is derived from S through the non-linear response function. Now the ratio is almost uniform in the image, and this means that the non-linearity compensation from S to I works well. 4.

Summary

We developed a practical method to derive non-linear response functions of cameras for scientific imaging, and presented a sample process of the derivation. Our method requires only the following: - A mechanism to control the incident light amount accurately, within a limited dynamic range, and at limited number of steps. - A light source with variable brightness control, which supplies the incident light to cameras; we do not need to know its accurate brightness. This means that our method does not require any special instruments which can control the input brightness over the wide dynamic range with high accuracy and with fine step widths. In our case, we needed a light modulator with two polarizers, but if the exposure time of the camera can be used to control the relative amount of the incident light, only a light source is needed to do measurements other than the camera. With such a simple setup, we can obtain accurate non-linear response functions. This work was supported by a Grant-in-Aid for Scientific Research (No.17204014, 20052008, P.I.: T. Sakurai) from the Ministry of Education, Culture, Sports, Science and Technology of Japan. References 1. I. K. Baldry, “Time-series spectroscopy of pulsating stars,” Thesis of the University of Sydney (1999). 2. R. L. Gilliland, T. M. Brown, H. Kjeldsen, J. K. McCarthy, M. L. Peri, J. A. Belmonte, I. Vidal, L. E. Cram, J. Palmer, S. Frandsen, M. Parthasarathy, L. Petro, H. Schneider, 9

P. B. Stetson, and W. W. Weiss, “A search for solar-like oscillations in the stars of M67 with CCD ensemble photometry on a network of 4 M telescopes,” Astron. J. 106, 2441-2476 (1993). 3. A. Tajitsu, W. Aoki, S. Kawanomoto, and N. Narita, “Nonlinearity in the detector used in the Subaru Telescope High Dispersion Spectrograph,” Publ. Natl. Astron. Obs. Japan 13, 1-8 (2010). 4. K. D. Mielenz and K. L. Eckerle, “Spectrophotometer linearity testing using the double aperture method,” Appl. Opt. 11, 2294-2303 (1972). 5. E. Theocharous, F. J. J. Clarke, L. J. Rogers, and N. P. Fox, “Latest measurement techniques at NPL for the characterization of infrared detectors and materials,” in Materials for Infrared Detectors III, Proc. SPIE 5209, 228-239 (2003). 6. E. Theocharous, J. Ishii, and N. P. Fox, “Absolute linearity measurements on HgCdTe detectors in the infrared,” Appl. Opt. 43, 4182-4188 (2004). 7. E. Theocharous, “Absolute linearity measurements on a gold-blackcoated deuterated Lalanine-doped triglycine sulfate pyroelectric detector,” Appl. Opt. 47, 3731-3736 (2008).

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List of Figure Captions Fig. 1. (a) Sample drawing of a non-linear response function between the amount of light input, I, and the output A/D count, S. (b) Drawing of the results of practically available measurements between the throughput, T , and the measured A/D count, S, under the brightness of the light source, B1 − B3 . (c) Enlargement of two of the measured points in panel (b). Fig. 2. Instrumental setup for the measurements. Between a light source and the tested camera, a light modulator, which consists of two linear polarizers, is placed. Fig. 3. (a) Sample image taken in our measurements. A rectangle at the center shows the area used for the analysis. (b) Enlargement of the central area shown in panel (a). (c) Abnormal pixels in panel (b) are shown in black, and remaining normal pixels are shown in white. Fig. 4. (a) Measured A/D counts S at various transmission values T are shown with plus signs. Each curve represents the measurements carried out under the same brightness of the light source. (b) Relation between three of the measured points and calculated derivative dS/dT . Fig. 5. Relation between the measured A/D count, S, and the T dS/dT value. For each point, the upper and lower limits of T dS/dT are connected by a straight line. (b) Measured points shown in panel (a) except for those showing large ambiguity are plotted. Result of the fitting for these points is shown with a solid grey curve. (c) Enlargement of the low S range of panel (b). The fitted curve, shown with a thick solid grey curve, consists of three different analytical functions. They are shown by a dotted curve, a dashed curve, and a dash-dot curve, respectively. Ranges where the three functions are valid are labeled as I, II, and III. Fig. 6. Errors between the data and the fitting result shown in Fig. 5(b) are plotted with plus signs. Each curve connecting plus signs corresponds to one of the curves in Fig. 4(a). Fig. 7. (a) Calculated non-linear response function between the amount of light input, I, and the output A/D count, S, is shown with a solid grey curve. The measured points shown in Fig. 4(a) are also plotted with plus signs, which are scaled to fit the response function. (b) Enlargement of the low-I range of panel (a). Fig. 8. Errors between the data and the fitting result shown in Fig. 7(a) are plotted with plus signs. Each curve connecting plus signs corresponds to one of the curves in Fig. 4(a). Fig. 9. (a)(b) Sample images taken with the tested camera at different transmission values, T (0.27 and 0.79, respectively). (c) Ratio of the A/D counts of the dark-corrected images in panels (a) and (b). The ratio within ±25% of the average is displayed. (d) Ratio of the non-linearity compensated results of the dark-corrected images in panels (a) and (b).

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Fig. 7. (a) Calculated non-linear response function between the amount of light input, I, and the output A/D count, S, is shown with a solid grey curve. The measured points shown in Fig. 4(a) are also plotted with plus signs, which are scaled to fit the response function. (b) Enlargement of the low-I range of panel (a). Hanaoka-f7.eps.

18

2000

20

Error

10

0

-10

-20 0

5.0•103

1.0•104 1.5•104 A/D Count S

2.0•104

Fig. 8. Errors between the data and the fitting result shown in Figure 7a are plotted with plus signs. Each curve connecting plus signs corresponds to one of the curves in Fig. 4(a). Hanaoka-f8.eps.

19

(a)

(c)

(b)

15000

(d)

0 1.25

0.75

Fig. 9. (a)(b) Sample images taken with the tested camera at different transmission values, T (0.27 and 0.79, respectively). (c) Ratio of the A/D counts of the dark-corrected images in panels (a) and (b). The ratio within ±25% of the average is displayed. (d) Ratio of the non-linearity compensated results of the dark-corrected images in panels (a) and (b). Hanaoka-f9.eps.

20