# Radiometric Calibration for AgCam

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## Abstract

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## 1. Introduction

^{4}α, where α is the angle between a ray and the camera optical axis. But for most imaging systems, the vignette effect is much more complex and is typically studied with raying tracing and modeled empirically using polynomial functions or hyperbolic cosine functions [4,5,6,7,8,9]. Because no two CCD detectors are the same, their differences in non-uniform quantum efficiency lead to differences in photo-electric responsivity. Dark current is associated with bias current or voltage that is inherent for the electronic components [10] and is expected to vary with each detector.

## 2. AgCam System

**Figure 1.**Electro-optical components of AgCam. (a) components integrated in vibration isolation and absorption frame, and (b) computer-aided-design model of internal optical beamsplitter and trim filters.

## 3. Laboratory Calibration

## 4. The Modeled AgCam System

- -
- DN(i) is the raw output as a digital number between 0~255 (8 bits digitization);
- -
- DN
_{0}(i) is the digital number dark current offset of CCD cells, a threshold below which no output will be triggered even there is impinging light; - -
- VE(i) represents the vignette effect;
- -
- QE(i) represents quantum efficiency coefficients, which determine the conversion of digital number values into radiant energy;
- -
- t is the integration time (unit is µs) and L(i)t would be the total energy received by the CCD cell (i) during the acquisition of an image.

_{0}(i), VE(i) and QE(i) for each CCD cell need to be determined. During the calibration, the incident light field is uniform, (denoted as L

_{0}), and Equation 1 becomes:

_{0}(i), VE(i) and QE(i), we expect the digital number values will vary for different CCD cells even under uniform illumination. The purpose of radiometric calibration that we have conducted was to derive parameters DN

_{0}(i), VE(i) and QE(i), so that an incident radiation field for an arbitrary operational image can be determined.

## 5. Calibration Analysis and Results

#### 5.1. Dark Current Offset

_{0}(i) and L

_{0}VE(i)QE(i) can be derived by applying a least squares regression on these five images, and the results are shown in Figure 3 [(a) for DN

_{0}(i) and (b) for L

_{0}VE(i)QE(i)]. For a perfect CCD detector, the DN

_{0}would be zero; no output would be generated when there is no incident light. Figure 3 (a) shows that DN

_{0}(i) varies roughly between −9 and 7 digital counts. The mean of DN

_{0}(i) determined using this least-square method is about 3.5. The mean value of DN

_{0}(i) measured when the images were taken with the lens capped was also approximately 3.5, an independent validation.

**Figure 3.**The terms of Equation 2 determined from the Least Square analysis of the 5 sets of images of a uniform target at different integration time, (a) DN

_{0}(i) and (b) L

_{0}VE(i) QE(i). The blue curve in (b) is cos

^{4}α and the red curve polynomial fit at 9th order.

#### 5.2. Vignetting Effect Correction

_{0}VE(i)QE(i) of Equation (2) exhibits two notable features: a rapid dropping in values towards the edges presumably due to the vignette effect and a seemingly random variation overlaid on the vignette effect trend. The latter random variation is due to the non-uniform quantum efficiency among CCD detectors. The blue curve in Figure 1(b) is the vignette effect modeled by the cos

^{4}α law using the AgCam optical parameters. The large discrepancy between the actual and the modeled vignette effect, 55% vs. 1.2% drop at the edges, suggests the AgCam optical system is more complex than the simple vignette model can predict. This complexity is certainly due to the optical components of the system in addition to the Mamiya lens, including the beam splitter, the filters, and the corrective cylindrical lens.

**Figure 4.**(a) Vignette effect correction coefficients—VE(i). (b) Quantum efficiency calibration coefficient—QE(i).

_{0}VE(i)QE(i). After testing with different orders between two to 12, we chose the ones that gave the minimal residual variances: nine for the NIR and six for the Red. The polynomial curve is shown in Figure 3(b) as a red curve. Assuming there is no vignette effect at the principal optical axis, namely, VE(i

_{0}) = 1, we derived the VE(i) [shown in Figure 4(a)] by normalizing the red curve in Figure 3(b) with its maximum, which is located at CCD cell No. 3304. The principal optical axis is thus not aligned at the center of the CCD array (i.e., No. 3072). This misalignment, however, will not affect the calibration or the quality of imaging. The ratio of the two curves, L

_{0}VE(i)QE(i) [the black line in Figure 3(b)] to that of VE(i) [the red line in Figure 4 (a)], gives L

_{0}QE(i); this will be used later to derive QE(i).

#### 5.3. Quantum Efficiency Calibration

- -
- i
_{0}denotes the location of the principal optical axis. Also we have assumed VE(i_{0}) = 1. - -
- the number of 100 in Equation (3) is the integration time which was fixed at 100 µs during the radiometric calibration.

_{0})] at the principal axis. With this and the curves shown in Figure 3(b) and Figure 4(a), straightforward derivation produces QE(i) for the entire CCD array, shown in Figure 4(b).

#### 5.4. Calibration Results

_{0}(i), VE(i), and QE(i), respectively. The results for the NIR band at 200 μs and RED band at 300 μs. are shown in Figure 6 and Figure 7, respectively [(a) for uncorrected and (b) for corrected images].

**Figure 6.**Results of 200μs NIR. (a) Uncorrected data; (b) Radiance of calibrated data. The coefficient of variation for (a) was estimated after the vignette effect was corrected.

**Figure 7.**Same as Figure 6 but for the Red band with an integration time of 300 μs.

## 6. Applications to AgCam Imagery

**Figure 8.**Results of calibrated AgCam image (RED, 200 μs). Data was acquired October 5th, 2006 at Grand Forks, ND.

**(a)**Uncorrected AgCam raw image;

**(b)**Radiometrically corrected images;

**(c)**A selected scan line from AgCam raw image (Red line in (a));

**(d)**The same scan line from radiometrically corrected images (Red line in (b)). The coefficient of variation for (c) was estimated after the vignette effect was corrected.

**Figure 9.**Same as Figure 8 but for AgCam image (NIR, 300 μs), acquired August 1st, 2006 at West Palm Beach, FL.

## 7. Conclusions

## Acknowledgements

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**MDPI and ACS Style**

Olsen, D.; Dou, C.; Zhang, X.; Hu, L.; Kim, H.; Hildum, E. Radiometric Calibration for AgCam. *Remote Sens.* **2010**, *2*, 464-477.
https://doi.org/10.3390/rs2020464

**AMA Style**

Olsen D, Dou C, Zhang X, Hu L, Kim H, Hildum E. Radiometric Calibration for AgCam. *Remote Sensing*. 2010; 2(2):464-477.
https://doi.org/10.3390/rs2020464

**Chicago/Turabian Style**

Olsen, Doug, Changyong Dou, Xiaodong Zhang, Lianbo Hu, Hojin Kim, and Edward Hildum. 2010. "Radiometric Calibration for AgCam" *Remote Sensing* 2, no. 2: 464-477.
https://doi.org/10.3390/rs2020464