A relative radiometric correction method for airborne SWIR hyperspectral image using the side-slither technique

Abstract

Stripe noise still remains in airborne short-wave infrared (SWIR) hyperspectral (HS) images after laboratory calibration due to the stray light of HS imager, nonlinear response of infrared focal plane array, and the distinct difference of equivalent color temperature between the integrating sphere and the sun. It is difficult to get a sun-like radiation source, and we apply the side-slither technique for relative radiometric correction of HS images. The calibration data corresponding to different irradiance were obtained by the side-slither technique of imager. Then, the two-point multi-section method is used for relative radiometric correction of HS images. This paper presents the principle, the experimental results, and the analysis of the proposed method. To validate the effectiveness of this method, it was compared with other methods and evaluated by quantitative quality indices. The results reveal that this method has a good performance in relative radiometric correction of HS image and is superior to the laboratory calibration based on integrating sphere. Consequently, the proposed method can successfully eliminate the adverse effect caused by the difference of equivalent color temperature between radiation sources, and also can improve the accuracy of HS applications such as absolute radiation correction and target recognition.

Appendix

In the process of laboratory calibration based on integrating sphere, the response DNs of detector pixel and the average response DNs for N different irradiance are denoted by (x₁, x₂,…, x_N) and (y₁, y₂,…, y_N), respectively. According to the linear model shown in formula (1) and the least square method, the corrected gain and offset are shown as following.

$$k = \frac{{N\left( {\sum {x_{i} y_{i} } } \right) - \left( {\sum {x_{i} } } \right)\left( {\sum {y_{i} } } \right)}}{{N\left( {\sum {x_{i}^{2} } } \right) - \left( {\sum {x_{i} } } \right)^{2} }}$$

(12)

$$b = \frac{{\left( {\sum {x_{i}^{2} } } \right)\left( {\sum {y_{i} } } \right) - \left( {\sum {x_{i} } } \right)\left( {\sum {x_{i} y_{i} } } \right)}}{{N\left( {\sum {x_{i}^{2} } } \right) - \left( {\sum {x_{i} } } \right)^{2} }}$$

(13)

However, there is stray light in the imager in imaging process. When stray light is considered as an additive noise, the response DN of detector pixel and average response DN can be expressed as $x_{i} + \Delta x_{i}$ and y_i+ Δyi, respectively. Therefore, the corrected gain can be expressed as:

$$\begin{aligned} k^{'} = \frac{{N\left( {\sum {\left( {x_{i} + \Delta x_{i} } \right)(y_{i} + \Delta y_{i} )} } \right) - \left( {\sum {\left( {x_{i} + \Delta x_{i} } \right)} } \right)\left( {\sum {(y_{i} + \Delta y_{i} )} } \right)}}{{N\left( {\sum {\left( {x_{i} + \Delta x_{i} } \right)^{2} } } \right) - \left( {\sum {\left( {x_{i} + \Delta x_{i} } \right)} } \right)^{2} }} \hfill \\ \, = \frac{{N\left( {\sum {x_{i} y_{i} } } \right) - \left( {\sum {x_{i} } } \right)\left( {\sum {y_{i} } } \right) + N\left( {\left( {\sum {x_{i} \Delta y_{i} } } \right) + \left( {\sum {y_{i} \Delta x_{i} } } \right) + (\sum {\Delta x_{i} \Delta y_{i} } )} \right) - \left( {\sum {x_{i} } } \right)\left( {\sum {\Delta y_{i} } } \right) - \left( {\sum {\Delta x_{i} } } \right)\left( {\sum {y_{i} } } \right) - \left( {\sum {\Delta x_{i} } } \right)\left( {\sum {\Delta y_{i} } } \right)}}{{N\left( {\sum {x_{i}^{2} } } \right) - \left( {\sum {x_{i} } } \right)^{2} + 2N\left( {\sum {x_{i} \Delta x_{i} } } \right) + N\left( {\sum {\Delta x_{i}^{2} } } \right) - 2\left( {\sum {x_{i} } } \right)\left( {\sum {\Delta x_{i} } } \right) - \left( {\sum {\Delta x_{i} } } \right)^{2} }} \hfill \\ \, = \frac{{N\left( {\sum {x_{i} y_{i} } } \right) - \left( {\sum {x_{i} } } \right)\left( {\sum {y_{i} } } \right) + \Delta \varepsilon_{1} }}{{N\left( {\sum {x_{i}^{2} } } \right) - \left( {\sum {x_{i} } } \right)^{2} + \Delta \varepsilon_{2} }} \hfill \\ \, = k + \Delta \delta_{k} \hfill \\ \end{aligned}$$

(14)

Similarly, the corrected offset can be expressed as:

$$b^{'} = b + \Delta \delta_{b}$$

(15)

Under a solar radiation, the response DN of the detector pixel is represented by $V + \Delta V$. As shown in Fig. 2, when the spectrum ranges from 1.15 to 2.5 μm, the radiance of sunlight reflection is smaller than that of integrating sphere. When the corrected gain and offset are calculated using the data obtained by the integrating sphere, it belongs to the linear interpolation as shown in Fig. 14a. The error of corrected gain and offset introduced by linear interpolation can be neglected. Thus, the response DN of the detector pixel can be corrected to:

Fig. 14

The relationship between the data obtained by imager under the sun radiation and integrating sphere radiation.‘*’represents the data acquired by imager under the integrating sphere radiation, and ‘o’ represents the data acquired by imager under the sun radiation. b Linear interpolation. b Linear extrapolation

$$\begin{aligned} V^{'} = k^{'} \times \left( {V + \Delta V} \right) + b^{'} \hfill \\ \, = k \times V + b + k \times \Delta V + V \times \Delta \delta_{k} + \Delta \delta_{k} \times \Delta V + \Delta \delta_{b} \hfill \\ \end{aligned}$$

(16)

Here, $k \times \Delta V + V \times \Delta \delta_{k} + \Delta \delta_{k} \times \Delta V + \Delta \delta_{b}$ is the correction error introduced by the stray light of imager.

In contrast, when the spectrum ranges from 0.9 to 1.15 μm, the radiance of sunlight reflection is larger than that of integrating sphere. When the gain and offset are calculated using the data obtained by the integrating sphere, it belongs to the linear extrapolation as shown in Fig. 14b. The errors introduced by linear extrapolation are non-negligible, and the corrected gain and offset can be expressed as k′+ Δσ_k and b′ + Δσ_b, respectively. Thus, the response DN of the detector pixel can be corrected to:

$$\begin{aligned} V^{'} = \left( {k^{'} + \Delta \sigma_{k} } \right) \times \left( {V + \Delta V} \right) + \left( {b^{'} + \Delta \sigma_{b} } \right) \hfill \\ \, = k \times V + b + k \times \Delta V + V \times \Delta \delta_{k} + \Delta \delta_{k} \times \Delta V + \Delta \delta_{b} + \Delta \sigma_{k} \times V + \Delta \sigma_{k} \times \Delta V + \Delta \sigma_{b} \hfill \\ \end{aligned}$$

(17)

Compared with Formula (16), the correction error $\Delta \sigma_{k} \times V + \Delta \sigma_{k} \times \Delta V + \Delta \sigma_{b}$ is introduced by the difference of equivalent color temperature between the integrating sphere and the sun. Here, Δσ_k× V increases with the response DN of detector pixel and is a non-negligible error term.