Abstract

A scene-based algorithm is developed to compensate for bias nonuniformity in focal-plane arrays. Nonuniformity can be extremely problematic, especially for mid- to far-infrared imaging systems. The technique is based on use of estimates of interframe subpixel shifts in an image sequence, in conjunction with a linear-interpolation model for the motion, to extract information on the bias nonuniformity algebraically. The performance of the proposed algorithm is analyzed by using real infrared and simulated data. One advantage of this technique is its simplicity; it requires relatively few frames to generate an effective correction matrix, thereby permitting the execution of frequent on-the-fly nonuniformity correction as drift occurs. Additionally, the performance is shown to exhibit considerable robustness with respect to lack of the common types of temporal and spatial irradiance diversity that are typically required by statistical scene-based nonuniformity correction techniques.

© 2002 Optical Society of America

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References

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  1. A. F. Milton, F. R. Barone, M. R. Kruer, “Influence of nonuniformity on infrared focal plane array performance,” Opt. Eng. 24, 855–862 (1985).
    [CrossRef]
  2. D. L. Perry, E. L. Dereniak, “Linear theory of nonuniformity correction in infrared staring sensors,” Opt. Eng. 32, 1853–1859 (1993).
    [CrossRef]
  3. P. M. Narendra, N. A. Foss, “Shutterless fixed pattern noise correction for infrared imaging arrays,” in Technical Issues in Focal Plane Development, W. S. Chan, E. Krikorian, eds., Proc. SPIE282, 44–51 (1981).
    [CrossRef]
  4. P. M. Narendra, “Reference-free nonuniformity compensation for IR imaging arrays,” in Smart Sensors II, D. F. Barbe, ed., Proc. SPIE252, 10–17 (1980).
    [CrossRef]
  5. J. G. Harris, “Continuous-time calibration of VLSI sensors for gain and offset variations,” in Smart Focal Plane Arrays and Focal Plane Array Testing, M. Wigdor, M. A. Massie, eds., Proc. SPIE2474, 23–33 (1995).
    [CrossRef]
  6. J. G. Harris, Y. M. Chiang, “Nonuniformity correction using constant average statistics constraint: Analog and digital implementations,” in Infrared Technology and Applications XXIII, B. F. Andersen, M. Strojnik, eds., Proc. SPIE3061, 895–905 (1997).
    [CrossRef]
  7. Y. M. Chiang, J. G. Harris, “An analog integrated circuit for continuous-time gain and offset calibration of sensor arrays,” J. Analog Integr. Circuits Signal Process. 12, 231–238 (1997).
    [CrossRef]
  8. W. F. O’Neil, “Dithered scan detector compensation,” presented at the 1993 International Meeting of the Infrared Information Symposium Specialty Group on Passive Sensors, Ann Arbor, Mich., 1993).
  9. W. F. O’Neil, “Experimental verification of dithered scan nonuniformity correction,” in Proceedings of the 1996 International Meeting of the Infrared Information Symposium Specialty Group on Passive Sensors (Infrared Information Analysis Center, Ann Arbor, Michigan, 1997) Vol. 1, pp. 329–339.
  10. R. C. Hardie, M. M. Hayat, E. E. Armstrong, B. Yasuda, “Scene-based nonuniformity correction using video sequences and registration,” Appl. Opt. 39, 1241–1250 (2000).
    [CrossRef]
  11. M. M. Hayat, S. N. Torres, E. E. Armstrong, B. Yasuda, “Statistical algorithm for nonuniformity correction in focal-plane arrays,” Appl. Opt. 38, 772–780 (1999).
    [CrossRef]
  12. S. N. Torres, M. M. Hayat, E. E. Armstrong, B. Yasuda, “A Kalman-filtering approach for nonuniformity correction in focal-plane array sensors,” in Infrared Imaging Systems: Design, Analysis, Modeling, and Testing XI, G. C. Holst and JCD Publishing, eds., Proc. SPIE4030, 196–203 (2000).
    [CrossRef]
  13. S. N. Torres, M. M. Hayat, “Compensation for gain and bias nonuniformity and drift in array detectors: A Kalman-filtering approach,” manuscript available from M. M. Hayat; hayat@eece.unm.edu.
  14. G. C. Holst, CCD Arrays, Cameras and Displays (SPIE Optical Engineering Press, Bellingham, Wash., 1996).
  15. M. Irani, S. Peleg, “Improving resolution by image registration,” CVGIP: Graph. Models Image Process. 53, 231–239 (1991).
  16. R. C. Hardie, K. J. Barnard, J. G. Bognar, E. A. Watson, “High-resolution image reconstruction from a sequence of rotated and translated frames and its application to an infrared imaging system,” Opt. Eng. 37, 247–260 (1998).
    [CrossRef]
  17. E. E. Armstrong, M. M. Hayat, R. C. Hardie, S. N. Torres, B. Yasuda, “Nonuniformity correction for improved registration and high-resolution image reconstruction in IR imagery,” in Application of Digital Image Processing XXII, A. G. Tescher and Lockheed Martin Missions Systems, eds., Proc. SPIE3808, 150–161 (1999).
    [CrossRef]
  18. S. C. Cain, M. M. Hayat, E. E. Armstrong, “Projection-based image registration in the presence of fixed-pattern noise,” IEEE Trans. Image Process. 10, 1860–1872 (2001).
    [CrossRef]

2001

S. C. Cain, M. M. Hayat, E. E. Armstrong, “Projection-based image registration in the presence of fixed-pattern noise,” IEEE Trans. Image Process. 10, 1860–1872 (2001).
[CrossRef]

2000

1999

1998

R. C. Hardie, K. J. Barnard, J. G. Bognar, E. A. Watson, “High-resolution image reconstruction from a sequence of rotated and translated frames and its application to an infrared imaging system,” Opt. Eng. 37, 247–260 (1998).
[CrossRef]

1997

Y. M. Chiang, J. G. Harris, “An analog integrated circuit for continuous-time gain and offset calibration of sensor arrays,” J. Analog Integr. Circuits Signal Process. 12, 231–238 (1997).
[CrossRef]

1993

D. L. Perry, E. L. Dereniak, “Linear theory of nonuniformity correction in infrared staring sensors,” Opt. Eng. 32, 1853–1859 (1993).
[CrossRef]

1991

M. Irani, S. Peleg, “Improving resolution by image registration,” CVGIP: Graph. Models Image Process. 53, 231–239 (1991).

1985

A. F. Milton, F. R. Barone, M. R. Kruer, “Influence of nonuniformity on infrared focal plane array performance,” Opt. Eng. 24, 855–862 (1985).
[CrossRef]

Armstrong, E. E.

S. C. Cain, M. M. Hayat, E. E. Armstrong, “Projection-based image registration in the presence of fixed-pattern noise,” IEEE Trans. Image Process. 10, 1860–1872 (2001).
[CrossRef]

R. C. Hardie, M. M. Hayat, E. E. Armstrong, B. Yasuda, “Scene-based nonuniformity correction using video sequences and registration,” Appl. Opt. 39, 1241–1250 (2000).
[CrossRef]

M. M. Hayat, S. N. Torres, E. E. Armstrong, B. Yasuda, “Statistical algorithm for nonuniformity correction in focal-plane arrays,” Appl. Opt. 38, 772–780 (1999).
[CrossRef]

S. N. Torres, M. M. Hayat, E. E. Armstrong, B. Yasuda, “A Kalman-filtering approach for nonuniformity correction in focal-plane array sensors,” in Infrared Imaging Systems: Design, Analysis, Modeling, and Testing XI, G. C. Holst and JCD Publishing, eds., Proc. SPIE4030, 196–203 (2000).
[CrossRef]

E. E. Armstrong, M. M. Hayat, R. C. Hardie, S. N. Torres, B. Yasuda, “Nonuniformity correction for improved registration and high-resolution image reconstruction in IR imagery,” in Application of Digital Image Processing XXII, A. G. Tescher and Lockheed Martin Missions Systems, eds., Proc. SPIE3808, 150–161 (1999).
[CrossRef]

Barnard, K. J.

R. C. Hardie, K. J. Barnard, J. G. Bognar, E. A. Watson, “High-resolution image reconstruction from a sequence of rotated and translated frames and its application to an infrared imaging system,” Opt. Eng. 37, 247–260 (1998).
[CrossRef]

Barone, F. R.

A. F. Milton, F. R. Barone, M. R. Kruer, “Influence of nonuniformity on infrared focal plane array performance,” Opt. Eng. 24, 855–862 (1985).
[CrossRef]

Bognar, J. G.

R. C. Hardie, K. J. Barnard, J. G. Bognar, E. A. Watson, “High-resolution image reconstruction from a sequence of rotated and translated frames and its application to an infrared imaging system,” Opt. Eng. 37, 247–260 (1998).
[CrossRef]

Cain, S. C.

S. C. Cain, M. M. Hayat, E. E. Armstrong, “Projection-based image registration in the presence of fixed-pattern noise,” IEEE Trans. Image Process. 10, 1860–1872 (2001).
[CrossRef]

Chiang, Y. M.

Y. M. Chiang, J. G. Harris, “An analog integrated circuit for continuous-time gain and offset calibration of sensor arrays,” J. Analog Integr. Circuits Signal Process. 12, 231–238 (1997).
[CrossRef]

J. G. Harris, Y. M. Chiang, “Nonuniformity correction using constant average statistics constraint: Analog and digital implementations,” in Infrared Technology and Applications XXIII, B. F. Andersen, M. Strojnik, eds., Proc. SPIE3061, 895–905 (1997).
[CrossRef]

Dereniak, E. L.

D. L. Perry, E. L. Dereniak, “Linear theory of nonuniformity correction in infrared staring sensors,” Opt. Eng. 32, 1853–1859 (1993).
[CrossRef]

Foss, N. A.

P. M. Narendra, N. A. Foss, “Shutterless fixed pattern noise correction for infrared imaging arrays,” in Technical Issues in Focal Plane Development, W. S. Chan, E. Krikorian, eds., Proc. SPIE282, 44–51 (1981).
[CrossRef]

Hardie, R. C.

R. C. Hardie, M. M. Hayat, E. E. Armstrong, B. Yasuda, “Scene-based nonuniformity correction using video sequences and registration,” Appl. Opt. 39, 1241–1250 (2000).
[CrossRef]

R. C. Hardie, K. J. Barnard, J. G. Bognar, E. A. Watson, “High-resolution image reconstruction from a sequence of rotated and translated frames and its application to an infrared imaging system,” Opt. Eng. 37, 247–260 (1998).
[CrossRef]

E. E. Armstrong, M. M. Hayat, R. C. Hardie, S. N. Torres, B. Yasuda, “Nonuniformity correction for improved registration and high-resolution image reconstruction in IR imagery,” in Application of Digital Image Processing XXII, A. G. Tescher and Lockheed Martin Missions Systems, eds., Proc. SPIE3808, 150–161 (1999).
[CrossRef]

Harris, J. G.

Y. M. Chiang, J. G. Harris, “An analog integrated circuit for continuous-time gain and offset calibration of sensor arrays,” J. Analog Integr. Circuits Signal Process. 12, 231–238 (1997).
[CrossRef]

J. G. Harris, “Continuous-time calibration of VLSI sensors for gain and offset variations,” in Smart Focal Plane Arrays and Focal Plane Array Testing, M. Wigdor, M. A. Massie, eds., Proc. SPIE2474, 23–33 (1995).
[CrossRef]

J. G. Harris, Y. M. Chiang, “Nonuniformity correction using constant average statistics constraint: Analog and digital implementations,” in Infrared Technology and Applications XXIII, B. F. Andersen, M. Strojnik, eds., Proc. SPIE3061, 895–905 (1997).
[CrossRef]

Hayat, M. M.

S. C. Cain, M. M. Hayat, E. E. Armstrong, “Projection-based image registration in the presence of fixed-pattern noise,” IEEE Trans. Image Process. 10, 1860–1872 (2001).
[CrossRef]

R. C. Hardie, M. M. Hayat, E. E. Armstrong, B. Yasuda, “Scene-based nonuniformity correction using video sequences and registration,” Appl. Opt. 39, 1241–1250 (2000).
[CrossRef]

M. M. Hayat, S. N. Torres, E. E. Armstrong, B. Yasuda, “Statistical algorithm for nonuniformity correction in focal-plane arrays,” Appl. Opt. 38, 772–780 (1999).
[CrossRef]

S. N. Torres, M. M. Hayat, E. E. Armstrong, B. Yasuda, “A Kalman-filtering approach for nonuniformity correction in focal-plane array sensors,” in Infrared Imaging Systems: Design, Analysis, Modeling, and Testing XI, G. C. Holst and JCD Publishing, eds., Proc. SPIE4030, 196–203 (2000).
[CrossRef]

E. E. Armstrong, M. M. Hayat, R. C. Hardie, S. N. Torres, B. Yasuda, “Nonuniformity correction for improved registration and high-resolution image reconstruction in IR imagery,” in Application of Digital Image Processing XXII, A. G. Tescher and Lockheed Martin Missions Systems, eds., Proc. SPIE3808, 150–161 (1999).
[CrossRef]

Holst, G. C.

G. C. Holst, CCD Arrays, Cameras and Displays (SPIE Optical Engineering Press, Bellingham, Wash., 1996).

Irani, M.

M. Irani, S. Peleg, “Improving resolution by image registration,” CVGIP: Graph. Models Image Process. 53, 231–239 (1991).

Kruer, M. R.

A. F. Milton, F. R. Barone, M. R. Kruer, “Influence of nonuniformity on infrared focal plane array performance,” Opt. Eng. 24, 855–862 (1985).
[CrossRef]

Milton, A. F.

A. F. Milton, F. R. Barone, M. R. Kruer, “Influence of nonuniformity on infrared focal plane array performance,” Opt. Eng. 24, 855–862 (1985).
[CrossRef]

Narendra, P. M.

P. M. Narendra, N. A. Foss, “Shutterless fixed pattern noise correction for infrared imaging arrays,” in Technical Issues in Focal Plane Development, W. S. Chan, E. Krikorian, eds., Proc. SPIE282, 44–51 (1981).
[CrossRef]

P. M. Narendra, “Reference-free nonuniformity compensation for IR imaging arrays,” in Smart Sensors II, D. F. Barbe, ed., Proc. SPIE252, 10–17 (1980).
[CrossRef]

O’Neil, W. F.

W. F. O’Neil, “Dithered scan detector compensation,” presented at the 1993 International Meeting of the Infrared Information Symposium Specialty Group on Passive Sensors, Ann Arbor, Mich., 1993).

W. F. O’Neil, “Experimental verification of dithered scan nonuniformity correction,” in Proceedings of the 1996 International Meeting of the Infrared Information Symposium Specialty Group on Passive Sensors (Infrared Information Analysis Center, Ann Arbor, Michigan, 1997) Vol. 1, pp. 329–339.

Peleg, S.

M. Irani, S. Peleg, “Improving resolution by image registration,” CVGIP: Graph. Models Image Process. 53, 231–239 (1991).

Perry, D. L.

D. L. Perry, E. L. Dereniak, “Linear theory of nonuniformity correction in infrared staring sensors,” Opt. Eng. 32, 1853–1859 (1993).
[CrossRef]

Torres, S. N.

M. M. Hayat, S. N. Torres, E. E. Armstrong, B. Yasuda, “Statistical algorithm for nonuniformity correction in focal-plane arrays,” Appl. Opt. 38, 772–780 (1999).
[CrossRef]

S. N. Torres, M. M. Hayat, E. E. Armstrong, B. Yasuda, “A Kalman-filtering approach for nonuniformity correction in focal-plane array sensors,” in Infrared Imaging Systems: Design, Analysis, Modeling, and Testing XI, G. C. Holst and JCD Publishing, eds., Proc. SPIE4030, 196–203 (2000).
[CrossRef]

E. E. Armstrong, M. M. Hayat, R. C. Hardie, S. N. Torres, B. Yasuda, “Nonuniformity correction for improved registration and high-resolution image reconstruction in IR imagery,” in Application of Digital Image Processing XXII, A. G. Tescher and Lockheed Martin Missions Systems, eds., Proc. SPIE3808, 150–161 (1999).
[CrossRef]

Watson, E. A.

R. C. Hardie, K. J. Barnard, J. G. Bognar, E. A. Watson, “High-resolution image reconstruction from a sequence of rotated and translated frames and its application to an infrared imaging system,” Opt. Eng. 37, 247–260 (1998).
[CrossRef]

Yasuda, B.

R. C. Hardie, M. M. Hayat, E. E. Armstrong, B. Yasuda, “Scene-based nonuniformity correction using video sequences and registration,” Appl. Opt. 39, 1241–1250 (2000).
[CrossRef]

M. M. Hayat, S. N. Torres, E. E. Armstrong, B. Yasuda, “Statistical algorithm for nonuniformity correction in focal-plane arrays,” Appl. Opt. 38, 772–780 (1999).
[CrossRef]

S. N. Torres, M. M. Hayat, E. E. Armstrong, B. Yasuda, “A Kalman-filtering approach for nonuniformity correction in focal-plane array sensors,” in Infrared Imaging Systems: Design, Analysis, Modeling, and Testing XI, G. C. Holst and JCD Publishing, eds., Proc. SPIE4030, 196–203 (2000).
[CrossRef]

E. E. Armstrong, M. M. Hayat, R. C. Hardie, S. N. Torres, B. Yasuda, “Nonuniformity correction for improved registration and high-resolution image reconstruction in IR imagery,” in Application of Digital Image Processing XXII, A. G. Tescher and Lockheed Martin Missions Systems, eds., Proc. SPIE3808, 150–161 (1999).
[CrossRef]

Appl. Opt.

CVGIP: Graph. Models Image Process.

M. Irani, S. Peleg, “Improving resolution by image registration,” CVGIP: Graph. Models Image Process. 53, 231–239 (1991).

IEEE Trans. Image Process.

S. C. Cain, M. M. Hayat, E. E. Armstrong, “Projection-based image registration in the presence of fixed-pattern noise,” IEEE Trans. Image Process. 10, 1860–1872 (2001).
[CrossRef]

J. Analog Integr. Circuits Signal Process.

Y. M. Chiang, J. G. Harris, “An analog integrated circuit for continuous-time gain and offset calibration of sensor arrays,” J. Analog Integr. Circuits Signal Process. 12, 231–238 (1997).
[CrossRef]

Opt. Eng.

R. C. Hardie, K. J. Barnard, J. G. Bognar, E. A. Watson, “High-resolution image reconstruction from a sequence of rotated and translated frames and its application to an infrared imaging system,” Opt. Eng. 37, 247–260 (1998).
[CrossRef]

A. F. Milton, F. R. Barone, M. R. Kruer, “Influence of nonuniformity on infrared focal plane array performance,” Opt. Eng. 24, 855–862 (1985).
[CrossRef]

D. L. Perry, E. L. Dereniak, “Linear theory of nonuniformity correction in infrared staring sensors,” Opt. Eng. 32, 1853–1859 (1993).
[CrossRef]

Other

P. M. Narendra, N. A. Foss, “Shutterless fixed pattern noise correction for infrared imaging arrays,” in Technical Issues in Focal Plane Development, W. S. Chan, E. Krikorian, eds., Proc. SPIE282, 44–51 (1981).
[CrossRef]

P. M. Narendra, “Reference-free nonuniformity compensation for IR imaging arrays,” in Smart Sensors II, D. F. Barbe, ed., Proc. SPIE252, 10–17 (1980).
[CrossRef]

J. G. Harris, “Continuous-time calibration of VLSI sensors for gain and offset variations,” in Smart Focal Plane Arrays and Focal Plane Array Testing, M. Wigdor, M. A. Massie, eds., Proc. SPIE2474, 23–33 (1995).
[CrossRef]

J. G. Harris, Y. M. Chiang, “Nonuniformity correction using constant average statistics constraint: Analog and digital implementations,” in Infrared Technology and Applications XXIII, B. F. Andersen, M. Strojnik, eds., Proc. SPIE3061, 895–905 (1997).
[CrossRef]

S. N. Torres, M. M. Hayat, E. E. Armstrong, B. Yasuda, “A Kalman-filtering approach for nonuniformity correction in focal-plane array sensors,” in Infrared Imaging Systems: Design, Analysis, Modeling, and Testing XI, G. C. Holst and JCD Publishing, eds., Proc. SPIE4030, 196–203 (2000).
[CrossRef]

S. N. Torres, M. M. Hayat, “Compensation for gain and bias nonuniformity and drift in array detectors: A Kalman-filtering approach,” manuscript available from M. M. Hayat; hayat@eece.unm.edu.

G. C. Holst, CCD Arrays, Cameras and Displays (SPIE Optical Engineering Press, Bellingham, Wash., 1996).

E. E. Armstrong, M. M. Hayat, R. C. Hardie, S. N. Torres, B. Yasuda, “Nonuniformity correction for improved registration and high-resolution image reconstruction in IR imagery,” in Application of Digital Image Processing XXII, A. G. Tescher and Lockheed Martin Missions Systems, eds., Proc. SPIE3808, 150–161 (1999).
[CrossRef]

W. F. O’Neil, “Dithered scan detector compensation,” presented at the 1993 International Meeting of the Infrared Information Symposium Specialty Group on Passive Sensors, Ann Arbor, Mich., 1993).

W. F. O’Neil, “Experimental verification of dithered scan nonuniformity correction,” in Proceedings of the 1996 International Meeting of the Infrared Information Symposium Specialty Group on Passive Sensors (Infrared Information Analysis Center, Ann Arbor, Michigan, 1997) Vol. 1, pp. 329–339.

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Figures (11)

Fig. 1
Fig. 1

Block diagram of the proposed NUC algorithm.

Fig. 2
Fig. 2

Average absolute error in the shift estimates as a function of the standard deviation of the bias nonuniformity for a 200-frame sequence. A true shift of 0.5 pixel was used in the simulations.

Fig. 3
Fig. 3

Average absolute error in the shift estimates as a function of bias nonuniformity standard deviation and the size R of the smoothing mask.

Fig. 4
Fig. 4

(a) Frame 1 from the down-sampled image sequence before addition of bias nonuniformity, (b) frame 1 from the down-sampled image sequence after the addition of bias nonuniformity, (c) frame 1 from the image sequence corrected by use of the algebraic NUC algorithm, (d) frame 1 from the image sequence corrected by use of Harris’s (bias-only version) constant-statistics NUC technique.

Fig. 5
Fig. 5

Average absolute error in the estimation of bias nonuniformity as a function of the number of frame pairs used in generating the correction matrix. In this example, the bias nonuniformity has a zero mean and a standard deviation of 20.

Fig. 6
Fig. 6

Average absolute error of the bias estimates as a function of bias nonuniformity standard deviation and shift value.

Fig. 7
Fig. 7

Average absolute error in the computed bias values as a function of the standard deviations of gain and bias. The correction algorithm incorporated the true shift values.

Fig. 8
Fig. 8

Average absolute error in the computed bias values as a function of the standard deviations of gain and bias. The correction algorithm incorporated the estimated shift values.

Fig. 9
Fig. 9

(a) Frame 1 from infrared data set 1, (b) correction by Harris’s method, (c) correction by the algebraic algorithm (unrestricted shifts), (d) correction by the algebraic technique with the shifts restricted to the interval [0.5, 1.0].

Fig. 10
Fig. 10

(a) Nonuniformity correction map associated with the image of Fig. 9(c), (b) nonuniformity correction map associated with the image of Fig. 9(d).

Fig. 11
Fig. 11

(a) Frame 100 from infrared data set 2, (b) correction by the algebraic technique (restricted shifts), (c) correction by Harris’s constant-statistics algorithm.

Equations (21)

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

yn(i, j)=a(i, j)zn(i, j)+b(i, j),
yn(i, j)=zn(i, j)+b(i, j).
B=b(1, 1)b(1, 2)b(1, N)b(2, 1)b(2, 2)b(2, N)b(M, 1)b(M, 2)b(M, N).
yk+1(i+1, j)=αzk(i, j)+(1-α)zk(i+1, j)+b(i+1, j),0<α1.
ym+1(i, j+1)=βzm(i, j)+(1-β)zm(i, j+1)+b(i, j+1),0<β1.
yn(1, 1)=zn(1, 1)+b(1, 1),
yn(2, 1)=zn(2, 1)+b(2, 1).
yn+1(2, 1)=αzn(1, 1)+(1-α)zn(2, 1)+b(2, 1).
V˜B(2, 1)=1α[αyn(1, 1)+(1-α)yn(2, 1)-yn+1(2, 1)],
V˜B(2, 1)=b(1, 1)-b(2, 1).
yn(2, 1)+V˜B(2, 1)=zn(2, 1)+b(1, 1).
V˜B(i, j)=1α[αyn(i-1, j)+(1-α)yn(i, j)-yn+1(i, j)]=b(i-1, j)-b(i, j).
V˜B=000b(1, 1)-b(2, 1)b(1, 2)-b(2, 2)b(1, N)-b(2, N)b(2, 1)-b(3, 1)b(2, 2)-b(3, 2)b(2, N)-b(3, N)b(M, 1, 1)-b(M, 1)b(M-1, 2)-b(M, 2)b(M-1, )-b(M, N).
VB(i, j)=r=2iV˜B(r, j)=b(1, j)-b(i, j),
VB=000b(1, 1)-b(2, 1)b(1, 2)-b(2, 2)b(1, N)-b(2, N)b(1, 1)-b(3, 1)b(1, 2)-b(3, 2)b(1, N)-b(3, N)b(1, 1)-b(M, 1)b(1, 2)-b(M, 2)b(1, N)-b(M, N).
yk+VB=zk(1, 1)+b(1, 1)zk(1, 2)+b(1, 2)zk(1, N)+b(1, N)zk(2, 1)+b(1, 1)zk(2, 2)+b(1, 2)zk(2, N)+b(1, N)zk(M, 1)+b(1, 1)zk(M, 2)+b(1, 2)zk(M, N)+b(1, N).
B=b(1, 1)b(1, 2)b(1, N)b(1, 1)b(1, 2)b(1, N),
H˜(i, j)=1β[βym(i, j-1)+(1-β)ym(i, j)-ym+i(i, j)].
HB=0b(1, 1)-b(1, 2)b(1, 1)-b(1, 3)b(1, 1)-b(1, N)0b(2, 1)-b(2, 2)b(2, 1)-b(2, 3)b(2, 1)-b(2, N)0b(M, 1)-b(M, 2)b(M, 1)-b(M, 3)b(M, 1)-b(M, N).
HB=0b(1, 1)-b(1, 2)b(1, 1)-b(1, 3)b(1, 1)-b(1, N)0b(1, 1)-b(1, 2)b(1, 1)-b(1, 3)b(1, 1)-b(1, N)0b(1, 1)-b(1, 2)b(1, 1)-b(1, 3)b(1, 1)-b(1, N),
C=VB+HB=0b(1, 1)-b(1, 2)b(1, 1)-b(1, 3)b(1, 1)-b(1, N)b(1, 1)-b(2, 1)b(1, 1)-b(2, 2)b(1, 1)-b(2, 3)b(1, 1)-b(2, N)b(1, 1)-b(M, 1)b(1, 1)-b(M, 2)b(1, 1)-b(M, 3)b(1, 1)-b(M, N),

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