Optimal out-of-band correction for multispectral ... - OSA Publishing
Optimal out-of-band correction for multispectral remote sensing Wei Chen Remote Sensing Division, Naval Research Laboratory, Washington, D.C. 20375, USA ([email protected]) Received 29 June 2012; revised 25 September 2012; accepted 14 October 2012; posted 15 October 2012 (Doc. ID 171608); published 19 November 2012
In this paper, an optimal out-of-band (OOB) correction transform (OOBCT) for dealing with onboard Visible/Infrared Imaging Radiometer Suite (VIIRS) OOB effects is proposed. This paper addresses the OOB response issue without consideration of the impact of other error sources on the correction processing. The OOBCT matrix is derived by minimizing an objective function of error summation between the expected and realistic recovered band-averaged spectral radiances. Using the VIIRS filter transmittance functions for all multiband sensors obtained from prelaunch laboratory measurements and a simulated dataset obtained from Airborne Visible InfraRed Imaging Spectrometer (AVIRIS) hyperspectral data, the OOBCT matrix is numerically computed. The processing of the OOB correction is straightforward and can be performed by a product between the OOBCT matrix and a measured multispectral image vector. The experimental results with both AVIRIS and Hyperspectral Imager for the Coastal Ocean datasets demonstrate that the ratios of average errors of recovered band-averaged spectral radiances divided by the measured radiances with the OOB responses are less than 4%. The average values of the relative errors for all pixels and bands indicate that the OOBCT method outperforms the works reported in literature. © 2012 Optical Society of America OCIS codes: 010.0280, 010.5630.
1. Introduction
The correction of remote sensing radiometry for ocean color applications requires a highly accurate characterization of the optical instrument performance. For example, the Visible/Infrared Imager Radiometer Suite (VIIRS) launched successfully on 28 October 2011, employs a filter radiometer to remotely sense the atmosphere in 22 visible and IR bands located in the 0.4–12.5 μm range. Products of VIIRS data are a crucial continuation for climate change study under NASA and support National Oceanic and Atmospheric Administration (NOAA) and Department of Defense operational objectives. The primary issues of the VIIRS instrument are out-of-band (OOB) response and optical cross talk. The OOB response is defined as the ratio of the integrated response outside the 1% of peak response points of a spectral band to the integrated response 1559-128X/12/337962-07$15.00/0 © 2012 Optical Society of America 7962
APPLIED OPTICS / Vol. 51, No. 33 / 20 November 2012
for the band. The OOB effect produces a radiometric bias that depends on the source of radiance being measured, and it could adversely impact VIIRS product quality. In ocean color applications, accurate and consistent sensor calibration is essential [1,2]. Several multispectral radiometric instruments such as Sea-viewing Wide Field-of-view Sensor (SeaWiFS) and VIIRS are known to exhibit a significant radiance contribution from the OOB spectral response [3–7]. To deal with the SeaWiFS OOB effects, Gordon developed a methodology [3] for SeaWiFS calibration. Based on the Gordon methodology, an improved correction method to remove the spectral band effects of the SeaWiFS on the derived normalized waterleaving radiance, which results in improved ocean near-surface chlorophyll concentration, is developed and implemented in the operational SeaWiFS data processing system [8]. These calibration methods adjust the measured radiances to correct OOB response for ease of comparison to in situ measured multispectral radiances. The SeaWiFS correction
scheme has been successfully applied to data products retrieved over case 1 ocean waters [8]. However, the correction scheme is inherently not useable for SeaWiFS data product corrections over case 2 turbid waters or over land. An alternative solution for recovering the in-band multispectral radiances for SeaWiFS and VIIRS instruments is developed by Chen and Gao [9] and Gao and Chen [10], respectively, using the multispectral decomposition transform (MDT) method. The OOB correction by the MDT method is based on the characteristics of the multiband responses, and the same OOB responses for a single sensor are also detected by other multiband sensors with different spectral passbands. The information of multiband radiances recorded by multispectral radiometers distributed at different bands provides a possibility for decomposition. The MDT approach uses the decomposition principle to recover the average narrowband signals from contaminated signals using filter transmittance functions instead of calibration methods [3,8]. For an N-channel multispectral sensor, OOB effects are corrected by applying an N × N MDT matrix to the measured signals. The MDT matrices for SeaWiFS and VIIRS instruments are also reported by Chen and Gao [9], and Gao and Chen [10], respectively. The characteristics of the VIIRS multiband response functions indicate that wavelength intervals of the expected band-averaged (or in-band averaged) spectral radiance are usually less than the partitioned narrowband intervals in the MDT method. For this reason, there is still some contamination in the recovered narrowband radiances by the MDT [11]. To address the issue and obtain a highly accurate OOB correction, an out-of-band correction transform (OBCT) method is developed by Chen and Lucke [11]. The OBCT method provides a novel approach for dealing with the VIIRS OOB effects and is a significant improvement in comparison to the MDT method. If the transform by the MDT method is a zero-order form for the OOB correction based on the bandgap treatments, then the transform by the OBCT method is a first-order form and the MDT matrix is a special case of the OBCT matrix when the bandgaps approach zero. The linear decomposition transforms derived by the MDT and OBCT methods show two different forms and exhibit recognizably different performances for the same VIIRS OOB correction. These results encourage us to believe that there may exist an optimal decomposition transform among all possible linear transforms between the measured (contaminated by the OOB effects) and recovered band-averaged spectral radiances. Our goal in this paper is to seek an optimal out-of-band correction transform (OOBCT) to reduce the errors between the expected and realistic recovered band-averaged spectral radiances for dealing with the OOB effects. This paper is organized as follows. In Section 2, an OOBCT is derived for recovering the band-averaged spectral radiance. Section 3, using the VIIRS filter
transmittance functions and multispectral simulation image data, the OOBCT matrix is numerically computed and demonstrated. In Section 4, the proposed method and the VIIRS OOBCT matrix for the OOB correction are tested by a set of simulation image data. Finally, conclusions are drawn in Section 5. 2. Optimal Out-of-Band Correction Transform A. Multiband Radiometric Instrument
VIIRS is a newly launched multispectral remote sensing instrument with high spectral resolution. The VIIRS visible to near-infrared (VisNIR) channel names and wavelengths of the nominal band centers are tabulated in the first two columns in Table 1 [12]. The VIIRS filter data are available from a public domain website [13]. The VIIRS spectral responses with 1 nm wavelength intervals are within a wavelength range from λmin 0.391 μm to λmax 1.001 μm. A set of spectrally contiguous VIIRS (V3) M1–M7 filter transmittance curves (normalized at the peak of the filter transmission) is shown in Fig. 1. It has been found that the seven channels located between 0.4 and 0.9 μm (M1–M7) in the VisNIR focal plane have problems in OOB responses. As can be seen in Fig. 1, the filter curves of the VIIRS bands M1 and M4 exhibit more significant OOB transmittances than the rest of the bands. Table 1. VIIRS VisNIR Channel Names, Positions, and Minimum and Maximum Wavelengths between In-Band and Out-of-Band Responses
VIIRS Channel
λ (μm)
λk min (μm)
λk max (μm)
M1 M2 M3 M4 M5 M6 M7
0.412 0.445 0.488 0.555 0.672 0.746 0.865
0.395 0.432 0.473 0.530 0.649 0.731 0.830
0.426 0.458 0.506 0.572 0.693 0.760 0.897
Fig. 1. VIIRS version 3 M1–M7 filter transmittance curves normalized at the peaks. 20 November 2012 / Vol. 51, No. 33 / APPLIED OPTICS
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The OOB response is defined as the ratio of the integrated response outside the 1% of peak response points (upper and lower) to the integrated response for the band. The wavelengths in the third and forth columns in Table 1 show the wavelength ranges of the in-band and OOB responses for the VIIRS instrument. The spectral response function shown in Fig. 1 is defined by H k λ, where k is an index of the kth band filter and λ is the wavelength. A normalized spectral response function hk λ between the full range of the λmin and λmax is given by hk λ R λ
H k λ max
λmin
B.
H k λdλ
:
(1)
Band-Averaged Spectral Radiance
The measured radiance at the sensor is an average value of the spectral response with the kth band filter. This system is linear, because the measured band-averaged spectral radiance sˆ k sˆ ijk (for a compact notation) on a pixel, where i and j are pixel indices, from a sensor with the kth band filter can be expressed by Zλ max ˆsk hk λsλdλ; (2) λmin
where sˆ k and sλ sij λ are measured (uncorrected) and original radiances, respectively. The full range integral in Eq. (2) between λmin and λmax indicates that the measured band-averaged spectral radiance sˆ k consists of two contributions: in-band averaged and contaminated OOB signals. The in-band averaged spectral radiance sk sijk on a pixel between k the wavelengths λk min and λmax in Table 1 is given by R λk max λk min
sk
H k λsλdλ
R λk max λk min
H k λdλ
:
(3)
The in-band averaged spectral radiance is a bandaveraged spectral radiance without the OOB effects. This in-band averaged spectral radiance sk is the expected recovered signal from the contaminated signal sˆ k . The integrals for computing the measured radiance and in-band averaged spectral radiance in Eqs. (2) and (3) can be approximately evaluated by summations with 1 nm wavelength resolution of the spectral radiances and responses. The summations are given by sˆ k
λmax −λmin X
P sk
p0
hk λmin psλmin p;
k λk max −λmin
q0
P
k H k λk min qsλmin q
k λk max −λmin
q0
7964
H k λk min q
;
where we have utilized the property λmax −λmin X p0
C.
hk λmin p 1.
Optimal Out-of-Band Correction Transform
An OOB correction can be performed using the MDT [9,10] or OBCT [11] method. The in-band averaged spectral radiances for all pixels can be recovered by an OOB correction transform T kl such as s¯ k
N X
T kl sˆ l ;
(6)
l1
where s¯ k s¯ ijk is the realistic recovered in-band averaged spectral radiance on a pixel and N is the number of multispectral bands for a particular instrument. The transform matrix in Eq. (6) for the OOB correction can be determined by the response functions that are dependent on the characteristics of the filters for a particular instrument and parameters of the multiple band partition [9,10] or inband and bandgap partitions [11]. A real multispectral radiance image may contain the impact of random and spatial structured noise in the scene. All the noises within the narrowband regions remain in the image recovered by the transform. In this paper, we seek an optimal transform among all possible linear transforms for the OOB correction so that the errors between the expected and realistic recovered in-band averaged spectral radiances approach a minimum for all bands and pixels. Using the least-squares principle, a cost function is given by χ2
N XX i;j k1
sijk − s¯ ijk 2 ;
where i and j go over all pixels in an N x × N y image (i ∈ 1; N x ∩ j ∈ 1; N y ). Substituting s¯ ijk in Eq. (6) into the cost function and minimizing it with respect to all elements T αβ of the transform matrix, we have X
sijα sˆ ijβ
i;j
N X
T αl
l1
X
sˆ ijl sˆ ijβ ;
i;j
or the matrix form ˆ S TS;
where matrices S Sαβ and Sˆ Sˆ αβ , and α and β are row and column indices, respectively. The elements of the matrices S and Sˆ can be expressed by correlation functions, respectively,
(4)
Sαβ
1 X s sˆ corrsα ; sˆ β ; N x N y i;j ijα ijβ
(5)
Sˆ αβ
1 X sˆ sˆ corrˆsα ; sˆ β : N x N y i;j ijα ijβ
APPLIED OPTICS / Vol. 51, No. 33 / 20 November 2012
(7)
It is clear that the OOBCT can be solved from Eq. (7), and it is given by T SSˆ −1 :
(8)
A matrix form of Eq. (6) for recovering the bandaveraged spectral radiance from the measured radiance is given by s¯ Tˆs;
(9)
where s¯ s¯ ij ¯sk and sˆ sˆij ˆsk are column vectors on each pixel, respectively. The OOBCT matrix is a function of the cross correlations and autocorrelations between two bands. To recover the in-band averaged spectral radiance, the MDT [9,10] and the OBCT [11] matrices for dealing with the OOB effects are derived based on the spectral response functions and the partition methodologies. The OOBCT matrix with Eqs. (4) and (5) depends on the spectral response functions for a particular instrument through the cross-correlation and autocorrelation functions. 3. VIIRS OOBCT Matrix
We are concerned with the computation of the OOBCT matrix for the VIIRS optical instrument in this section. To evaluate the cross correlation and autocorrelation between the two bands in Eq. (7), the measured radiance and expected band-averaged spectral radiance must be known. Using Eqs. (4) and (5) and a real measured hyperspectral image dataset, the measured radiances and expected band-averaged spectral radiances can be obtained by a simulation method [9–11,14]. The zero-order and first-order transforms [9–11] can be calculated based only on the filter transmittance functions and partition parameters. The OOBCT matrix is a function of both the response functions and the measured signals. To find a transform in a variety of applications, the OOBCT matrix calculated by a specific dataset with minimized errors should be statistically optimized for other datasets. For a specific optical system such as the VIIRS instrument, the computation of the OOBCT matrix is a system training process before the procedure of the VIIRS OOB correction is operated. If the types of spectra in the employed hyperspectral image dataset are rich enough, the minimized OOBCT for a particular dataset is optimized for other measured or simulation multispectral datasets. The training dataset must cover all matters of interest and typical spectra such as deep water, turbid water, land, and vegetation. The Airborne Visible InfraRed Imaging Spectrometer (AVIRIS) [15] is used extensively for Earth remote sensing. It is a unique optical sensor that delivers calibrated images of the upwelling spectral radiance in 224 contiguous spectral bands with wavelengths from 370 to 2500 nm. The Hyerspectral Imager for the Coastal Ocean (HICO) sensor is the first hyperspectral imager designed specifically for remote sensing of the coastal environment [16] from
the International Space Station. It has 128 narrow channels (5.7 nm wide) covering the contiguous spectral range between 350 and 1080 nm with a spatial resolution of 100 m. AVIRIS and HICO data are used to simulate VIIRS data, with and without OOB effects, using the VIIRS (V3) filter transmittance functions in Fig. 1 and wavelength in-band intervals in Table 1. The multispectral simulation images with and without the OOB response are synthesized by Eqs. (4) and (5), where sλ is supplied by either AVIRIS or HICO hyperspectral image data. Two AVIRIS and one HICO datasets with 512 × 2000 resolution, shown in Figs. 2(a)–2(c) are chosen for the training process and experimental test data. The AVIRIS and HICO hyperspectral images acquired over the coastal area of New Jersey on 31 July 2001, the Kennedy Space Center, Florida, on 26 March 1996, and Freshwater Beach, Australia, on 14 May 2010, respectively, are displayed in true color in Fig. 2. The function sλ with 1 nm resolution is obtained by a interpolation between the two nearest hyperspectral bands from the AVIRIS and HICO datasets. The dataset of the AVIRIS Kennedy Space Center in Fig. 2(b) is employed as the training data for the computation of the OOBCT matrix. Two typical radiance spectra over areas covered by clear water and land in the AVIRIS Kennedy Space Center image are shown in Fig. 3. The VIIRS OOBCT (7 × 7) matrix T based on Eqs. (8), (4), and (5), the transmittance functions of
Fig. 2. (a) and (b) AVIRIS images acquired over the coastal area of New Jersey on 31 July 2001 and the Kennedy Space Center, Florida, on 26 March 1996, and (c) HICO image acquired over Freshwater Beach, Australia, 14 May 2010. The radiance spectral profiles and the relative errors at positions marked by A–D are shown in Figs. 4 to 7. 20 November 2012 / Vol. 51, No. 33 / APPLIED OPTICS
7965
the filters in Fig. 1, the in-band wavelength intervals in Table 1, and a training dataset in Fig. 2(b) is given by 0
1.02768 B −2.00174 × 10−3 B B −1.41508 × 10−3 B B −2.51098 × 10−3 B B −2.60251 × 10−3 B @ −1.57124 × 10−3 −4.51680 × 10−4
−1.6661 × 10−3 2.64793 × 10−3 1.00942 1.03182 × 10−3 −1.18287 × 10−3 1.01633 −6.65125 × 10−3 −3.80949 × 10−3 −2.84681 × 10−3 6.27085 × 10−3 −1.06805 × 10−4 1.35511 × 10−3 3.16511 × 10−5 3.12846 × 10−5
−2.98118 × 10−3 −2.81024 × 10−3 −3.57091 × 10−3 1.02722 −1.11028 × 10−2 −2.31164 × 10−3 −3.75632 × 10−4
All nonmain diagonal elements in the VIIRS OOBCT matrix are the OOB correction terms, and main diagonal elements are greater than but close to 1. The first and fourth main diagonal elements with larger correction amounts (relative errors
In this paper, an optimal out-of-band (OOB) correction transform (OOBCT) for dealing with onboard Visible/Infrared Imaging Radiometer Suite (VIIRS) OOB effects is proposed. This paper addresses the OOB response issue without consideration of the impact of other error sources on the correction processing. The OOBCT matrix is derived by minimizing an objective function of error summation between the expected and realistic recovered band-averaged spectral radiances. Using the VIIRS filter transmittance functions for all multiband sensors obtained from prelaunch laboratory measurements and a simulated dataset obtained from Airborne Visible InfraRed Imaging Spectrometer (AVIRIS) hyperspectral data, the OOBCT matrix is numerically computed. The processing of the OOB correction is straightforward and can be performed by a product between the OOBCT matrix and a measured multispectral image vector. The experimental results with both AVIRIS and Hyperspectral Imager for the Coastal Ocean datasets demonstrate that the ratios of average errors of recovered band-averaged spectral radiances divided by the measured radiances with the OOB responses are less than 4%. The average values of the relative errors for all pixels and bands indicate that the OOBCT method outperforms the works reported in literature. © 2012 Optical Society of America OCIS codes: 010.0280, 010.5630.
1. Introduction
The correction of remote sensing radiometry for ocean color applications requires a highly accurate characterization of the optical instrument performance. For example, the Visible/Infrared Imager Radiometer Suite (VIIRS) launched successfully on 28 October 2011, employs a filter radiometer to remotely sense the atmosphere in 22 visible and IR bands located in the 0.4–12.5 μm range. Products of VIIRS data are a crucial continuation for climate change study under NASA and support National Oceanic and Atmospheric Administration (NOAA) and Department of Defense operational objectives. The primary issues of the VIIRS instrument are out-of-band (OOB) response and optical cross talk. The OOB response is defined as the ratio of the integrated response outside the 1% of peak response points of a spectral band to the integrated response 1559-128X/12/337962-07$15.00/0 © 2012 Optical Society of America 7962
APPLIED OPTICS / Vol. 51, No. 33 / 20 November 2012
for the band. The OOB effect produces a radiometric bias that depends on the source of radiance being measured, and it could adversely impact VIIRS product quality. In ocean color applications, accurate and consistent sensor calibration is essential [1,2]. Several multispectral radiometric instruments such as Sea-viewing Wide Field-of-view Sensor (SeaWiFS) and VIIRS are known to exhibit a significant radiance contribution from the OOB spectral response [3–7]. To deal with the SeaWiFS OOB effects, Gordon developed a methodology [3] for SeaWiFS calibration. Based on the Gordon methodology, an improved correction method to remove the spectral band effects of the SeaWiFS on the derived normalized waterleaving radiance, which results in improved ocean near-surface chlorophyll concentration, is developed and implemented in the operational SeaWiFS data processing system [8]. These calibration methods adjust the measured radiances to correct OOB response for ease of comparison to in situ measured multispectral radiances. The SeaWiFS correction
scheme has been successfully applied to data products retrieved over case 1 ocean waters [8]. However, the correction scheme is inherently not useable for SeaWiFS data product corrections over case 2 turbid waters or over land. An alternative solution for recovering the in-band multispectral radiances for SeaWiFS and VIIRS instruments is developed by Chen and Gao [9] and Gao and Chen [10], respectively, using the multispectral decomposition transform (MDT) method. The OOB correction by the MDT method is based on the characteristics of the multiband responses, and the same OOB responses for a single sensor are also detected by other multiband sensors with different spectral passbands. The information of multiband radiances recorded by multispectral radiometers distributed at different bands provides a possibility for decomposition. The MDT approach uses the decomposition principle to recover the average narrowband signals from contaminated signals using filter transmittance functions instead of calibration methods [3,8]. For an N-channel multispectral sensor, OOB effects are corrected by applying an N × N MDT matrix to the measured signals. The MDT matrices for SeaWiFS and VIIRS instruments are also reported by Chen and Gao [9], and Gao and Chen [10], respectively. The characteristics of the VIIRS multiband response functions indicate that wavelength intervals of the expected band-averaged (or in-band averaged) spectral radiance are usually less than the partitioned narrowband intervals in the MDT method. For this reason, there is still some contamination in the recovered narrowband radiances by the MDT [11]. To address the issue and obtain a highly accurate OOB correction, an out-of-band correction transform (OBCT) method is developed by Chen and Lucke [11]. The OBCT method provides a novel approach for dealing with the VIIRS OOB effects and is a significant improvement in comparison to the MDT method. If the transform by the MDT method is a zero-order form for the OOB correction based on the bandgap treatments, then the transform by the OBCT method is a first-order form and the MDT matrix is a special case of the OBCT matrix when the bandgaps approach zero. The linear decomposition transforms derived by the MDT and OBCT methods show two different forms and exhibit recognizably different performances for the same VIIRS OOB correction. These results encourage us to believe that there may exist an optimal decomposition transform among all possible linear transforms between the measured (contaminated by the OOB effects) and recovered band-averaged spectral radiances. Our goal in this paper is to seek an optimal out-of-band correction transform (OOBCT) to reduce the errors between the expected and realistic recovered band-averaged spectral radiances for dealing with the OOB effects. This paper is organized as follows. In Section 2, an OOBCT is derived for recovering the band-averaged spectral radiance. Section 3, using the VIIRS filter
transmittance functions and multispectral simulation image data, the OOBCT matrix is numerically computed and demonstrated. In Section 4, the proposed method and the VIIRS OOBCT matrix for the OOB correction are tested by a set of simulation image data. Finally, conclusions are drawn in Section 5. 2. Optimal Out-of-Band Correction Transform A. Multiband Radiometric Instrument
VIIRS is a newly launched multispectral remote sensing instrument with high spectral resolution. The VIIRS visible to near-infrared (VisNIR) channel names and wavelengths of the nominal band centers are tabulated in the first two columns in Table 1 [12]. The VIIRS filter data are available from a public domain website [13]. The VIIRS spectral responses with 1 nm wavelength intervals are within a wavelength range from λmin 0.391 μm to λmax 1.001 μm. A set of spectrally contiguous VIIRS (V3) M1–M7 filter transmittance curves (normalized at the peak of the filter transmission) is shown in Fig. 1. It has been found that the seven channels located between 0.4 and 0.9 μm (M1–M7) in the VisNIR focal plane have problems in OOB responses. As can be seen in Fig. 1, the filter curves of the VIIRS bands M1 and M4 exhibit more significant OOB transmittances than the rest of the bands. Table 1. VIIRS VisNIR Channel Names, Positions, and Minimum and Maximum Wavelengths between In-Band and Out-of-Band Responses
VIIRS Channel
λ (μm)
λk min (μm)
λk max (μm)
M1 M2 M3 M4 M5 M6 M7
0.412 0.445 0.488 0.555 0.672 0.746 0.865
0.395 0.432 0.473 0.530 0.649 0.731 0.830
0.426 0.458 0.506 0.572 0.693 0.760 0.897
Fig. 1. VIIRS version 3 M1–M7 filter transmittance curves normalized at the peaks. 20 November 2012 / Vol. 51, No. 33 / APPLIED OPTICS
7963
The OOB response is defined as the ratio of the integrated response outside the 1% of peak response points (upper and lower) to the integrated response for the band. The wavelengths in the third and forth columns in Table 1 show the wavelength ranges of the in-band and OOB responses for the VIIRS instrument. The spectral response function shown in Fig. 1 is defined by H k λ, where k is an index of the kth band filter and λ is the wavelength. A normalized spectral response function hk λ between the full range of the λmin and λmax is given by hk λ R λ
H k λ max
λmin
B.
H k λdλ
:
(1)
Band-Averaged Spectral Radiance
The measured radiance at the sensor is an average value of the spectral response with the kth band filter. This system is linear, because the measured band-averaged spectral radiance sˆ k sˆ ijk (for a compact notation) on a pixel, where i and j are pixel indices, from a sensor with the kth band filter can be expressed by Zλ max ˆsk hk λsλdλ; (2) λmin
where sˆ k and sλ sij λ are measured (uncorrected) and original radiances, respectively. The full range integral in Eq. (2) between λmin and λmax indicates that the measured band-averaged spectral radiance sˆ k consists of two contributions: in-band averaged and contaminated OOB signals. The in-band averaged spectral radiance sk sijk on a pixel between k the wavelengths λk min and λmax in Table 1 is given by R λk max λk min
sk
H k λsλdλ
R λk max λk min
H k λdλ
:
(3)
The in-band averaged spectral radiance is a bandaveraged spectral radiance without the OOB effects. This in-band averaged spectral radiance sk is the expected recovered signal from the contaminated signal sˆ k . The integrals for computing the measured radiance and in-band averaged spectral radiance in Eqs. (2) and (3) can be approximately evaluated by summations with 1 nm wavelength resolution of the spectral radiances and responses. The summations are given by sˆ k
λmax −λmin X
P sk
p0
hk λmin psλmin p;
k λk max −λmin
q0
P
k H k λk min qsλmin q
k λk max −λmin
q0
7964
H k λk min q
;
where we have utilized the property λmax −λmin X p0
C.
hk λmin p 1.
Optimal Out-of-Band Correction Transform
An OOB correction can be performed using the MDT [9,10] or OBCT [11] method. The in-band averaged spectral radiances for all pixels can be recovered by an OOB correction transform T kl such as s¯ k
N X
T kl sˆ l ;
(6)
l1
where s¯ k s¯ ijk is the realistic recovered in-band averaged spectral radiance on a pixel and N is the number of multispectral bands for a particular instrument. The transform matrix in Eq. (6) for the OOB correction can be determined by the response functions that are dependent on the characteristics of the filters for a particular instrument and parameters of the multiple band partition [9,10] or inband and bandgap partitions [11]. A real multispectral radiance image may contain the impact of random and spatial structured noise in the scene. All the noises within the narrowband regions remain in the image recovered by the transform. In this paper, we seek an optimal transform among all possible linear transforms for the OOB correction so that the errors between the expected and realistic recovered in-band averaged spectral radiances approach a minimum for all bands and pixels. Using the least-squares principle, a cost function is given by χ2
N XX i;j k1
sijk − s¯ ijk 2 ;
where i and j go over all pixels in an N x × N y image (i ∈ 1; N x ∩ j ∈ 1; N y ). Substituting s¯ ijk in Eq. (6) into the cost function and minimizing it with respect to all elements T αβ of the transform matrix, we have X
sijα sˆ ijβ
i;j
N X
T αl
l1
X
sˆ ijl sˆ ijβ ;
i;j
or the matrix form ˆ S TS;
where matrices S Sαβ and Sˆ Sˆ αβ , and α and β are row and column indices, respectively. The elements of the matrices S and Sˆ can be expressed by correlation functions, respectively,
(4)
Sαβ
1 X s sˆ corrsα ; sˆ β ; N x N y i;j ijα ijβ
(5)
Sˆ αβ
1 X sˆ sˆ corrˆsα ; sˆ β : N x N y i;j ijα ijβ
APPLIED OPTICS / Vol. 51, No. 33 / 20 November 2012
(7)
It is clear that the OOBCT can be solved from Eq. (7), and it is given by T SSˆ −1 :
(8)
A matrix form of Eq. (6) for recovering the bandaveraged spectral radiance from the measured radiance is given by s¯ Tˆs;
(9)
where s¯ s¯ ij ¯sk and sˆ sˆij ˆsk are column vectors on each pixel, respectively. The OOBCT matrix is a function of the cross correlations and autocorrelations between two bands. To recover the in-band averaged spectral radiance, the MDT [9,10] and the OBCT [11] matrices for dealing with the OOB effects are derived based on the spectral response functions and the partition methodologies. The OOBCT matrix with Eqs. (4) and (5) depends on the spectral response functions for a particular instrument through the cross-correlation and autocorrelation functions. 3. VIIRS OOBCT Matrix
We are concerned with the computation of the OOBCT matrix for the VIIRS optical instrument in this section. To evaluate the cross correlation and autocorrelation between the two bands in Eq. (7), the measured radiance and expected band-averaged spectral radiance must be known. Using Eqs. (4) and (5) and a real measured hyperspectral image dataset, the measured radiances and expected band-averaged spectral radiances can be obtained by a simulation method [9–11,14]. The zero-order and first-order transforms [9–11] can be calculated based only on the filter transmittance functions and partition parameters. The OOBCT matrix is a function of both the response functions and the measured signals. To find a transform in a variety of applications, the OOBCT matrix calculated by a specific dataset with minimized errors should be statistically optimized for other datasets. For a specific optical system such as the VIIRS instrument, the computation of the OOBCT matrix is a system training process before the procedure of the VIIRS OOB correction is operated. If the types of spectra in the employed hyperspectral image dataset are rich enough, the minimized OOBCT for a particular dataset is optimized for other measured or simulation multispectral datasets. The training dataset must cover all matters of interest and typical spectra such as deep water, turbid water, land, and vegetation. The Airborne Visible InfraRed Imaging Spectrometer (AVIRIS) [15] is used extensively for Earth remote sensing. It is a unique optical sensor that delivers calibrated images of the upwelling spectral radiance in 224 contiguous spectral bands with wavelengths from 370 to 2500 nm. The Hyerspectral Imager for the Coastal Ocean (HICO) sensor is the first hyperspectral imager designed specifically for remote sensing of the coastal environment [16] from
the International Space Station. It has 128 narrow channels (5.7 nm wide) covering the contiguous spectral range between 350 and 1080 nm with a spatial resolution of 100 m. AVIRIS and HICO data are used to simulate VIIRS data, with and without OOB effects, using the VIIRS (V3) filter transmittance functions in Fig. 1 and wavelength in-band intervals in Table 1. The multispectral simulation images with and without the OOB response are synthesized by Eqs. (4) and (5), where sλ is supplied by either AVIRIS or HICO hyperspectral image data. Two AVIRIS and one HICO datasets with 512 × 2000 resolution, shown in Figs. 2(a)–2(c) are chosen for the training process and experimental test data. The AVIRIS and HICO hyperspectral images acquired over the coastal area of New Jersey on 31 July 2001, the Kennedy Space Center, Florida, on 26 March 1996, and Freshwater Beach, Australia, on 14 May 2010, respectively, are displayed in true color in Fig. 2. The function sλ with 1 nm resolution is obtained by a interpolation between the two nearest hyperspectral bands from the AVIRIS and HICO datasets. The dataset of the AVIRIS Kennedy Space Center in Fig. 2(b) is employed as the training data for the computation of the OOBCT matrix. Two typical radiance spectra over areas covered by clear water and land in the AVIRIS Kennedy Space Center image are shown in Fig. 3. The VIIRS OOBCT (7 × 7) matrix T based on Eqs. (8), (4), and (5), the transmittance functions of
Fig. 2. (a) and (b) AVIRIS images acquired over the coastal area of New Jersey on 31 July 2001 and the Kennedy Space Center, Florida, on 26 March 1996, and (c) HICO image acquired over Freshwater Beach, Australia, 14 May 2010. The radiance spectral profiles and the relative errors at positions marked by A–D are shown in Figs. 4 to 7. 20 November 2012 / Vol. 51, No. 33 / APPLIED OPTICS
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the filters in Fig. 1, the in-band wavelength intervals in Table 1, and a training dataset in Fig. 2(b) is given by 0
1.02768 B −2.00174 × 10−3 B B −1.41508 × 10−3 B B −2.51098 × 10−3 B B −2.60251 × 10−3 B @ −1.57124 × 10−3 −4.51680 × 10−4
−1.6661 × 10−3 2.64793 × 10−3 1.00942 1.03182 × 10−3 −1.18287 × 10−3 1.01633 −6.65125 × 10−3 −3.80949 × 10−3 −2.84681 × 10−3 6.27085 × 10−3 −1.06805 × 10−4 1.35511 × 10−3 3.16511 × 10−5 3.12846 × 10−5
−2.98118 × 10−3 −2.81024 × 10−3 −3.57091 × 10−3 1.02722 −1.11028 × 10−2 −2.31164 × 10−3 −3.75632 × 10−4
All nonmain diagonal elements in the VIIRS OOBCT matrix are the OOB correction terms, and main diagonal elements are greater than but close to 1. The first and fourth main diagonal elements with larger correction amounts (relative errors
