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ТЕХНОСФЕРА_РИЦ
Issue #1/2024
H. G. Asadov, R. O. Guseinova, A. J. Alieva, D. A. Gumbatov
Issues of Empirical Radiometric Calibration of Spectrometric Equipment Installed on the UAVs
Issues of Empirical Radiometric Calibration of Spectrometric Equipment Installed on the UAVs
DOI: 10.22184/1993-7296.FRos.2024.18.1.64.70
The article is devoted to consideration of the reflection factor dependence of ground-based calibration test panels used in the vicarious empirical calibration of radiometric spectral equipment installed on the UAVs on the radiation wavelength. Having considered the saturation effect of digital readouts (DN) of the target object image due to the DN logarithmic dependence on the reflection factor of real-life objects, the peak achievement conditions for the DN value averaged over the entire wavelength range have been studied. It is shown that such a peak value can be achieved if there is an inverse relation between the slope of reflective specifications of test panels and the wavelength. It is concluded that the actually applied test panels better comply with this requirement at high values of the “gray level” indicator.
The article is devoted to consideration of the reflection factor dependence of ground-based calibration test panels used in the vicarious empirical calibration of radiometric spectral equipment installed on the UAVs on the radiation wavelength. Having considered the saturation effect of digital readouts (DN) of the target object image due to the DN logarithmic dependence on the reflection factor of real-life objects, the peak achievement conditions for the DN value averaged over the entire wavelength range have been studied. It is shown that such a peak value can be achieved if there is an inverse relation between the slope of reflective specifications of test panels and the wavelength. It is concluded that the actually applied test panels better comply with this requirement at high values of the “gray level” indicator.
Теги: gray level ground-based test objects radiometric calibration reflection factor uav platform бпла-платформа коэффициент отражения наземные тест-объекты радиометрическая калибровка уровень серого
Issues of Empirical Radiometric Calibration of Spectrometric Equipment Installed on the UAVs
H. G. Asadov¹, R. O. Guseinova², A. J. Alieva¹, D. A. Gumbatov ¹ ¹National Aerospace Agency, Baku, Republic of Azerbaijan
²Azerbaijan University of Architecture and Construction, Baku, Republic of Azerbaijan
The article is devoted to consideration of the reflection factor dependence of ground-based calibration test panels used in the vicarious empirical calibration of radiometric spectral equipment installed on the UAVs on the radiation wavelength. Having considered the saturation effect of digital readouts (DN) of the target object image due to the DN logarithmic dependence on the reflection factor of real-life objects, the peak achievement conditions for the DN value averaged over the entire wavelength range have been studied. It is shown that such a peak value can be achieved if there is an inverse relation between the slope of reflective specifications of test panels and the wavelength. It is concluded that the actually applied test panels better comply with this requirement at high values of the “gray level” indicator.
Keywords: radiometric calibration, ground-based test objects, UAV platform, gray level, reflection factor
Article received: December 04, 2023
Article accepted: January 12, 2024
Introduction
The modern remote study methods for the ground vegetation condition, the spectral cameras mounted on the UAVs are widely used [1–3]. The advantages of UAVs in comparison to other carriers of remote sensing equipment in this regard are quite obvious: the UAVs provide the higher spatial and temporal resolution, and are also the more cost-effective remote monitoring tools [4–6]. Therefore, the UAVs with spectral equipment are used in the precision agriculture [7–9], forestry [10], land management [11] and other related areas.
Moreover, the images obtained using the tools located on the UAVs are subject to the influence of various sources of radiation and optical radiation re-emission [12, 13]. Such influences, as well as the effects of various noise characteristics of the spectral equipment applied, determine the need to establish the functional relations between the calibration tool indicators/readings and actual remote sensing images, obtained in particular using the UAVs. In practice, the connection establishment procedure between the reflection indices of test panels and remotely received data is widely used [14]. The radiometric vicarious calibration performed for this purpose provides for the establishment of functional relations between the digital readouts (DN) of images and reflection indices of the calibration test panel surfaces under study [15, 16].
According to [17], for radiometric vicarious calibration it is advisable to use the special ground-based panels that are homogeneous and demonstrate the properties of a Lambertian reflector when reflecting the probing signal. Such panels are made in the form of mosaics with the red, green, blue and black sections and can be made of masonite [16] or polyvinyl chloride [17].
As it was shown in [16], the relations between DN and reflective indices of the ground-based objects is exponential. The ELM method (empirical linearization method) involves logarithmation of the specified dependence and bringing it into the linear form. The general equation is as follows:
. (1)
where y – reflective index of a ground-based object; m – graded index for each spectral area that determines the texture of a ground-based object;
с – calibration constant. The relevant equations for test objects of red, green and blue colors are given in Table 1.
However, it is reasonable to assume that the graded index m in the equation (1) also depends on the “gray level” of the calibration reflector used. The fact is that the calibration reflector used has various reflective index values at a fixed wavelength in the wavelength range 400–700 nm depending on the “gray level” of the test panel surface. In this case, as the reflective index is decreased along the wavelengths, the slope of functional dependence is increased with the increase in the “gray level” of the calibration panel.
The purpose of this paper is to clarify influence of the “gray level” of calibration panels on the achievable DN values in the resulting images.
Materials and methods
To solve the problem of determining functional dependence of the reflective index R of the calibration panel on the wavelength and “gray level”, it shall be approximated with the curves, slope and constant component of which is the “gray level” function. Moreover, the slope of curves is also changed depending on the specific part of the radiation wavelengths:
, (2)
where
, (3)
s – gray level;
λ – wavelength;
λ0 = 430 nm;
R – reflective index;
ϕ(s) – value R at λ = λ0;
k – slope of the linear dependence R = ψ(l, s) in specific part of the radiation wavelengths.
Moreover, according to [18], the dependence between reflective index and DN is exponential:
. (4)
The data for specific spectral areas is shown in Table 1.
With due regard to the formula (4), during the linear variation of reflective index, DN is saturated according to the logarithmic law. In order to avoid a decrease in the signal-to-noise ratio in the saturation area, the research problem is specified as follows: what type of dependence k(s, λ) on the wavelength at a fixed value s of can provide the maximum average integral value of DN.
Prior to this problem solution, the formula (2) at s = const shall be provided in the following form
. (5)
Having considered (4) and (5), it is possible to make up an equality expression between the reflective indices R1 and R2:
. (6)
On the basis of (6) it is possible to determine the following
. (7)
As it can be seen from the resulting expression (6), with an increase of λ, the coefficient k shall be decreased.
To achieve the above research objective, the following problem shall be considered: it is necessary to determine the optimal shape of the function k(λ) as a solution when imposing some integral constraint on this function in the following form
. (8)
The mathematical sense of the restrictive condition (10) is that the optimal function k(λ) shall be found not within the entire set of continuous and twice differentiable functions, but within the subclass of functions that comply with the condition (8). The physical sense is an application of such a set of calibration panels for which the condition (8) is met. A general view of some of these functions is shown in the figure.
Next, the formula (6) can be used to determine the following
. (9)
Having considered k = k(λ) , it is possible to integrate (9) within λ1 = λ2.
. (10)
The expression (10) represents the objective functional F to be further used to calculate the optimal form of k(λ).
Based on the expressions (8) and (10), the objective functional of unconditional variational optimization can be prepared:
. (11)
The solution to the problem (11) according to [19] shall meet the following condition:
. (12)
The formula (12) allows to obtain the following
. (13)
It should be noted that the value γ can be calculated using the expressions (8) and (13). For sake of brevity, it is possible to assume that the Lagrangian multiplier value calculated in this way is equal to γ0. The final solution to the problem can be made as follows
. (14)
It can be shown that when solving (14), the objective functional (11) reaches a maximum value that corresponds to an increase in the small values of DN with increasing λ.
Conclusion
The issues of empirical radiometric calibration of spectrometric equipment installed on the UAVs are considered. It is considered that the reflective ground-based test panels used in the vicarious calibration procedure have a reflective index that depends on the wavelength in the form of a descending curve, the slope of which is decreased with the decreasing gray level. It is shown that upon imposing some restrictive condition on the function, the average integral value with inverse relations between and can reach a maximum level. However, in the actual calibration panels with a small “gray level” value, the inverse relations between and appears to be significantly weakened.
In practice, this means that during calibration, it shall be necessary to switch to the reflective ground-based panels with a high gray level, i. e. to use the whiter panels.
Authors
Asadov Kh. G., Dr of Sc.(Engi.), professor, National aerospace agency, Baku, Republic of Azerbaijan.
ORCID: 0000-0003-1186-1535
Guseinova R. O., Cand. of Sc. (Engin.), associate professor, Azerbaijan University of Architecture and construction, Baku, Republic of Azerbaijan.
ORCID: 0000-0002-7686-7503
Alieva A. J., Cand. of Sc. (Engin.), National aerospace agency, Baku, Republic of Azerbaijan.
ORCID: 0000-0002-7686-7503
Gumbatov D. A., Ph. D. candidate, National aerospace agency, Baku, Republic of Azerbaijan.
ORCID: 0009-0008-1345-1771
H. G. Asadov¹, R. O. Guseinova², A. J. Alieva¹, D. A. Gumbatov ¹ ¹National Aerospace Agency, Baku, Republic of Azerbaijan
²Azerbaijan University of Architecture and Construction, Baku, Republic of Azerbaijan
The article is devoted to consideration of the reflection factor dependence of ground-based calibration test panels used in the vicarious empirical calibration of radiometric spectral equipment installed on the UAVs on the radiation wavelength. Having considered the saturation effect of digital readouts (DN) of the target object image due to the DN logarithmic dependence on the reflection factor of real-life objects, the peak achievement conditions for the DN value averaged over the entire wavelength range have been studied. It is shown that such a peak value can be achieved if there is an inverse relation between the slope of reflective specifications of test panels and the wavelength. It is concluded that the actually applied test panels better comply with this requirement at high values of the “gray level” indicator.
Keywords: radiometric calibration, ground-based test objects, UAV platform, gray level, reflection factor
Article received: December 04, 2023
Article accepted: January 12, 2024
Introduction
The modern remote study methods for the ground vegetation condition, the spectral cameras mounted on the UAVs are widely used [1–3]. The advantages of UAVs in comparison to other carriers of remote sensing equipment in this regard are quite obvious: the UAVs provide the higher spatial and temporal resolution, and are also the more cost-effective remote monitoring tools [4–6]. Therefore, the UAVs with spectral equipment are used in the precision agriculture [7–9], forestry [10], land management [11] and other related areas.
Moreover, the images obtained using the tools located on the UAVs are subject to the influence of various sources of radiation and optical radiation re-emission [12, 13]. Such influences, as well as the effects of various noise characteristics of the spectral equipment applied, determine the need to establish the functional relations between the calibration tool indicators/readings and actual remote sensing images, obtained in particular using the UAVs. In practice, the connection establishment procedure between the reflection indices of test panels and remotely received data is widely used [14]. The radiometric vicarious calibration performed for this purpose provides for the establishment of functional relations between the digital readouts (DN) of images and reflection indices of the calibration test panel surfaces under study [15, 16].
According to [17], for radiometric vicarious calibration it is advisable to use the special ground-based panels that are homogeneous and demonstrate the properties of a Lambertian reflector when reflecting the probing signal. Such panels are made in the form of mosaics with the red, green, blue and black sections and can be made of masonite [16] or polyvinyl chloride [17].
As it was shown in [16], the relations between DN and reflective indices of the ground-based objects is exponential. The ELM method (empirical linearization method) involves logarithmation of the specified dependence and bringing it into the linear form. The general equation is as follows:
. (1)
where y – reflective index of a ground-based object; m – graded index for each spectral area that determines the texture of a ground-based object;
с – calibration constant. The relevant equations for test objects of red, green and blue colors are given in Table 1.
However, it is reasonable to assume that the graded index m in the equation (1) also depends on the “gray level” of the calibration reflector used. The fact is that the calibration reflector used has various reflective index values at a fixed wavelength in the wavelength range 400–700 nm depending on the “gray level” of the test panel surface. In this case, as the reflective index is decreased along the wavelengths, the slope of functional dependence is increased with the increase in the “gray level” of the calibration panel.
The purpose of this paper is to clarify influence of the “gray level” of calibration panels on the achievable DN values in the resulting images.
Materials and methods
To solve the problem of determining functional dependence of the reflective index R of the calibration panel on the wavelength and “gray level”, it shall be approximated with the curves, slope and constant component of which is the “gray level” function. Moreover, the slope of curves is also changed depending on the specific part of the radiation wavelengths:
, (2)
where
, (3)
s – gray level;
λ – wavelength;
λ0 = 430 nm;
R – reflective index;
ϕ(s) – value R at λ = λ0;
k – slope of the linear dependence R = ψ(l, s) in specific part of the radiation wavelengths.
Moreover, according to [18], the dependence between reflective index and DN is exponential:
. (4)
The data for specific spectral areas is shown in Table 1.
With due regard to the formula (4), during the linear variation of reflective index, DN is saturated according to the logarithmic law. In order to avoid a decrease in the signal-to-noise ratio in the saturation area, the research problem is specified as follows: what type of dependence k(s, λ) on the wavelength at a fixed value s of can provide the maximum average integral value of DN.
Prior to this problem solution, the formula (2) at s = const shall be provided in the following form
. (5)
Having considered (4) and (5), it is possible to make up an equality expression between the reflective indices R1 and R2:
. (6)
On the basis of (6) it is possible to determine the following
. (7)
As it can be seen from the resulting expression (6), with an increase of λ, the coefficient k shall be decreased.
To achieve the above research objective, the following problem shall be considered: it is necessary to determine the optimal shape of the function k(λ) as a solution when imposing some integral constraint on this function in the following form
. (8)
The mathematical sense of the restrictive condition (10) is that the optimal function k(λ) shall be found not within the entire set of continuous and twice differentiable functions, but within the subclass of functions that comply with the condition (8). The physical sense is an application of such a set of calibration panels for which the condition (8) is met. A general view of some of these functions is shown in the figure.
Next, the formula (6) can be used to determine the following
. (9)
Having considered k = k(λ) , it is possible to integrate (9) within λ1 = λ2.
. (10)
The expression (10) represents the objective functional F to be further used to calculate the optimal form of k(λ).
Based on the expressions (8) and (10), the objective functional of unconditional variational optimization can be prepared:
. (11)
The solution to the problem (11) according to [19] shall meet the following condition:
. (12)
The formula (12) allows to obtain the following
. (13)
It should be noted that the value γ can be calculated using the expressions (8) and (13). For sake of brevity, it is possible to assume that the Lagrangian multiplier value calculated in this way is equal to γ0. The final solution to the problem can be made as follows
. (14)
It can be shown that when solving (14), the objective functional (11) reaches a maximum value that corresponds to an increase in the small values of DN with increasing λ.
Conclusion
The issues of empirical radiometric calibration of spectrometric equipment installed on the UAVs are considered. It is considered that the reflective ground-based test panels used in the vicarious calibration procedure have a reflective index that depends on the wavelength in the form of a descending curve, the slope of which is decreased with the decreasing gray level. It is shown that upon imposing some restrictive condition on the function, the average integral value with inverse relations between and can reach a maximum level. However, in the actual calibration panels with a small “gray level” value, the inverse relations between and appears to be significantly weakened.
In practice, this means that during calibration, it shall be necessary to switch to the reflective ground-based panels with a high gray level, i. e. to use the whiter panels.
Authors
Asadov Kh. G., Dr of Sc.(Engi.), professor, National aerospace agency, Baku, Republic of Azerbaijan.
ORCID: 0000-0003-1186-1535
Guseinova R. O., Cand. of Sc. (Engin.), associate professor, Azerbaijan University of Architecture and construction, Baku, Republic of Azerbaijan.
ORCID: 0000-0002-7686-7503
Alieva A. J., Cand. of Sc. (Engin.), National aerospace agency, Baku, Republic of Azerbaijan.
ORCID: 0000-0002-7686-7503
Gumbatov D. A., Ph. D. candidate, National aerospace agency, Baku, Republic of Azerbaijan.
ORCID: 0009-0008-1345-1771
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