Abstract

Space smart optical orbiting payloads integrated with attitude and position (SSPIAP) are emerging as an essential tool that is extensively used in microsatellites. The on-orbit imaging link of SSPIAPs includes atmospheric disturbances, defocusing, and relative motion, and other noises, thereby resulting in low modulation transfer function (MTF) and poor image quality. The introduction of MTF compensations has pushed the limits of optical imaging, enabling high-resolution on-orbit dynamic imaging. However, the external targets for compensating MTF are limited by space and time because the availability and access to external targets are infrequently easy when a remote sensor is working on-orbit. Here, a new and robust MTF self-compensation method for a SSPIAP is proposed. In comparison with conventional methods with external targets, this method utilizes multiple natural sub-resolution features (SRFs), occupying several pixels on a uniform background, as observation targets which makes MTFC more maneuverable, robust and authentic. A mathematical morphology algorithm is used to extract SRFs. Moreover, the method relies on a regularization total variation energy function, a sparse prior framework, to invert the MTF. Experimental measurements confirm that the proposed method is effective and convenient to implement. This technique does not rely on specific external targets to compensate the MTF, making it potentially suitable for on-orbit dynamic long-range imaging.

© 2017 Optical Society of America

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Efficient assessment method of on-board modulation transfer function of optical remote sensing sensors

Jin Li, Fei Xing, Ting Sun, and Zheng You
Opt. Express 23(5) 6187-6208 (2015)

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    [Crossref] [PubMed]

2016 (2)

J. Li, F. Xing, D. Chu, and Z. Liu, “High-Accuracy Self-Calibration for Smart, Optical Orbiting Payloads Integrated with Attitude and Position Determination,” Sensors (Basel) 16(8), 1176–1195 (2016).
[Crossref] [PubMed]

J. Li, Y. Zhang, S. Liu, and Z. Wang, “Self-Calibration Method Based on Surface Micromaching of Light Transceiver Focal Plane for Optical Camera,” Remote Sens. 8(12), 893–913 (2016).
[Crossref]

2015 (1)

J. Li, F. Xing, T. Sun, Z. Liu, and Z. You, “Space high-accuracy intelligence payload system with integrated attitude and position determination,” Instrumentation 2(1), 3–15 (2015).

2014 (5)

E. Oh and J. K. Choi, “GOCI image enhancement using an MTF compensation technique for coastal water applications,” Opt. Express 22(22), 26908–26918 (2014).
[Crossref] [PubMed]

T. Choi, X. Xiong, and Z. Wang, “On-Orbit lunar modulation transfer function measurements for the moderate resolution imaging spectroradiometer,” IEEE Trans. Geosci. Remote Sens. 52(1), 270–277 (2014).
[Crossref]

Z. Wang and X. Xiong, “VIIRS on-orbit spatial characterization using the Moon,” IEEE Geosci. Remote Sens. Lett. 11(6), 1116–1120 (2014).
[Crossref]

K. Masaoka, T. Yamashita, Y. Nishida, and M. Sugawara, “Modified slanted-edge method and multidirectional modulation transfer function estimation,” Opt. Express 22(5), 6040–6046 (2014).
[Crossref] [PubMed]

T. Yuan, X. Zheng, X. Hu, W. Zhou, and W. Wang, “A method for the evaluation of image quality according to the recognition effectiveness of objects in the optical remote sensing image using machine learning algorithm,” PLoS One 9(1), e86528 (2014).
[Crossref] [PubMed]

2013 (1)

X. Li, X. Gu, Q. Fu, T. Yu, H. Gao, J. Li, and L. Liu, “Removing atmospheric MTF and establishing an MTFcompensation filter for the HJ-1A CCD camera,” Int. J. Remote Sens. 34(4), 1413–1427 (2013).
[Crossref]

2012 (2)

E. Oh, S.-W. Kim, S.-I. Cho, J.-H. Ryu, and Y.-H. Ahn, “Initial on-orbit modulation transfer function performance analysis for Geostationary Ocean Color Imager,” J. Astron. Space Sci. 29(2), 199–208 (2012).
[Crossref]

X. Mu, S. Xu, G. Li, and J. Hu, “Remote sensing image restoration with modulation transfer function compensation technology in-orbit,” Proc. SPIE 8768, 87681K (2012).

2011 (1)

2010 (2)

2008 (3)

H. Hwang, Y. Choi, S. Kwak, M. Kim, and W. Park, “MTF assessment of high resolution satellite images using ISO 12233 slanted-edge method,” Proc. SPIE 7109, 710905 (2008).
[Crossref]

X. Chen, N. George, G. Agranov, C. Liu, and B. Gravelle, “Sensor modulation transfer function measurement using band-limited laser speckle,” Opt. Express 16(24), 20047–20059 (2008).
[Crossref] [PubMed]

D. Y. Tsai, Y. Lee, and E. Matsuyama, “Information entropy measure for evaluation of image quality,” J. Digit. Imaging 21(3), 338–347 (2008).
[Crossref] [PubMed]

2007 (1)

Z. Deng, Z. Yin, and Y. Xiong, “High probability impulse noise-removing algorithm based on mathematical morphology,” IEEE Signal Process. Lett. 14(1), 31–34 (2007).
[Crossref]

2006 (1)

2004 (2)

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13(4), 600–612 (2004).
[Crossref] [PubMed]

C. Latry, V. Despringre, and C. Valorge, “Automatic MTF measurement through a least square method,” Proc. SPIE 5570, 233–244 (2004).
[Crossref]

2003 (2)

D. Leger, F. Viallefont, E. Hillairet, and A. Meygret, “In-flight refocusing and MTF assessment of SPOT5 HRGand HRS cameras,” Proc. SPIE 4881, 224–231 (2003).
[Crossref]

R. Ryan, B. Baldridge, R. A. Schowengerdt, T. Choi, D. L. Helder, and S. Blonski, “IKONOS spatial resolution and image interpretability characterization,” Remote Sens. Environ. 88(1–2), 37–52 (2003).
[Crossref]

2001 (2)

J. C. Storey, “Landsat 7 on-orbit modulation transfer function estimation,” Proc. SPIE 4540, 50–61 (2001).
[Crossref]

M. K. Cook, B. A. Peterson, G. Dial, L. Gibson, F. W. Gerlach, K. S. Hutchins, R. Kudola, and H. S. Bowen, “IKONOS technical performance assessment,” Proc. SPIE 4381, 94–108 (2001).
[Crossref]

1998 (1)

T. F. Chan and C.-K. Wong, “Total variation blind deconvolution,” IEEE Trans. Image Process. 7(3), 370–375 (1998).
[Crossref] [PubMed]

1997 (1)

Y. Yitzhaky, I. Dror, and N. S. Kopeika, “Restoration of atmospherically blurred images according to weather-predicted atmospheric modulation transfer functions,” Opt. Eng. 36(11), 3064–3072 (1997).
[Crossref]

1995 (1)

S. E. Reichenbach, D. E. Koehler, and D. W. Strelow, “Restoration and reconstruction of AVHRR images,” IEEE Trans. Geosci. Remote Sens. 33(4), 997–1007 (1995).
[Crossref]

1992 (2)

H.-S. Wong, Y. L. Yao, and E. S. Schlig, “TDI charge-coupled devices: design and application,” IBM J. Res. Develop. 36(1), 83–106 (1992).
[Crossref]

L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60(1-4), 259–268 (1992).
[Crossref]

1956 (1)

W. H. Steel, “The defocused image of sinusoidal gratings,” J. Mod. Opt. 3, 65–74 (1956).

Agranov, G.

Ahn, Y.-H.

E. Oh, S.-W. Kim, S.-I. Cho, J.-H. Ryu, and Y.-H. Ahn, “Initial on-orbit modulation transfer function performance analysis for Geostationary Ocean Color Imager,” J. Astron. Space Sci. 29(2), 199–208 (2012).
[Crossref]

Anderson, C.

D. Helder, J. Choi, and C. Anderson, “On-orbit modulation transfer function (MTF) measurements for IKONOS and QuickBird,” In Proceedings of the JACIE 2006 Civil Commercial Imagery Evaluation Workshop,Brookings, SD, USA, pp.14–16(2006).

Bailly, B.

C. Gaudin-Delrieu, J. Lamard, P. Cheroutre, B. Bailly, P. Dhuicq, and O. Puig, “The High Resolution Optical Instruments for the Pleiades HR Earth Observation Satellites,” in Proceedings of the 7th ICSO (International Conference on Space Optics), pp. 14–17(2008).

Baldridge, B.

R. Ryan, B. Baldridge, R. A. Schowengerdt, T. Choi, D. L. Helder, and S. Blonski, “IKONOS spatial resolution and image interpretability characterization,” Remote Sens. Environ. 88(1–2), 37–52 (2003).
[Crossref]

Blonski, S.

R. Ryan, B. Baldridge, R. A. Schowengerdt, T. Choi, D. L. Helder, and S. Blonski, “IKONOS spatial resolution and image interpretability characterization,” Remote Sens. Environ. 88(1–2), 37–52 (2003).
[Crossref]

Bovik, A. C.

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13(4), 600–612 (2004).
[Crossref] [PubMed]

Bowen, H. S.

M. K. Cook, B. A. Peterson, G. Dial, L. Gibson, F. W. Gerlach, K. S. Hutchins, R. Kudola, and H. S. Bowen, “IKONOS technical performance assessment,” Proc. SPIE 4381, 94–108 (2001).
[Crossref]

Campos, J.

Chan, T. F.

T. F. Chan and C.-K. Wong, “Total variation blind deconvolution,” IEEE Trans. Image Process. 7(3), 370–375 (1998).
[Crossref] [PubMed]

Chen, X.

Cheroutre, P.

C. Gaudin-Delrieu, J. Lamard, P. Cheroutre, B. Bailly, P. Dhuicq, and O. Puig, “The High Resolution Optical Instruments for the Pleiades HR Earth Observation Satellites,” in Proceedings of the 7th ICSO (International Conference on Space Optics), pp. 14–17(2008).

Cho, S.-I.

E. Oh, S.-W. Kim, S.-I. Cho, J.-H. Ryu, and Y.-H. Ahn, “Initial on-orbit modulation transfer function performance analysis for Geostationary Ocean Color Imager,” J. Astron. Space Sci. 29(2), 199–208 (2012).
[Crossref]

Choi, J.

D. Helder, J. Choi, and C. Anderson, “On-orbit modulation transfer function (MTF) measurements for IKONOS and QuickBird,” In Proceedings of the JACIE 2006 Civil Commercial Imagery Evaluation Workshop,Brookings, SD, USA, pp.14–16(2006).

Choi, J. K.

Choi, T.

T. Choi, X. Xiong, and Z. Wang, “On-Orbit lunar modulation transfer function measurements for the moderate resolution imaging spectroradiometer,” IEEE Trans. Geosci. Remote Sens. 52(1), 270–277 (2014).
[Crossref]

R. Ryan, B. Baldridge, R. A. Schowengerdt, T. Choi, D. L. Helder, and S. Blonski, “IKONOS spatial resolution and image interpretability characterization,” Remote Sens. Environ. 88(1–2), 37–52 (2003).
[Crossref]

Choi, Y.

H. Hwang, Y. Choi, S. Kwak, M. Kim, and W. Park, “MTF assessment of high resolution satellite images using ISO 12233 slanted-edge method,” Proc. SPIE 7109, 710905 (2008).
[Crossref]

Chu, D.

J. Li, F. Xing, D. Chu, and Z. Liu, “High-Accuracy Self-Calibration for Smart, Optical Orbiting Payloads Integrated with Attitude and Position Determination,” Sensors (Basel) 16(8), 1176–1195 (2016).
[Crossref] [PubMed]

Cook, M. K.

M. K. Cook, B. A. Peterson, G. Dial, L. Gibson, F. W. Gerlach, K. S. Hutchins, R. Kudola, and H. S. Bowen, “IKONOS technical performance assessment,” Proc. SPIE 4381, 94–108 (2001).
[Crossref]

Deng, Z.

Z. Deng, Z. Yin, and Y. Xiong, “High probability impulse noise-removing algorithm based on mathematical morphology,” IEEE Signal Process. Lett. 14(1), 31–34 (2007).
[Crossref]

Despringre, V.

C. Latry, V. Despringre, and C. Valorge, “Automatic MTF measurement through a least square method,” Proc. SPIE 5570, 233–244 (2004).
[Crossref]

Dhuicq, P.

C. Gaudin-Delrieu, J. Lamard, P. Cheroutre, B. Bailly, P. Dhuicq, and O. Puig, “The High Resolution Optical Instruments for the Pleiades HR Earth Observation Satellites,” in Proceedings of the 7th ICSO (International Conference on Space Optics), pp. 14–17(2008).

Dial, G.

M. K. Cook, B. A. Peterson, G. Dial, L. Gibson, F. W. Gerlach, K. S. Hutchins, R. Kudola, and H. S. Bowen, “IKONOS technical performance assessment,” Proc. SPIE 4381, 94–108 (2001).
[Crossref]

Dror, I.

Y. Yitzhaky, I. Dror, and N. S. Kopeika, “Restoration of atmospherically blurred images according to weather-predicted atmospheric modulation transfer functions,” Opt. Eng. 36(11), 3064–3072 (1997).
[Crossref]

Duffaut, J.

D. Leger, J. Duffaut, and F. Robinet, “MTF measurement using spotlight,”in Proceedings of IEEE International Geoscience and Remote Sesning Symposium, pp. 2010–2012 (1994).

Fatemi, E.

L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60(1-4), 259–268 (1992).
[Crossref]

Ferrero, A.

Fu, Q.

X. Li, X. Gu, Q. Fu, T. Yu, H. Gao, J. Li, and L. Liu, “Removing atmospheric MTF and establishing an MTFcompensation filter for the HJ-1A CCD camera,” Int. J. Remote Sens. 34(4), 1413–1427 (2013).
[Crossref]

Gao, H.

X. Li, X. Gu, Q. Fu, T. Yu, H. Gao, J. Li, and L. Liu, “Removing atmospheric MTF and establishing an MTFcompensation filter for the HJ-1A CCD camera,” Int. J. Remote Sens. 34(4), 1413–1427 (2013).
[Crossref]

Gaudin-Delrieu, C.

C. Gaudin-Delrieu, J. Lamard, P. Cheroutre, B. Bailly, P. Dhuicq, and O. Puig, “The High Resolution Optical Instruments for the Pleiades HR Earth Observation Satellites,” in Proceedings of the 7th ICSO (International Conference on Space Optics), pp. 14–17(2008).

George, N.

Gerlach, F. W.

M. K. Cook, B. A. Peterson, G. Dial, L. Gibson, F. W. Gerlach, K. S. Hutchins, R. Kudola, and H. S. Bowen, “IKONOS technical performance assessment,” Proc. SPIE 4381, 94–108 (2001).
[Crossref]

Gibson, L.

M. K. Cook, B. A. Peterson, G. Dial, L. Gibson, F. W. Gerlach, K. S. Hutchins, R. Kudola, and H. S. Bowen, “IKONOS technical performance assessment,” Proc. SPIE 4381, 94–108 (2001).
[Crossref]

Gravelle, B.

Gu, M.

Gu, X.

X. Li, X. Gu, Q. Fu, T. Yu, H. Gao, J. Li, and L. Liu, “Removing atmospheric MTF and establishing an MTFcompensation filter for the HJ-1A CCD camera,” Int. J. Remote Sens. 34(4), 1413–1427 (2013).
[Crossref]

Guo, X. J.

Helder, D.

D. Helder, J. Choi, and C. Anderson, “On-orbit modulation transfer function (MTF) measurements for IKONOS and QuickBird,” In Proceedings of the JACIE 2006 Civil Commercial Imagery Evaluation Workshop,Brookings, SD, USA, pp.14–16(2006).

Helder, D. L.

R. Ryan, B. Baldridge, R. A. Schowengerdt, T. Choi, D. L. Helder, and S. Blonski, “IKONOS spatial resolution and image interpretability characterization,” Remote Sens. Environ. 88(1–2), 37–52 (2003).
[Crossref]

Hillairet, E.

D. Leger, F. Viallefont, E. Hillairet, and A. Meygret, “In-flight refocusing and MTF assessment of SPOT5 HRGand HRS cameras,” Proc. SPIE 4881, 224–231 (2003).
[Crossref]

Hu, J.

X. Mu, S. Xu, G. Li, and J. Hu, “Remote sensing image restoration with modulation transfer function compensation technology in-orbit,” Proc. SPIE 8768, 87681K (2012).

Hu, X.

T. Yuan, X. Zheng, X. Hu, W. Zhou, and W. Wang, “A method for the evaluation of image quality according to the recognition effectiveness of objects in the optical remote sensing image using machine learning algorithm,” PLoS One 9(1), e86528 (2014).
[Crossref] [PubMed]

Huang, S. M.

Hutchins, K. S.

M. K. Cook, B. A. Peterson, G. Dial, L. Gibson, F. W. Gerlach, K. S. Hutchins, R. Kudola, and H. S. Bowen, “IKONOS technical performance assessment,” Proc. SPIE 4381, 94–108 (2001).
[Crossref]

Hwang, H.

H. Hwang, Y. Choi, S. Kwak, M. Kim, and W. Park, “MTF assessment of high resolution satellite images using ISO 12233 slanted-edge method,” Proc. SPIE 7109, 710905 (2008).
[Crossref]

Kaveh, M.

Y.-L. You and M. Kaveh, “Anisotropic blind image restoration,” in International Conference on Image Processing, pp. 461–464(1996).

Kim, M.

H. Hwang, Y. Choi, S. Kwak, M. Kim, and W. Park, “MTF assessment of high resolution satellite images using ISO 12233 slanted-edge method,” Proc. SPIE 7109, 710905 (2008).
[Crossref]

Kim, S.-W.

E. Oh, S.-W. Kim, S.-I. Cho, J.-H. Ryu, and Y.-H. Ahn, “Initial on-orbit modulation transfer function performance analysis for Geostationary Ocean Color Imager,” J. Astron. Space Sci. 29(2), 199–208 (2012).
[Crossref]

Koehler, D. E.

S. E. Reichenbach, D. E. Koehler, and D. W. Strelow, “Restoration and reconstruction of AVHRR images,” IEEE Trans. Geosci. Remote Sens. 33(4), 997–1007 (1995).
[Crossref]

Kohm, K.

K. Kohm, “Modulation transfer function measurement method and results for the Orbview-3 high resolution imaging satellite,” in Proc. Geo-Imagery Bridging Continents XXth ISPRS Congr., pp. 7–12 (2004).

Kopeika, N. S.

Y. Yitzhaky, I. Dror, and N. S. Kopeika, “Restoration of atmospherically blurred images according to weather-predicted atmospheric modulation transfer functions,” Opt. Eng. 36(11), 3064–3072 (1997).
[Crossref]

Kudola, R.

M. K. Cook, B. A. Peterson, G. Dial, L. Gibson, F. W. Gerlach, K. S. Hutchins, R. Kudola, and H. S. Bowen, “IKONOS technical performance assessment,” Proc. SPIE 4381, 94–108 (2001).
[Crossref]

Kwak, S.

H. Hwang, Y. Choi, S. Kwak, M. Kim, and W. Park, “MTF assessment of high resolution satellite images using ISO 12233 slanted-edge method,” Proc. SPIE 7109, 710905 (2008).
[Crossref]

Lamard, J.

C. Gaudin-Delrieu, J. Lamard, P. Cheroutre, B. Bailly, P. Dhuicq, and O. Puig, “The High Resolution Optical Instruments for the Pleiades HR Earth Observation Satellites,” in Proceedings of the 7th ICSO (International Conference on Space Optics), pp. 14–17(2008).

Latry, C.

C. Latry, V. Despringre, and C. Valorge, “Automatic MTF measurement through a least square method,” Proc. SPIE 5570, 233–244 (2004).
[Crossref]

Lee, Y.

D. Y. Tsai, Y. Lee, and E. Matsuyama, “Information entropy measure for evaluation of image quality,” J. Digit. Imaging 21(3), 338–347 (2008).
[Crossref] [PubMed]

Leger, D.

D. Leger, F. Viallefont, E. Hillairet, and A. Meygret, “In-flight refocusing and MTF assessment of SPOT5 HRGand HRS cameras,” Proc. SPIE 4881, 224–231 (2003).
[Crossref]

D. Leger, J. Duffaut, and F. Robinet, “MTF measurement using spotlight,”in Proceedings of IEEE International Geoscience and Remote Sesning Symposium, pp. 2010–2012 (1994).

Léger, D.

Li, G.

X. Mu, S. Xu, G. Li, and J. Hu, “Remote sensing image restoration with modulation transfer function compensation technology in-orbit,” Proc. SPIE 8768, 87681K (2012).

Li, J.

J. Li, F. Xing, D. Chu, and Z. Liu, “High-Accuracy Self-Calibration for Smart, Optical Orbiting Payloads Integrated with Attitude and Position Determination,” Sensors (Basel) 16(8), 1176–1195 (2016).
[Crossref] [PubMed]

J. Li, Y. Zhang, S. Liu, and Z. Wang, “Self-Calibration Method Based on Surface Micromaching of Light Transceiver Focal Plane for Optical Camera,” Remote Sens. 8(12), 893–913 (2016).
[Crossref]

J. Li, F. Xing, T. Sun, Z. Liu, and Z. You, “Space high-accuracy intelligence payload system with integrated attitude and position determination,” Instrumentation 2(1), 3–15 (2015).

X. Li, X. Gu, Q. Fu, T. Yu, H. Gao, J. Li, and L. Liu, “Removing atmospheric MTF and establishing an MTFcompensation filter for the HJ-1A CCD camera,” Int. J. Remote Sens. 34(4), 1413–1427 (2013).
[Crossref]

Li, X.

X. Li, X. Gu, Q. Fu, T. Yu, H. Gao, J. Li, and L. Liu, “Removing atmospheric MTF and establishing an MTFcompensation filter for the HJ-1A CCD camera,” Int. J. Remote Sens. 34(4), 1413–1427 (2013).
[Crossref]

Liu, B.

Liu, C.

Liu, L.

X. Li, X. Gu, Q. Fu, T. Yu, H. Gao, J. Li, and L. Liu, “Removing atmospheric MTF and establishing an MTFcompensation filter for the HJ-1A CCD camera,” Int. J. Remote Sens. 34(4), 1413–1427 (2013).
[Crossref]

Liu, S.

J. Li, Y. Zhang, S. Liu, and Z. Wang, “Self-Calibration Method Based on Surface Micromaching of Light Transceiver Focal Plane for Optical Camera,” Remote Sens. 8(12), 893–913 (2016).
[Crossref]

Liu, X. L.

Liu, Z.

J. Li, F. Xing, D. Chu, and Z. Liu, “High-Accuracy Self-Calibration for Smart, Optical Orbiting Payloads Integrated with Attitude and Position Determination,” Sensors (Basel) 16(8), 1176–1195 (2016).
[Crossref] [PubMed]

J. Li, F. Xing, T. Sun, Z. Liu, and Z. You, “Space high-accuracy intelligence payload system with integrated attitude and position determination,” Instrumentation 2(1), 3–15 (2015).

Masaoka, K.

Matsuyama, E.

D. Y. Tsai, Y. Lee, and E. Matsuyama, “Information entropy measure for evaluation of image quality,” J. Digit. Imaging 21(3), 338–347 (2008).
[Crossref] [PubMed]

Meygret, A.

D. Leger, F. Viallefont, E. Hillairet, and A. Meygret, “In-flight refocusing and MTF assessment of SPOT5 HRGand HRS cameras,” Proc. SPIE 4881, 224–231 (2003).
[Crossref]

Mu, X.

X. Mu, S. Xu, G. Li, and J. Hu, “Remote sensing image restoration with modulation transfer function compensation technology in-orbit,” Proc. SPIE 8768, 87681K (2012).

Ni, C.

Nishida, Y.

Oh, E.

E. Oh and J. K. Choi, “GOCI image enhancement using an MTF compensation technique for coastal water applications,” Opt. Express 22(22), 26908–26918 (2014).
[Crossref] [PubMed]

E. Oh, S.-W. Kim, S.-I. Cho, J.-H. Ryu, and Y.-H. Ahn, “Initial on-orbit modulation transfer function performance analysis for Geostationary Ocean Color Imager,” J. Astron. Space Sci. 29(2), 199–208 (2012).
[Crossref]

Osher, S.

L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60(1-4), 259–268 (1992).
[Crossref]

Park, W.

H. Hwang, Y. Choi, S. Kwak, M. Kim, and W. Park, “MTF assessment of high resolution satellite images using ISO 12233 slanted-edge method,” Proc. SPIE 7109, 710905 (2008).
[Crossref]

Peterson, B. A.

M. K. Cook, B. A. Peterson, G. Dial, L. Gibson, F. W. Gerlach, K. S. Hutchins, R. Kudola, and H. S. Bowen, “IKONOS technical performance assessment,” Proc. SPIE 4381, 94–108 (2001).
[Crossref]

Pons, A.

Pozo, A. M.

Puig, O.

C. Gaudin-Delrieu, J. Lamard, P. Cheroutre, B. Bailly, P. Dhuicq, and O. Puig, “The High Resolution Optical Instruments for the Pleiades HR Earth Observation Satellites,” in Proceedings of the 7th ICSO (International Conference on Space Optics), pp. 14–17(2008).

Reichenbach, S. E.

S. E. Reichenbach, D. E. Koehler, and D. W. Strelow, “Restoration and reconstruction of AVHRR images,” IEEE Trans. Geosci. Remote Sens. 33(4), 997–1007 (1995).
[Crossref]

Robinet, F.

D. Leger, J. Duffaut, and F. Robinet, “MTF measurement using spotlight,”in Proceedings of IEEE International Geoscience and Remote Sesning Symposium, pp. 2010–2012 (1994).

Rubiño, M.

Rudin, L.

L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60(1-4), 259–268 (1992).
[Crossref]

Ryan, R.

R. Ryan, B. Baldridge, R. A. Schowengerdt, T. Choi, D. L. Helder, and S. Blonski, “IKONOS spatial resolution and image interpretability characterization,” Remote Sens. Environ. 88(1–2), 37–52 (2003).
[Crossref]

Ryu, J.-H.

E. Oh, S.-W. Kim, S.-I. Cho, J.-H. Ryu, and Y.-H. Ahn, “Initial on-orbit modulation transfer function performance analysis for Geostationary Ocean Color Imager,” J. Astron. Space Sci. 29(2), 199–208 (2012).
[Crossref]

Schlig, E. S.

H.-S. Wong, Y. L. Yao, and E. S. Schlig, “TDI charge-coupled devices: design and application,” IBM J. Res. Develop. 36(1), 83–106 (1992).
[Crossref]

Schowengerdt, R. A.

R. Ryan, B. Baldridge, R. A. Schowengerdt, T. Choi, D. L. Helder, and S. Blonski, “IKONOS spatial resolution and image interpretability characterization,” Remote Sens. Environ. 88(1–2), 37–52 (2003).
[Crossref]

Sheikh, H. R.

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13(4), 600–612 (2004).
[Crossref] [PubMed]

Simoncelli, E. P.

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13(4), 600–612 (2004).
[Crossref] [PubMed]

Steel, W. H.

W. H. Steel, “The defocused image of sinusoidal gratings,” J. Mod. Opt. 3, 65–74 (1956).

Storey, J. C.

J. C. Storey, “Landsat 7 on-orbit modulation transfer function estimation,” Proc. SPIE 4540, 50–61 (2001).
[Crossref]

Strelow, D. W.

S. E. Reichenbach, D. E. Koehler, and D. W. Strelow, “Restoration and reconstruction of AVHRR images,” IEEE Trans. Geosci. Remote Sens. 33(4), 997–1007 (1995).
[Crossref]

Sugawara, M.

Sun, T.

J. Li, F. Xing, T. Sun, Z. Liu, and Z. You, “Space high-accuracy intelligence payload system with integrated attitude and position determination,” Instrumentation 2(1), 3–15 (2015).

Tsai, D. Y.

D. Y. Tsai, Y. Lee, and E. Matsuyama, “Information entropy measure for evaluation of image quality,” J. Digit. Imaging 21(3), 338–347 (2008).
[Crossref] [PubMed]

Valorge, C.

C. Latry, V. Despringre, and C. Valorge, “Automatic MTF measurement through a least square method,” Proc. SPIE 5570, 233–244 (2004).
[Crossref]

Viallefont, F.

D. Leger, F. Viallefont, E. Hillairet, and A. Meygret, “In-flight refocusing and MTF assessment of SPOT5 HRGand HRS cameras,” Proc. SPIE 4881, 224–231 (2003).
[Crossref]

Viallefont-Robinet, F.

Wang, W.

T. Yuan, X. Zheng, X. Hu, W. Zhou, and W. Wang, “A method for the evaluation of image quality according to the recognition effectiveness of objects in the optical remote sensing image using machine learning algorithm,” PLoS One 9(1), e86528 (2014).
[Crossref] [PubMed]

Wang, Z.

J. Li, Y. Zhang, S. Liu, and Z. Wang, “Self-Calibration Method Based on Surface Micromaching of Light Transceiver Focal Plane for Optical Camera,” Remote Sens. 8(12), 893–913 (2016).
[Crossref]

T. Choi, X. Xiong, and Z. Wang, “On-Orbit lunar modulation transfer function measurements for the moderate resolution imaging spectroradiometer,” IEEE Trans. Geosci. Remote Sens. 52(1), 270–277 (2014).
[Crossref]

Z. Wang and X. Xiong, “VIIRS on-orbit spatial characterization using the Moon,” IEEE Geosci. Remote Sens. Lett. 11(6), 1116–1120 (2014).
[Crossref]

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13(4), 600–612 (2004).
[Crossref] [PubMed]

Wong, C.-K.

T. F. Chan and C.-K. Wong, “Total variation blind deconvolution,” IEEE Trans. Image Process. 7(3), 370–375 (1998).
[Crossref] [PubMed]

Wong, H.-S.

H.-S. Wong, Y. L. Yao, and E. S. Schlig, “TDI charge-coupled devices: design and application,” IBM J. Res. Develop. 36(1), 83–106 (1992).
[Crossref]

Xing, F.

J. Li, F. Xing, D. Chu, and Z. Liu, “High-Accuracy Self-Calibration for Smart, Optical Orbiting Payloads Integrated with Attitude and Position Determination,” Sensors (Basel) 16(8), 1176–1195 (2016).
[Crossref] [PubMed]

J. Li, F. Xing, T. Sun, Z. Liu, and Z. You, “Space high-accuracy intelligence payload system with integrated attitude and position determination,” Instrumentation 2(1), 3–15 (2015).

Xiong, X.

Z. Wang and X. Xiong, “VIIRS on-orbit spatial characterization using the Moon,” IEEE Geosci. Remote Sens. Lett. 11(6), 1116–1120 (2014).
[Crossref]

T. Choi, X. Xiong, and Z. Wang, “On-Orbit lunar modulation transfer function measurements for the moderate resolution imaging spectroradiometer,” IEEE Trans. Geosci. Remote Sens. 52(1), 270–277 (2014).
[Crossref]

Xiong, Y.

Z. Deng, Z. Yin, and Y. Xiong, “High probability impulse noise-removing algorithm based on mathematical morphology,” IEEE Signal Process. Lett. 14(1), 31–34 (2007).
[Crossref]

Xu, S.

X. Mu, S. Xu, G. Li, and J. Hu, “Remote sensing image restoration with modulation transfer function compensation technology in-orbit,” Proc. SPIE 8768, 87681K (2012).

Yamashita, T.

Yao, Y. L.

H.-S. Wong, Y. L. Yao, and E. S. Schlig, “TDI charge-coupled devices: design and application,” IBM J. Res. Develop. 36(1), 83–106 (1992).
[Crossref]

Yin, Z.

Z. Deng, Z. Yin, and Y. Xiong, “High probability impulse noise-removing algorithm based on mathematical morphology,” IEEE Signal Process. Lett. 14(1), 31–34 (2007).
[Crossref]

Yitzhaky, Y.

Y. Yitzhaky, I. Dror, and N. S. Kopeika, “Restoration of atmospherically blurred images according to weather-predicted atmospheric modulation transfer functions,” Opt. Eng. 36(11), 3064–3072 (1997).
[Crossref]

You, Y.-L.

Y.-L. You and M. Kaveh, “Anisotropic blind image restoration,” in International Conference on Image Processing, pp. 461–464(1996).

You, Z.

J. Li, F. Xing, T. Sun, Z. Liu, and Z. You, “Space high-accuracy intelligence payload system with integrated attitude and position determination,” Instrumentation 2(1), 3–15 (2015).

Yu, T.

X. Li, X. Gu, Q. Fu, T. Yu, H. Gao, J. Li, and L. Liu, “Removing atmospheric MTF and establishing an MTFcompensation filter for the HJ-1A CCD camera,” Int. J. Remote Sens. 34(4), 1413–1427 (2013).
[Crossref]

Yuan, T.

T. Yuan, X. Zheng, X. Hu, W. Zhou, and W. Wang, “A method for the evaluation of image quality according to the recognition effectiveness of objects in the optical remote sensing image using machine learning algorithm,” PLoS One 9(1), e86528 (2014).
[Crossref] [PubMed]

Zhang, Y.

J. Li, Y. Zhang, S. Liu, and Z. Wang, “Self-Calibration Method Based on Surface Micromaching of Light Transceiver Focal Plane for Optical Camera,” Remote Sens. 8(12), 893–913 (2016).
[Crossref]

Zheng, X.

T. Yuan, X. Zheng, X. Hu, W. Zhou, and W. Wang, “A method for the evaluation of image quality according to the recognition effectiveness of objects in the optical remote sensing image using machine learning algorithm,” PLoS One 9(1), e86528 (2014).
[Crossref] [PubMed]

Zhou, W.

T. Yuan, X. Zheng, X. Hu, W. Zhou, and W. Wang, “A method for the evaluation of image quality according to the recognition effectiveness of objects in the optical remote sensing image using machine learning algorithm,” PLoS One 9(1), e86528 (2014).
[Crossref] [PubMed]

IBM J. Res. Develop. (1)

H.-S. Wong, Y. L. Yao, and E. S. Schlig, “TDI charge-coupled devices: design and application,” IBM J. Res. Develop. 36(1), 83–106 (1992).
[Crossref]

IEEE Geosci. Remote Sens. Lett. (1)

Z. Wang and X. Xiong, “VIIRS on-orbit spatial characterization using the Moon,” IEEE Geosci. Remote Sens. Lett. 11(6), 1116–1120 (2014).
[Crossref]

IEEE Signal Process. Lett. (1)

Z. Deng, Z. Yin, and Y. Xiong, “High probability impulse noise-removing algorithm based on mathematical morphology,” IEEE Signal Process. Lett. 14(1), 31–34 (2007).
[Crossref]

IEEE Trans. Geosci. Remote Sens. (2)

S. E. Reichenbach, D. E. Koehler, and D. W. Strelow, “Restoration and reconstruction of AVHRR images,” IEEE Trans. Geosci. Remote Sens. 33(4), 997–1007 (1995).
[Crossref]

T. Choi, X. Xiong, and Z. Wang, “On-Orbit lunar modulation transfer function measurements for the moderate resolution imaging spectroradiometer,” IEEE Trans. Geosci. Remote Sens. 52(1), 270–277 (2014).
[Crossref]

IEEE Trans. Image Process. (2)

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13(4), 600–612 (2004).
[Crossref] [PubMed]

T. F. Chan and C.-K. Wong, “Total variation blind deconvolution,” IEEE Trans. Image Process. 7(3), 370–375 (1998).
[Crossref] [PubMed]

Instrumentation (1)

J. Li, F. Xing, T. Sun, Z. Liu, and Z. You, “Space high-accuracy intelligence payload system with integrated attitude and position determination,” Instrumentation 2(1), 3–15 (2015).

Int. J. Remote Sens. (1)

X. Li, X. Gu, Q. Fu, T. Yu, H. Gao, J. Li, and L. Liu, “Removing atmospheric MTF and establishing an MTFcompensation filter for the HJ-1A CCD camera,” Int. J. Remote Sens. 34(4), 1413–1427 (2013).
[Crossref]

J. Astron. Space Sci. (1)

E. Oh, S.-W. Kim, S.-I. Cho, J.-H. Ryu, and Y.-H. Ahn, “Initial on-orbit modulation transfer function performance analysis for Geostationary Ocean Color Imager,” J. Astron. Space Sci. 29(2), 199–208 (2012).
[Crossref]

J. Digit. Imaging (1)

D. Y. Tsai, Y. Lee, and E. Matsuyama, “Information entropy measure for evaluation of image quality,” J. Digit. Imaging 21(3), 338–347 (2008).
[Crossref] [PubMed]

J. Mod. Opt. (1)

W. H. Steel, “The defocused image of sinusoidal gratings,” J. Mod. Opt. 3, 65–74 (1956).

Opt. Eng. (1)

Y. Yitzhaky, I. Dror, and N. S. Kopeika, “Restoration of atmospherically blurred images according to weather-predicted atmospheric modulation transfer functions,” Opt. Eng. 36(11), 3064–3072 (1997).
[Crossref]

Opt. Express (7)

Physica D (1)

L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60(1-4), 259–268 (1992).
[Crossref]

PLoS One (1)

T. Yuan, X. Zheng, X. Hu, W. Zhou, and W. Wang, “A method for the evaluation of image quality according to the recognition effectiveness of objects in the optical remote sensing image using machine learning algorithm,” PLoS One 9(1), e86528 (2014).
[Crossref] [PubMed]

Proc. SPIE (6)

C. Latry, V. Despringre, and C. Valorge, “Automatic MTF measurement through a least square method,” Proc. SPIE 5570, 233–244 (2004).
[Crossref]

H. Hwang, Y. Choi, S. Kwak, M. Kim, and W. Park, “MTF assessment of high resolution satellite images using ISO 12233 slanted-edge method,” Proc. SPIE 7109, 710905 (2008).
[Crossref]

J. C. Storey, “Landsat 7 on-orbit modulation transfer function estimation,” Proc. SPIE 4540, 50–61 (2001).
[Crossref]

D. Leger, F. Viallefont, E. Hillairet, and A. Meygret, “In-flight refocusing and MTF assessment of SPOT5 HRGand HRS cameras,” Proc. SPIE 4881, 224–231 (2003).
[Crossref]

M. K. Cook, B. A. Peterson, G. Dial, L. Gibson, F. W. Gerlach, K. S. Hutchins, R. Kudola, and H. S. Bowen, “IKONOS technical performance assessment,” Proc. SPIE 4381, 94–108 (2001).
[Crossref]

X. Mu, S. Xu, G. Li, and J. Hu, “Remote sensing image restoration with modulation transfer function compensation technology in-orbit,” Proc. SPIE 8768, 87681K (2012).

Remote Sens. (1)

J. Li, Y. Zhang, S. Liu, and Z. Wang, “Self-Calibration Method Based on Surface Micromaching of Light Transceiver Focal Plane for Optical Camera,” Remote Sens. 8(12), 893–913 (2016).
[Crossref]

Remote Sens. Environ. (1)

R. Ryan, B. Baldridge, R. A. Schowengerdt, T. Choi, D. L. Helder, and S. Blonski, “IKONOS spatial resolution and image interpretability characterization,” Remote Sens. Environ. 88(1–2), 37–52 (2003).
[Crossref]

Sensors (Basel) (1)

J. Li, F. Xing, D. Chu, and Z. Liu, “High-Accuracy Self-Calibration for Smart, Optical Orbiting Payloads Integrated with Attitude and Position Determination,” Sensors (Basel) 16(8), 1176–1195 (2016).
[Crossref] [PubMed]

Other (6)

Y.-L. You and M. Kaveh, “Anisotropic blind image restoration,” in International Conference on Image Processing, pp. 461–464(1996).

K. Kohm, “Modulation transfer function measurement method and results for the Orbview-3 high resolution imaging satellite,” in Proc. Geo-Imagery Bridging Continents XXth ISPRS Congr., pp. 7–12 (2004).

P. Damilano, “Pleiades high resolution satellite: a solution for military and civilian needs in metric-class optical observation,” in AIAA/ USU Conference on Small Satellites, pp.1–9 (2001).

D. Helder, J. Choi, and C. Anderson, “On-orbit modulation transfer function (MTF) measurements for IKONOS and QuickBird,” In Proceedings of the JACIE 2006 Civil Commercial Imagery Evaluation Workshop,Brookings, SD, USA, pp.14–16(2006).

C. Gaudin-Delrieu, J. Lamard, P. Cheroutre, B. Bailly, P. Dhuicq, and O. Puig, “The High Resolution Optical Instruments for the Pleiades HR Earth Observation Satellites,” in Proceedings of the 7th ICSO (International Conference on Space Optics), pp. 14–17(2008).

D. Leger, J. Duffaut, and F. Robinet, “MTF measurement using spotlight,”in Proceedings of IEEE International Geoscience and Remote Sesning Symposium, pp. 2010–2012 (1994).

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Figures (15)

Fig. 1
Fig. 1 Imaging link model of the optical remote sensors.
Fig. 2
Fig. 2 Sketch map of the regularization term R.
Fig. 3
Fig. 3 Sub-resolution features in two remote images.
Fig. 4
Fig. 4 The entire MTF compensation procedure.
Fig. 5
Fig. 5 Gray value before and after mathematical morphology are (a) and (d), respectively; (b), (c), (e), (f) are the gray scale, color scale, and 3D plots of the same detailed view of a one star point.
Fig. 6
Fig. 6 Theoretical MTF in the X direction.
Fig. 7
Fig. 7 Theoretical2D MTF and recovered PSF;(a) and (b) are the 3D plots of MTF and2D color scale, respectively, of the same detailed view;(c),(d), and (e) are the gray scale, color scale, and 3D plots, respectively, of the same detailed view of PSF.
Fig. 8
Fig. 8 Simulated SRFs images, (a) original image with seven SRFs, (b) degraded image by MTF model, (c) magnified original SRFs, and (d) magnified degraded SRFs.
Fig. 9
Fig. 9 Mean squared error of MTF at different steps.
Fig. 10
Fig. 10 Reconstructed PSF and SRFs:(a),(b), and (c) are the gray scale, color scale, and 3D plots, respectively, of the same detailed view of the reconstructed PSF; and(d) is the reconstructed SRFs.
Fig. 11
Fig. 11 Experimental system: (a) is the schematic drawing of experiment setup; (b) is the prototype of the SSPIAP integrated optical camera with attitude sensor and (c) is the experiment setup.
Fig. 12
Fig. 12 Experimentaltesting targets:(a) is a slant edge target, (b) is the profile of edge, (c) is the point-star-like image, and (d) is the profile of the three points.
Fig. 13
Fig. 13 MTF before and after compensation, the MTF before compensation is measured by the edge-based method, and the MTF after compensation is inverted by the proposed method.
Fig. 14
Fig. 14 Compensated images:(a), (b), and (c) are the gray scale, color scale, and 3D plots, respectively, of the same detailed view of the captured image before compensation; and(d), (e), and (f) are the gray scale, color scale, and 3D plots, respectively, of the same detailed view of the captured image after compensation.
Fig. 15
Fig. 15 Recovered image using different methods: (a) is original image of the camera output, (b) is the recovered image using the edge method, and (c) is the compensated image using our method

Tables (3)

Tables Icon

Table 1 MTF values at the Nyquist frequency

Tables Icon

Table 2 Evaluation of the original and restored images

Tables Icon

Table 3 Evaluation of the original and restored images

Equations (31)

Equations on this page are rendered with MathJax. Learn more.

G ( ξ , η ) = F ( ξ , η ) H t o t a l ( ξ , η ) + N ( ξ , η ) H t o t a l ( ξ , η ) = H o p t i c s ( ξ , η ) H d e f o c u s ( ξ , η ) H det e c t o r ( ξ , η ) H i m a g e _ m o t i o n ( ξ , η ) H e l e c t r o n i c s ( ξ , η ) ,
G ( ξ , η ) = G ( ξ , η ) H ( ξ , η ) = F ( ξ , η ) H ( ξ , η ) H ( ξ , η ) + N ( ξ , η ) .
g ω ( x , y ) = f ω ( x , y ) h ( x , y ) + n ω ( x , y ) .
G ω ( ξ , η ) = F ω ( ξ , η ) H ( ξ , η ) + N ω ( ξ , η ) .
G ω = F ω H + N ω .
E ( f ) = T ( f ) + λ 2 U ( g f h ) 2 d U ,
T ( f ) = U | f | d U = U ( f y ) 2 + ( f x ) 2 d U .
E ω ( F ω ) = min F ω ( G ω H F ω p 2 + λ 1 T ( F ω ) + λ 2 F ω H G ω 2 2 ) ,
T ( F ω ) = | x F ω | + | y F ω | = i = 1 M 1 | f ( i , N ) f ( i + 1 , N ) | + j = 1 N 1 | f ( M , j ) f ( M , j + 1 ) | + i = 1 M 1 j = 1 N 1 | f ω ( i , j ) f ω ( i + 1 , j ) | 2 + | f ω ( i , j ) f ω ( i , j + 1 ) | 2 .
E ( H , F 1 , F 2 , ... , F ω ) = Q ( H , F 1 , F 2 , ... , F ω , G 1 , G 2 , ... , G ω ) + λ 1 T ( H ) + λ 2 W ( F 1 , F 2 , ... , F ω ) + λ 3 R ( F 1 , F 2 , ... , F ω , G 1 , G 2 , ... , G ω ) .
Q ( F 1 , F 2 , ... , F ω ) = ω = 1 Ω H F ω G ω 2 2 .
W ( F 1 , F 2 , ... , F ω ) = ω = 1 Ω F ω 2 2 .
G 1 F 2 = ( F 1 H ) F 2 = ( F 1 F 2 ) H G 2 F 1 = ( F 2 H ) F 1 = ( F 1 F 2 ) H .
R ( F 1 , F 2 , ... , F ω , G 1 , G 2 , ... , G ω ) = 1 2 1 i j Ω G i F j - G j F i 2 2 .
j = 1 N 1 i = 1 M 1 H ( i , j ) = 1 .
E ( H , F 1 , F 2 , ... , F ω ) = ω = 1 Ω H F ω G ω 2 2 + λ 1 T ( H ) + λ 2 ω = 1 Ω F ω 2 2 + λ 3 1 2 1 i j Ω G i F j - G j F i 2 2 .
g ω = [ g ω b ... g ω b ... g ω p ... g ω b ... g ω b ] = g ω p + g ω b = [ 0 0 g ω p g ω b 0 0 ] + [ g ω b ... g ω b ... g ω b ... g ω b ... g ω b ] ,
G ω - G ω b = F ω H + N - G ω b .
G ω - G ω b = F ω H + N - G ω b H = ( F ω G ω b ) H + N .
a ( x , y ) = τ Θ δ = min ( i , j ) D δ ( x + i , y + j ) D τ [ τ ( x + i , y + j ) δ ( i , j ) ] .
b ( x , y ) = τ δ = min ( i , j ) D δ ( x + i , y + j ) D τ [ τ ( x i , y j ) + δ ( i , j ) ] .
c = τ δ = ( τ Θ δ ) δ ,
d = τ ^ δ = ( τ δ ) Θ δ .
g ω = F ω b Q ω .
B ( x , y ) = 1 2 ( K 1 ) + 1 × 1 2 ( L 1 ) + 1 × i = ( K 1 ) K 1 j = ( L 1 ) L 1 o ( x + i , y + j ) .
T ( x , y ) = f ( x , y ) B ( x , y ) ,
g ( x , y ) = { 0 | g ( x , y ) | T g ( x , y ) | g ( x , y ) | > T .
{ E H = ω F ω ( H F ω G ω ) + λ 1 ( H | H | ) E F ω = ω H ( H F ω G ω ) + λ 2 ω = 1 Ω F ω + λ 3 1 i j Ω G i ( G i F j - G j F i ) ,
H ( n ) = H ( n 1 ) α E ( n 1 ) H ( n 1 ) ,
F ω ( n ) = F ω ( n 1 ) β E ( n 1 ) F ω ( n 1 ) ,
R m s e ( H t ) = H t H t 1 H t ,

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