Abstract

Hyperspectral imaging sensors suffer from spectral and spatial misregistrations due to optical-system aberrations and misalignments. These artifacts distort spectral signatures that are specific to target objects and thus reduce classification accuracy. The main objective of this work is to detect and correct spectral and spatial misregistrations of hyperspectral images. The Hyperion visible near-infrared subsystem is used as an example. An image registration method based on phase correlation demonstrates the accurate detection of the spectral and spatial misregistrations. Cubic spline interpolation using estimated properties makes it possible to modify the spectral signatures. The accuracy of the proposed postlaunch estimation of the Hyperion characteristics is comparable to that of the prelaunch measurements, which enables the accurate onboard calibration of hyperspectral sensors.

© 2010 Optical Society of America

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2010 (1)

A. Dadon, E. Ben-Dor, and A. Karnieli, “Use of derivative calculations and minimum noise fraction transform for detecting and correcting the spectral curvature effect (smile) in Hyperion images,” IEEE Trans. Geosci. Remote Sens. 48, 2603–2612 (2010).
[CrossRef]

2008 (1)

2007 (1)

2005 (3)

J. Chen and J. Katz, “Elimination of peak-locking error in PIV analysis using the correlation mapping method,” Meas. Sci. Technol. 16, 1605–1618 (2005).
[CrossRef]

J. T. Mundt, N. F. Glenn, K. T. Weber, T. S. Prather, L. W. Lass, and J. Pettingill, “Discrimination of hoary cress and determination of its detection limits via hyperspectral image processing and accuracy assessment techniques,” Remote Sens. Environ. 96, 509–517 (2005).
[CrossRef]

W. Liao, X. Chen, Y. Chen, and Y. Xia, “Index profiling of anisotropic graded-index planar waveguides from effective indices,” J. Opt. Soc. Am. A 22, 1334–1340 (2005).
[CrossRef]

2004 (1)

R. A. Neville, L. Sun, and K. Staenz, “Detection of keystone in imaging spectrometer data,” Proc. SPIE 5425, 208–217 (2004).
[CrossRef]

2003 (6)

W. S. Hoge, “Subspace identification extension to the phase correlation method,” IEEE Trans. Med. Imaging 22, 277–280(2003).
[CrossRef] [PubMed]

F. A. Kruse, J. W. Boardman, and J. F. Huntington, “Comparison of airborne hyperspectral data and EO-1 Hyperion for mineral mapping,” IEEE Trans. Geosci. Remote Sens. 41, 1388–1400 (2003).
[CrossRef]

J. S. Pearlman, P. S. Barry, C. C. Segal, J. Shepanski, D. Beiso, and S. L. Carman, “Hyperion, a space-based imaging spectrometer,” IEEE Trans. Geosci. Remote Sens. 41, 1160–1173(2003).
[CrossRef]

R. A. Neville, L. Sun, and K. Staenz, “Detection of spectral line curvature in imaging spectrometer data,” Proc. SPIE 5093, 144–154 (2003).
[CrossRef]

D. G. Goodenough, A. Dyk, K. O. Niemann, J. S. Pearlman, H. Chen, T. Han, M. Murdoch, and C. West, “Processing Hyperion and ALI for forest classification,” IEEE Trans. Geosci. Remote Sens. 41, 1321–1331 (2003).
[CrossRef]

E. Underwood, S. Ustin, and D. DiPietro, “Mapping nonnative plants using hyperspectral imagery,” Remote Sens. Environ. 86, 150–161 (2003).
[CrossRef]

2000 (2)

1998 (1)

1992 (1)

G. Behforooz, “The not-a-knot piecewise interpolatory cubic polynomial,” Appl. Math. Comput. 52, 29–35 (1992).
[CrossRef]

1988 (1)

A. A. Green, M. Berman, P. Switzer, and M. D. Craig, “A transformation for ordering multispectral data in terms of image quality with implications for noise removal,” IEEE Trans. Geosci. Remote Sens. 26, 65–74 (1988).
[CrossRef]

1985 (1)

1984 (1)

1977 (1)

Aoki, T.

S. Nagashima, T. Aoki, T. Higuchi, and K. Kobayashi, “A subpixel image matching technique using phase-only correlation,” in Proceedings of 2006 International Symposium on Intelligent Signal Processing and Communication Systems (IEEE, 2006), pp. 701–704.

Barry, P. S.

J. S. Pearlman, P. S. Barry, C. C. Segal, J. Shepanski, D. Beiso, and S. L. Carman, “Hyperion, a space-based imaging spectrometer,” IEEE Trans. Geosci. Remote Sens. 41, 1160–1173(2003).
[CrossRef]

Bartelt, H.

Behforooz, G.

G. Behforooz, “The not-a-knot piecewise interpolatory cubic polynomial,” Appl. Math. Comput. 52, 29–35 (1992).
[CrossRef]

Beiso, D.

J. S. Pearlman, P. S. Barry, C. C. Segal, J. Shepanski, D. Beiso, and S. L. Carman, “Hyperion, a space-based imaging spectrometer,” IEEE Trans. Geosci. Remote Sens. 41, 1160–1173(2003).
[CrossRef]

Ben-Dor, E.

A. Dadon, E. Ben-Dor, and A. Karnieli, “Use of derivative calculations and minimum noise fraction transform for detecting and correcting the spectral curvature effect (smile) in Hyperion images,” IEEE Trans. Geosci. Remote Sens. 48, 2603–2612 (2010).
[CrossRef]

Berman, M.

A. A. Green, M. Berman, P. Switzer, and M. D. Craig, “A transformation for ordering multispectral data in terms of image quality with implications for noise removal,” IEEE Trans. Geosci. Remote Sens. 26, 65–74 (1988).
[CrossRef]

Berthod, M.

H. Shekarforoush, M. Berthod, and J. Zerubia, “Subpixel image registration by estimating the polyphase decomposition of the cross power spectrum,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 1996), pp. 532–537.

Boardman, J. W.

F. A. Kruse, J. W. Boardman, and J. F. Huntington, “Comparison of airborne hyperspectral data and EO-1 Hyperion for mineral mapping,” IEEE Trans. Geosci. Remote Sens. 41, 1388–1400 (2003).
[CrossRef]

Carman, S. L.

J. S. Pearlman, P. S. Barry, C. C. Segal, J. Shepanski, D. Beiso, and S. L. Carman, “Hyperion, a space-based imaging spectrometer,” IEEE Trans. Geosci. Remote Sens. 41, 1160–1173(2003).
[CrossRef]

Chen, H.

D. G. Goodenough, A. Dyk, K. O. Niemann, J. S. Pearlman, H. Chen, T. Han, M. Murdoch, and C. West, “Processing Hyperion and ALI for forest classification,” IEEE Trans. Geosci. Remote Sens. 41, 1321–1331 (2003).
[CrossRef]

Chen, J.

J. Chen and J. Katz, “Elimination of peak-locking error in PIV analysis using the correlation mapping method,” Meas. Sci. Technol. 16, 1605–1618 (2005).
[CrossRef]

Chen, X.

Chen, Y.

Chrien, T. G.

Clancy, P.

P. Clancy, EO-1/Hyperion Early Orbit Checkout Report Part 2: On-orbit Performance Verification and Calibration (TRW, 2002), pp. 107–111.

Craig, M. D.

A. A. Green, M. Berman, P. Switzer, and M. D. Craig, “A transformation for ordering multispectral data in terms of image quality with implications for noise removal,” IEEE Trans. Geosci. Remote Sens. 26, 65–74 (1988).
[CrossRef]

Dadon, A.

A. Dadon, E. Ben-Dor, and A. Karnieli, “Use of derivative calculations and minimum noise fraction transform for detecting and correcting the spectral curvature effect (smile) in Hyperion images,” IEEE Trans. Geosci. Remote Sens. 48, 2603–2612 (2010).
[CrossRef]

Dell’Endice, F.

DiPietro, D.

E. Underwood, S. Ustin, and D. DiPietro, “Mapping nonnative plants using hyperspectral imagery,” Remote Sens. Environ. 86, 150–161 (2003).
[CrossRef]

Dyk, A.

D. G. Goodenough, A. Dyk, K. O. Niemann, J. S. Pearlman, H. Chen, T. Han, M. Murdoch, and C. West, “Processing Hyperion and ALI for forest classification,” IEEE Trans. Geosci. Remote Sens. 41, 1321–1331 (2003).
[CrossRef]

Feng, Y.

Gianino, P. D.

Glenn, N. F.

J. T. Mundt, N. F. Glenn, K. T. Weber, T. S. Prather, L. W. Lass, and J. Pettingill, “Discrimination of hoary cress and determination of its detection limits via hyperspectral image processing and accuracy assessment techniques,” Remote Sens. Environ. 96, 509–517 (2005).
[CrossRef]

Goodenough, D. G.

D. G. Goodenough, A. Dyk, K. O. Niemann, J. S. Pearlman, H. Chen, T. Han, M. Murdoch, and C. West, “Processing Hyperion and ALI for forest classification,” IEEE Trans. Geosci. Remote Sens. 41, 1321–1331 (2003).
[CrossRef]

Green, A. A.

A. A. Green, M. Berman, P. Switzer, and M. D. Craig, “A transformation for ordering multispectral data in terms of image quality with implications for noise removal,” IEEE Trans. Geosci. Remote Sens. 26, 65–74 (1988).
[CrossRef]

Green, R. O.

Han, T.

D. G. Goodenough, A. Dyk, K. O. Niemann, J. S. Pearlman, H. Chen, T. Han, M. Murdoch, and C. West, “Processing Hyperion and ALI for forest classification,” IEEE Trans. Geosci. Remote Sens. 41, 1321–1331 (2003).
[CrossRef]

Higuchi, T.

S. Nagashima, T. Aoki, T. Higuchi, and K. Kobayashi, “A subpixel image matching technique using phase-only correlation,” in Proceedings of 2006 International Symposium on Intelligent Signal Processing and Communication Systems (IEEE, 2006), pp. 701–704.

Hines, D. C.

C. D. Kuglin and D. C. Hines, “The phase correlation image alignment method,” in Proceedings of IEEE Conference on Cybernetics and Society (IEEE, 1975), pp. 163–165.

Hoge, W. S.

W. S. Hoge, “Subspace identification extension to the phase correlation method,” IEEE Trans. Med. Imaging 22, 277–280(2003).
[CrossRef] [PubMed]

Horner, J. L.

Huck, F. O.

Huntington, J. F.

F. A. Kruse, J. W. Boardman, and J. F. Huntington, “Comparison of airborne hyperspectral data and EO-1 Hyperion for mineral mapping,” IEEE Trans. Geosci. Remote Sens. 41, 1388–1400 (2003).
[CrossRef]

Itten, K. I.

Karnieli, A.

A. Dadon, E. Ben-Dor, and A. Karnieli, “Use of derivative calculations and minimum noise fraction transform for detecting and correcting the spectral curvature effect (smile) in Hyperion images,” IEEE Trans. Geosci. Remote Sens. 48, 2603–2612 (2010).
[CrossRef]

Katz, J.

J. Chen and J. Katz, “Elimination of peak-locking error in PIV analysis using the correlation mapping method,” Meas. Sci. Technol. 16, 1605–1618 (2005).
[CrossRef]

Kobayashi, K.

S. Nagashima, T. Aoki, T. Higuchi, and K. Kobayashi, “A subpixel image matching technique using phase-only correlation,” in Proceedings of 2006 International Symposium on Intelligent Signal Processing and Communication Systems (IEEE, 2006), pp. 701–704.

Kruse, F. A.

F. A. Kruse, J. W. Boardman, and J. F. Huntington, “Comparison of airborne hyperspectral data and EO-1 Hyperion for mineral mapping,” IEEE Trans. Geosci. Remote Sens. 41, 1388–1400 (2003).
[CrossRef]

Kuglin, C. D.

C. D. Kuglin and D. C. Hines, “The phase correlation image alignment method,” in Proceedings of IEEE Conference on Cybernetics and Society (IEEE, 1975), pp. 163–165.

Lass, L. W.

J. T. Mundt, N. F. Glenn, K. T. Weber, T. S. Prather, L. W. Lass, and J. Pettingill, “Discrimination of hoary cress and determination of its detection limits via hyperspectral image processing and accuracy assessment techniques,” Remote Sens. Environ. 96, 509–517 (2005).
[CrossRef]

Liao, W.

Mouroulis, P.

Mundt, J. T.

J. T. Mundt, N. F. Glenn, K. T. Weber, T. S. Prather, L. W. Lass, and J. Pettingill, “Discrimination of hoary cress and determination of its detection limits via hyperspectral image processing and accuracy assessment techniques,” Remote Sens. Environ. 96, 509–517 (2005).
[CrossRef]

Murdoch, M.

D. G. Goodenough, A. Dyk, K. O. Niemann, J. S. Pearlman, H. Chen, T. Han, M. Murdoch, and C. West, “Processing Hyperion and ALI for forest classification,” IEEE Trans. Geosci. Remote Sens. 41, 1321–1331 (2003).
[CrossRef]

Nagashima, S.

S. Nagashima, T. Aoki, T. Higuchi, and K. Kobayashi, “A subpixel image matching technique using phase-only correlation,” in Proceedings of 2006 International Symposium on Intelligent Signal Processing and Communication Systems (IEEE, 2006), pp. 701–704.

Neville, R. A.

R. A. Neville, L. Sun, and K. Staenz, “Detection of keystone in imaging spectrometer data,” Proc. SPIE 5425, 208–217 (2004).
[CrossRef]

R. A. Neville, L. Sun, and K. Staenz, “Detection of spectral line curvature in imaging spectrometer data,” Proc. SPIE 5093, 144–154 (2003).
[CrossRef]

Nieke, J.

Niemann, K. O.

D. G. Goodenough, A. Dyk, K. O. Niemann, J. S. Pearlman, H. Chen, T. Han, M. Murdoch, and C. West, “Processing Hyperion and ALI for forest classification,” IEEE Trans. Geosci. Remote Sens. 41, 1321–1331 (2003).
[CrossRef]

Park, S. K.

Pearlman, J. S.

D. G. Goodenough, A. Dyk, K. O. Niemann, J. S. Pearlman, H. Chen, T. Han, M. Murdoch, and C. West, “Processing Hyperion and ALI for forest classification,” IEEE Trans. Geosci. Remote Sens. 41, 1321–1331 (2003).
[CrossRef]

J. S. Pearlman, P. S. Barry, C. C. Segal, J. Shepanski, D. Beiso, and S. L. Carman, “Hyperion, a space-based imaging spectrometer,” IEEE Trans. Geosci. Remote Sens. 41, 1160–1173(2003).
[CrossRef]

Pettingill, J.

J. T. Mundt, N. F. Glenn, K. T. Weber, T. S. Prather, L. W. Lass, and J. Pettingill, “Discrimination of hoary cress and determination of its detection limits via hyperspectral image processing and accuracy assessment techniques,” Remote Sens. Environ. 96, 509–517 (2005).
[CrossRef]

Prather, T. S.

J. T. Mundt, N. F. Glenn, K. T. Weber, T. S. Prather, L. W. Lass, and J. Pettingill, “Discrimination of hoary cress and determination of its detection limits via hyperspectral image processing and accuracy assessment techniques,” Remote Sens. Environ. 96, 509–517 (2005).
[CrossRef]

Sasian, J. M.

Schlapfer, D.

Segal, C. C.

J. S. Pearlman, P. S. Barry, C. C. Segal, J. Shepanski, D. Beiso, and S. L. Carman, “Hyperion, a space-based imaging spectrometer,” IEEE Trans. Geosci. Remote Sens. 41, 1160–1173(2003).
[CrossRef]

Shekarforoush, H.

H. Shekarforoush, M. Berthod, and J. Zerubia, “Subpixel image registration by estimating the polyphase decomposition of the cross power spectrum,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 1996), pp. 532–537.

Shepanski, J.

J. S. Pearlman, P. S. Barry, C. C. Segal, J. Shepanski, D. Beiso, and S. L. Carman, “Hyperion, a space-based imaging spectrometer,” IEEE Trans. Geosci. Remote Sens. 41, 1160–1173(2003).
[CrossRef]

Staenz, K.

R. A. Neville, L. Sun, and K. Staenz, “Detection of keystone in imaging spectrometer data,” Proc. SPIE 5425, 208–217 (2004).
[CrossRef]

R. A. Neville, L. Sun, and K. Staenz, “Detection of spectral line curvature in imaging spectrometer data,” Proc. SPIE 5093, 144–154 (2003).
[CrossRef]

Sun, L.

R. A. Neville, L. Sun, and K. Staenz, “Detection of keystone in imaging spectrometer data,” Proc. SPIE 5425, 208–217 (2004).
[CrossRef]

R. A. Neville, L. Sun, and K. Staenz, “Detection of spectral line curvature in imaging spectrometer data,” Proc. SPIE 5093, 144–154 (2003).
[CrossRef]

Switzer, P.

A. A. Green, M. Berman, P. Switzer, and M. D. Craig, “A transformation for ordering multispectral data in terms of image quality with implications for noise removal,” IEEE Trans. Geosci. Remote Sens. 26, 65–74 (1988).
[CrossRef]

Underwood, E.

E. Underwood, S. Ustin, and D. DiPietro, “Mapping nonnative plants using hyperspectral imagery,” Remote Sens. Environ. 86, 150–161 (2003).
[CrossRef]

Ustin, S.

E. Underwood, S. Ustin, and D. DiPietro, “Mapping nonnative plants using hyperspectral imagery,” Remote Sens. Environ. 86, 150–161 (2003).
[CrossRef]

Weber, K. T.

J. T. Mundt, N. F. Glenn, K. T. Weber, T. S. Prather, L. W. Lass, and J. Pettingill, “Discrimination of hoary cress and determination of its detection limits via hyperspectral image processing and accuracy assessment techniques,” Remote Sens. Environ. 96, 509–517 (2005).
[CrossRef]

West, C.

D. G. Goodenough, A. Dyk, K. O. Niemann, J. S. Pearlman, H. Chen, T. Han, M. Murdoch, and C. West, “Processing Hyperion and ALI for forest classification,” IEEE Trans. Geosci. Remote Sens. 41, 1321–1331 (2003).
[CrossRef]

Westerweel, J.

J. Westerweel, “Effect of sensor geometry on the performance of PIV interrogation,” in Laser Techniques Applied to Fluid Mechanics, R.J.Adrian, D.F. G.Durão, F.Durst, M.V.Heitor, M.Maeda, and J.H.Whitelaw, eds. (Springer-Verlag, 2000), pp. 37–55.
[CrossRef]

Xia, Y.

Xiang, Y.

Zerubia, J.

H. Shekarforoush, M. Berthod, and J. Zerubia, “Subpixel image registration by estimating the polyphase decomposition of the cross power spectrum,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 1996), pp. 532–537.

Appl. Math. Comput. (1)

G. Behforooz, “The not-a-knot piecewise interpolatory cubic polynomial,” Appl. Math. Comput. 52, 29–35 (1992).
[CrossRef]

Appl. Opt. (7)

IEEE Trans. Geosci. Remote Sens. (5)

F. A. Kruse, J. W. Boardman, and J. F. Huntington, “Comparison of airborne hyperspectral data and EO-1 Hyperion for mineral mapping,” IEEE Trans. Geosci. Remote Sens. 41, 1388–1400 (2003).
[CrossRef]

J. S. Pearlman, P. S. Barry, C. C. Segal, J. Shepanski, D. Beiso, and S. L. Carman, “Hyperion, a space-based imaging spectrometer,” IEEE Trans. Geosci. Remote Sens. 41, 1160–1173(2003).
[CrossRef]

D. G. Goodenough, A. Dyk, K. O. Niemann, J. S. Pearlman, H. Chen, T. Han, M. Murdoch, and C. West, “Processing Hyperion and ALI for forest classification,” IEEE Trans. Geosci. Remote Sens. 41, 1321–1331 (2003).
[CrossRef]

A. Dadon, E. Ben-Dor, and A. Karnieli, “Use of derivative calculations and minimum noise fraction transform for detecting and correcting the spectral curvature effect (smile) in Hyperion images,” IEEE Trans. Geosci. Remote Sens. 48, 2603–2612 (2010).
[CrossRef]

A. A. Green, M. Berman, P. Switzer, and M. D. Craig, “A transformation for ordering multispectral data in terms of image quality with implications for noise removal,” IEEE Trans. Geosci. Remote Sens. 26, 65–74 (1988).
[CrossRef]

IEEE Trans. Med. Imaging (1)

W. S. Hoge, “Subspace identification extension to the phase correlation method,” IEEE Trans. Med. Imaging 22, 277–280(2003).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A (1)

Meas. Sci. Technol. (1)

J. Chen and J. Katz, “Elimination of peak-locking error in PIV analysis using the correlation mapping method,” Meas. Sci. Technol. 16, 1605–1618 (2005).
[CrossRef]

Opt. Express (1)

Proc. SPIE (2)

R. A. Neville, L. Sun, and K. Staenz, “Detection of keystone in imaging spectrometer data,” Proc. SPIE 5425, 208–217 (2004).
[CrossRef]

R. A. Neville, L. Sun, and K. Staenz, “Detection of spectral line curvature in imaging spectrometer data,” Proc. SPIE 5093, 144–154 (2003).
[CrossRef]

Remote Sens. Environ. (2)

E. Underwood, S. Ustin, and D. DiPietro, “Mapping nonnative plants using hyperspectral imagery,” Remote Sens. Environ. 86, 150–161 (2003).
[CrossRef]

J. T. Mundt, N. F. Glenn, K. T. Weber, T. S. Prather, L. W. Lass, and J. Pettingill, “Discrimination of hoary cress and determination of its detection limits via hyperspectral image processing and accuracy assessment techniques,” Remote Sens. Environ. 96, 509–517 (2005).
[CrossRef]

Other (5)

P. Clancy, EO-1/Hyperion Early Orbit Checkout Report Part 2: On-orbit Performance Verification and Calibration (TRW, 2002), pp. 107–111.

C. D. Kuglin and D. C. Hines, “The phase correlation image alignment method,” in Proceedings of IEEE Conference on Cybernetics and Society (IEEE, 1975), pp. 163–165.

H. Shekarforoush, M. Berthod, and J. Zerubia, “Subpixel image registration by estimating the polyphase decomposition of the cross power spectrum,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 1996), pp. 532–537.

J. Westerweel, “Effect of sensor geometry on the performance of PIV interrogation,” in Laser Techniques Applied to Fluid Mechanics, R.J.Adrian, D.F. G.Durão, F.Durst, M.V.Heitor, M.Maeda, and J.H.Whitelaw, eds. (Springer-Verlag, 2000), pp. 37–55.
[CrossRef]

S. Nagashima, T. Aoki, T. Higuchi, and K. Kobayashi, “A subpixel image matching technique using phase-only correlation,” in Proceedings of 2006 International Symposium on Intelligent Signal Processing and Communication Systems (IEEE, 2006), pp. 701–704.

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Figures (8)

Fig. 1
Fig. 1

(a) Original band 31, (b) MNF band 1 obtained from bands 8 to 57 for a desert scene in Chile, (c) spectral profile of the scene, (d) weighting factors that contribute to MNF1.

Fig. 2
Fig. 2

Spectral distortion of the oxygen absorption line detected by subpixel image registration based on (a) the NCC method and (b) the PC method. Gray area represents 1 standard deviation.

Fig. 3
Fig. 3

Comparison of prelaunch and estimated smile properties.

Fig. 4
Fig. 4

Radiance difference between bands 40 and 42 for (a) the original data and the data modified using (b) the prelaunch smile property and the properties estimated by (c) the NCC method and (d) the PC method.

Fig. 5
Fig. 5

Profiles of radiance difference between bands 40 and 42. The original data and the data modified using the prelaunch smile property and the properties estimated by the NCC method and the PC method are shown.

Fig. 6
Fig. 6

Keystone property estimated by the PC method for six cross-track numbers. Error bars represent 1 standard deviation.

Fig. 7
Fig. 7

Average keystone property estimated (a) before and (b) after correction. Error bars represent 1 standard deviation.

Fig. 8
Fig. 8

Spatial misregistration between bands 13 and 50 estimated from various spatially and temporally different scenes. Error bars represent 1 standard deviation.

Tables (1)

Tables Icon

Table 1 Quadratic Coefficient, x Coordinate of Axis of Smile Effect, and RSS of Quadratic Fitting

Equations (14)

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C ( m , n ) = i j { f ( i , j ) f ¯ } × { g ( i m , j n ) g ¯ } ( i j { f ( i , j ) f ¯ } 2 × i j { g ( i m , j n ) g ¯ } 2 ) 1 / 2 ,
( δ 1 , δ 2 ) = ( C ( m 0 1 , n 0 ) C ( m 0 + 1 , n 0 ) 2 C ( m 0 1 , n 0 ) 4 C ( m 0 , n 0 ) + 2 C ( m 0 + 1 , n 0 ) , C ( m 0 , n 0 1 ) C ( m 0 , n 0 + 1 ) 2 C ( m 0 , n 0 1 ) 4 C ( m 0 , n 0 ) + 2 C ( m 0 , n 0 + 1 ) ) .
R ( ω 1 , ω 2 ) = F ( ω 1 , ω 2 ) G * ( ω 1 , ω 2 ) F ( ω 1 , ω 2 ) G * ( ω 1 , ω 2 ) .
G ( ω 1 , ω 2 ) F ( ω 1 , ω 2 ) · e j 2 π ( ω 1 N 1 δ 1 + ω 2 N 2 δ 2 ) .
r ( n 1 , n 2 ) = F 1 [ R ( ω 1 , ω 2 ) ] α · sin π ( n 1 + δ 1 ) π ( n 1 + δ 1 ) sin π ( n 2 + δ 2 ) π ( n 2 + δ 2 ) ,
w ( n 1 , n 2 ) = 1 + cos ( π n 1 M 1 ) 2 1 + cos ( π n 2 M 2 ) 2 .
H ( ω 1 , ω 2 ) = { 1 ω 1 U 1 , ω 2 U 2 0 otherwise ,
r ( n 1 , n 2 ) α · sin V 1 N 1 π ( n 1 + δ 1 ) π ( n 1 + δ 1 ) sin V 2 N 2 π ( n 2 + δ 2 ) π ( n 2 + δ 2 ) ,
r ( n ) β · sin V N π ( n + δ ) π ( n + δ ) ,
δ = r ( n 0 1 ) r ( n 0 + 1 ) r ( n 0 1 ) 2 cos ( π V N ) r ( n 0 ) + r ( n 0 + 1 ) .
s i ( x ) = M i ( x i + 1 x ) 3 6 h i + M i + 1 ( x x i ) 3 6 h i + ( y i M i h i 2 6 ) x i + 1 x h i + ( y i + 1 M i + 1 h i 2 6 ) x x i h i ( x i x < x i + 1 , i = 1 , 2 , , n 1 ) ,
α 2 + 2 β 2 β 2 2 α 2 2 α 3 2 β 3 α n 2 2 β n 2 α n 1 2 β n 1 2 2 α n 1 + β n 1 M 2 M 3 M n 2 M n 1 = β 2 d 2 d 3 d n 2 α n 1 d n 1 ,
α i = h i 1 h i 1 + h i , β i = 1 α i , d i = 6 y i y i 1 h i 1 y i + 1 y i h i x i 1 x i + 1 ( i = 2 , 3 , , n 1 ) .
s ^ x , y , λ = { a s x + 1 , y , λ ( x < W ) a s x 1 , y , λ ( x = W ) ( 1 y H , 1 λ N ) .

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