Abstract

Images acquired by airborne infrared search and track (IRST) systems are often characterized by nonuniform noise. In this paper, a scene-based nonuniformity correction method for infrared focal-plane arrays (FPAs) is proposed based on the constant statistics of the received radiation ratios of adjacent pixels. The gain of each pixel is computed recursively based on the ratios between adjacent pixels, which are estimated through a median operation. Then, an elaborate mathematical model describing the error propagation, derived from random noise and the recursive calculation procedure, is established. The proposed method maintains the characteristics of traditional methods in calibrating the whole electro-optics chain, in compensating for temporal drifts, and in not preserving the radiometric accuracy of the system. Moreover, the proposed method is robust since the frame number is the only variant, and is suitable for real-time applications owing to its low computational complexity and simplicity of implementation. The experimental results, on different scenes from a proof-of-concept point target detection system with a long-wave Sofradir FPA, demonstrate the compelling performance of the proposed method.

© 2017 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Staircase-scene-based nonuniformity correction in aerial point target detection systems

Lijun Huo, Dabiao Zhou, Dejiang Wang, Rang Liu, and Bin He
Appl. Opt. 55(25) 7149-7156 (2016)

Scene-based nonuniformity correction algorithm based on interframe registration

Chao Zuo, Qian Chen, Guohua Gu, and Xiubao Sui
J. Opt. Soc. Am. A 28(6) 1164-1176 (2011)

Scene-based nonuniformity correction with video sequences and registration

Russell C. Hardie, Majeed M. Hayat, Earnest Armstrong, and Brian Yasuda
Appl. Opt. 39(8) 1241-1250 (2000)

References

  • View by:
  • |
  • |
  • |

  1. L. Fortunato, A. Ondini, C. Quaranta, and C. Giunti, “SKYWARD: the next generation airborne infrared search and track,” Proc. SPIE 9819, 98190K (2016).
  2. A. Rogalski, Infrared Detectors (CRC, 2011), 2nd ed.
  3. G. A. Page, B. D. Carroll, A. E. Pratt, and P. N. Randall, “Long-range target detection algorithms for infrared search and track,” Proc. SPIE 3698, 48–57 (1999).
    [Crossref]
  4. D. L. Perry and E. L. Dereniak, “Linear theory of nonuniformity correction in infrared staring sensors,” Opt. Eng. 32, 1854–1859 (1993).
    [Crossref]
  5. Z. Sun, S. Chang, and W. Zhu, “Radiometric calibration method for large aperture infrared system with broad dynamic range,” Appl. Opt. 54, 4659–4666 (2015).
    [Crossref] [PubMed]
  6. P. W. Nugent, J. A. Shaw, and N. J. Pust, “Radiometric calibration of infrared imagers using an internal shutter as an equivalent external blackbody,” Opt. Eng. 53, 123106 (2014).
    [Crossref]
  7. L. Huo, D. Zhou, D. Wang, R. Liu, and B. He, “Staircase-scene-based nonuniformity correction in aerial point target detection systems,” Appl. Opt. 55, 7149–7156 (2016).
    [Crossref] [PubMed]
  8. M. M. Hayat, S. N. Torres, E. Armstrong, S. C. Cain, and B. Yasuda, “Statistical algorithm for nonuniformity correction in focal-plane arrays,” Appl. Opt. 38, 772–780 (1999).
    [Crossref]
  9. R. C. Hardie, F. Baxley, B. Brys, and P. Hytla, “Scene-based nonuniformity correction with reduced ghosting using a gated LMS algorithm,” Opt. Express 17, 14918–14933 (2009).
    [Crossref] [PubMed]
  10. C. Zhang and W. Zhao, “Scene-based nonuniformity correction using local constant statistics,” J. Opt. Soc. Am. A 25, 1444–1453 (2008).
    [Crossref]
  11. R. A. Leathers, T. V. Downes, and R. G. Priest, “Scene-based nonuniformity corrections for optical and SWIR pushbroom sensors,” Opt. Express 13, 5136–5150 (2005).
    [Crossref] [PubMed]
  12. L. Geng, Q. Chen, and W. Qian, “An adjacent differential statistics method for IRFPA nonuniformity correction,” IEEE Photon. J. 5, 6801615 (2013).
    [Crossref]
  13. L. Rui, Y. Yang, Z. Duan, and Y. Li, “Improved neural network based scene-adaptive nonuniformity correction method for infrared focal plane arrays,” Appl. Opt. 47, 4331–4335 (2008).
    [Crossref]
  14. Y. Cao and C.-L. Tisse, “Single-image-based solution for optics temperature-dependent nonuniformity correction in an uncooled long-wave infrared camera,” Opt. Lett. 39, 646–648 (2014).
    [Crossref] [PubMed]
  15. E. Vera, P. Meza, and S. Torres, “Total variation approach for adaptive nonuniformity correction in focal-plane arrays,” Opt. Lett. 36, 172–174 (2011).
    [Crossref] [PubMed]
  16. L. Liu and T. Zhang, “Optics temperature-dependent nonuniformity correction via ℓ0-regularized prior for airborne infrared imaging systems,” IEEE Photon. J. 7, 6803016 (2016).
  17. J. E. Pezoa, M. M. Hayat, S. N. Torres, and M. S. Rahman, “Multimodel Kalman filtering for adaptive nonuniformity correction in infrared sensors,” J. Opt. Soc. Am. A 23, 1282–1291 (2006).
    [Crossref]
  18. B. Yasuda, E. Armstrong, M. M. Hayat, and R. C. Hardie, “Scene-based nonuniformity correction with video sequences and registration,” Appl. Opt. 39, 1241–1250 (2000).
    [Crossref]
  19. C. Zuo, Q. Chen, G. Gu, and X. Sui, “Scene-based nonuniformity correction algorithm based on interframe registration,” J. Opt. Soc. Am. A 28, 1164–1176 (2011).
    [Crossref]
  20. J. Zeng, X. Sui, and H. Gao, “Adaptive image-registration-based nonuniformity correction algorithm with ghost artifacts eliminating for infrared focal plane arrays,” IEEE Photon. J. 7, 1–16 (2015).
  21. D. Wang, T. Zhang, and H. Kuang, “Relationship between the charge-coupled device signal-to-noise ratio and dynamic range with respect to the analog gain,” Appl. Opt. 51, 7103–7114 (2012).
    [Crossref] [PubMed]
  22. G. E. Healey and R. Kondepudy, “Radiometric CCD camera calibration and noise estimation,” IEEE Trans. Pattern Anal. Mach. Intell. 16, 267–276 (1994).
    [Crossref]
  23. A. Ferrero, J. Campos, and A. Pons, “Correction of photoresponse nonuniformity for matrix detectors based on prior compensation for their nonlinear behavior,” Appl. Opt. 45, 2422–2427 (2006).
    [Crossref] [PubMed]
  24. Texas Instruments, TMS320C645x DSP External Memory Interface (EMIF) User’s Guide (2005).
  25. M. Blum, R. W. Floyd, V. Pratt, R. L. Rivest, and R. E. Tarjan, “Time bounds for selection,” J. Comput. Syst. Sci. 7, 448–461 (1973).
    [Crossref]

2016 (3)

L. Fortunato, A. Ondini, C. Quaranta, and C. Giunti, “SKYWARD: the next generation airborne infrared search and track,” Proc. SPIE 9819, 98190K (2016).

L. Huo, D. Zhou, D. Wang, R. Liu, and B. He, “Staircase-scene-based nonuniformity correction in aerial point target detection systems,” Appl. Opt. 55, 7149–7156 (2016).
[Crossref] [PubMed]

L. Liu and T. Zhang, “Optics temperature-dependent nonuniformity correction via ℓ0-regularized prior for airborne infrared imaging systems,” IEEE Photon. J. 7, 6803016 (2016).

2015 (2)

J. Zeng, X. Sui, and H. Gao, “Adaptive image-registration-based nonuniformity correction algorithm with ghost artifacts eliminating for infrared focal plane arrays,” IEEE Photon. J. 7, 1–16 (2015).

Z. Sun, S. Chang, and W. Zhu, “Radiometric calibration method for large aperture infrared system with broad dynamic range,” Appl. Opt. 54, 4659–4666 (2015).
[Crossref] [PubMed]

2014 (2)

P. W. Nugent, J. A. Shaw, and N. J. Pust, “Radiometric calibration of infrared imagers using an internal shutter as an equivalent external blackbody,” Opt. Eng. 53, 123106 (2014).
[Crossref]

Y. Cao and C.-L. Tisse, “Single-image-based solution for optics temperature-dependent nonuniformity correction in an uncooled long-wave infrared camera,” Opt. Lett. 39, 646–648 (2014).
[Crossref] [PubMed]

2013 (1)

L. Geng, Q. Chen, and W. Qian, “An adjacent differential statistics method for IRFPA nonuniformity correction,” IEEE Photon. J. 5, 6801615 (2013).
[Crossref]

2012 (1)

2011 (2)

2009 (1)

2008 (2)

2006 (2)

2005 (1)

2000 (1)

1999 (2)

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

G. A. Page, B. D. Carroll, A. E. Pratt, and P. N. Randall, “Long-range target detection algorithms for infrared search and track,” Proc. SPIE 3698, 48–57 (1999).
[Crossref]

1994 (1)

G. E. Healey and R. Kondepudy, “Radiometric CCD camera calibration and noise estimation,” IEEE Trans. Pattern Anal. Mach. Intell. 16, 267–276 (1994).
[Crossref]

1993 (1)

D. L. Perry and E. L. Dereniak, “Linear theory of nonuniformity correction in infrared staring sensors,” Opt. Eng. 32, 1854–1859 (1993).
[Crossref]

1973 (1)

M. Blum, R. W. Floyd, V. Pratt, R. L. Rivest, and R. E. Tarjan, “Time bounds for selection,” J. Comput. Syst. Sci. 7, 448–461 (1973).
[Crossref]

Armstrong, E.

Baxley, F.

Blum, M.

M. Blum, R. W. Floyd, V. Pratt, R. L. Rivest, and R. E. Tarjan, “Time bounds for selection,” J. Comput. Syst. Sci. 7, 448–461 (1973).
[Crossref]

Brys, B.

Cain, S. C.

Campos, J.

Cao, Y.

Carroll, B. D.

G. A. Page, B. D. Carroll, A. E. Pratt, and P. N. Randall, “Long-range target detection algorithms for infrared search and track,” Proc. SPIE 3698, 48–57 (1999).
[Crossref]

Chang, S.

Chen, Q.

L. Geng, Q. Chen, and W. Qian, “An adjacent differential statistics method for IRFPA nonuniformity correction,” IEEE Photon. J. 5, 6801615 (2013).
[Crossref]

C. Zuo, Q. Chen, G. Gu, and X. Sui, “Scene-based nonuniformity correction algorithm based on interframe registration,” J. Opt. Soc. Am. A 28, 1164–1176 (2011).
[Crossref]

Dereniak, E. L.

D. L. Perry and E. L. Dereniak, “Linear theory of nonuniformity correction in infrared staring sensors,” Opt. Eng. 32, 1854–1859 (1993).
[Crossref]

Downes, T. V.

Duan, Z.

Ferrero, A.

Floyd, R. W.

M. Blum, R. W. Floyd, V. Pratt, R. L. Rivest, and R. E. Tarjan, “Time bounds for selection,” J. Comput. Syst. Sci. 7, 448–461 (1973).
[Crossref]

Fortunato, L.

L. Fortunato, A. Ondini, C. Quaranta, and C. Giunti, “SKYWARD: the next generation airborne infrared search and track,” Proc. SPIE 9819, 98190K (2016).

Gao, H.

J. Zeng, X. Sui, and H. Gao, “Adaptive image-registration-based nonuniformity correction algorithm with ghost artifacts eliminating for infrared focal plane arrays,” IEEE Photon. J. 7, 1–16 (2015).

Geng, L.

L. Geng, Q. Chen, and W. Qian, “An adjacent differential statistics method for IRFPA nonuniformity correction,” IEEE Photon. J. 5, 6801615 (2013).
[Crossref]

Giunti, C.

L. Fortunato, A. Ondini, C. Quaranta, and C. Giunti, “SKYWARD: the next generation airborne infrared search and track,” Proc. SPIE 9819, 98190K (2016).

Gu, G.

Hardie, R. C.

Hayat, M. M.

He, B.

Healey, G. E.

G. E. Healey and R. Kondepudy, “Radiometric CCD camera calibration and noise estimation,” IEEE Trans. Pattern Anal. Mach. Intell. 16, 267–276 (1994).
[Crossref]

Huo, L.

Hytla, P.

Kondepudy, R.

G. E. Healey and R. Kondepudy, “Radiometric CCD camera calibration and noise estimation,” IEEE Trans. Pattern Anal. Mach. Intell. 16, 267–276 (1994).
[Crossref]

Kuang, H.

Leathers, R. A.

Li, Y.

Liu, L.

L. Liu and T. Zhang, “Optics temperature-dependent nonuniformity correction via ℓ0-regularized prior for airborne infrared imaging systems,” IEEE Photon. J. 7, 6803016 (2016).

Liu, R.

Meza, P.

Nugent, P. W.

P. W. Nugent, J. A. Shaw, and N. J. Pust, “Radiometric calibration of infrared imagers using an internal shutter as an equivalent external blackbody,” Opt. Eng. 53, 123106 (2014).
[Crossref]

Ondini, A.

L. Fortunato, A. Ondini, C. Quaranta, and C. Giunti, “SKYWARD: the next generation airborne infrared search and track,” Proc. SPIE 9819, 98190K (2016).

Page, G. A.

G. A. Page, B. D. Carroll, A. E. Pratt, and P. N. Randall, “Long-range target detection algorithms for infrared search and track,” Proc. SPIE 3698, 48–57 (1999).
[Crossref]

Perry, D. L.

D. L. Perry and E. L. Dereniak, “Linear theory of nonuniformity correction in infrared staring sensors,” Opt. Eng. 32, 1854–1859 (1993).
[Crossref]

Pezoa, J. E.

Pons, A.

Pratt, A. E.

G. A. Page, B. D. Carroll, A. E. Pratt, and P. N. Randall, “Long-range target detection algorithms for infrared search and track,” Proc. SPIE 3698, 48–57 (1999).
[Crossref]

Pratt, V.

M. Blum, R. W. Floyd, V. Pratt, R. L. Rivest, and R. E. Tarjan, “Time bounds for selection,” J. Comput. Syst. Sci. 7, 448–461 (1973).
[Crossref]

Priest, R. G.

Pust, N. J.

P. W. Nugent, J. A. Shaw, and N. J. Pust, “Radiometric calibration of infrared imagers using an internal shutter as an equivalent external blackbody,” Opt. Eng. 53, 123106 (2014).
[Crossref]

Qian, W.

L. Geng, Q. Chen, and W. Qian, “An adjacent differential statistics method for IRFPA nonuniformity correction,” IEEE Photon. J. 5, 6801615 (2013).
[Crossref]

Quaranta, C.

L. Fortunato, A. Ondini, C. Quaranta, and C. Giunti, “SKYWARD: the next generation airborne infrared search and track,” Proc. SPIE 9819, 98190K (2016).

Rahman, M. S.

Randall, P. N.

G. A. Page, B. D. Carroll, A. E. Pratt, and P. N. Randall, “Long-range target detection algorithms for infrared search and track,” Proc. SPIE 3698, 48–57 (1999).
[Crossref]

Rivest, R. L.

M. Blum, R. W. Floyd, V. Pratt, R. L. Rivest, and R. E. Tarjan, “Time bounds for selection,” J. Comput. Syst. Sci. 7, 448–461 (1973).
[Crossref]

Rogalski, A.

A. Rogalski, Infrared Detectors (CRC, 2011), 2nd ed.

Rui, L.

Shaw, J. A.

P. W. Nugent, J. A. Shaw, and N. J. Pust, “Radiometric calibration of infrared imagers using an internal shutter as an equivalent external blackbody,” Opt. Eng. 53, 123106 (2014).
[Crossref]

Sui, X.

J. Zeng, X. Sui, and H. Gao, “Adaptive image-registration-based nonuniformity correction algorithm with ghost artifacts eliminating for infrared focal plane arrays,” IEEE Photon. J. 7, 1–16 (2015).

C. Zuo, Q. Chen, G. Gu, and X. Sui, “Scene-based nonuniformity correction algorithm based on interframe registration,” J. Opt. Soc. Am. A 28, 1164–1176 (2011).
[Crossref]

Sun, Z.

Tarjan, R. E.

M. Blum, R. W. Floyd, V. Pratt, R. L. Rivest, and R. E. Tarjan, “Time bounds for selection,” J. Comput. Syst. Sci. 7, 448–461 (1973).
[Crossref]

Tisse, C.-L.

Torres, S.

Torres, S. N.

Vera, E.

Wang, D.

Yang, Y.

Yasuda, B.

Zeng, J.

J. Zeng, X. Sui, and H. Gao, “Adaptive image-registration-based nonuniformity correction algorithm with ghost artifacts eliminating for infrared focal plane arrays,” IEEE Photon. J. 7, 1–16 (2015).

Zhang, C.

Zhang, T.

L. Liu and T. Zhang, “Optics temperature-dependent nonuniformity correction via ℓ0-regularized prior for airborne infrared imaging systems,” IEEE Photon. J. 7, 6803016 (2016).

D. Wang, T. Zhang, and H. Kuang, “Relationship between the charge-coupled device signal-to-noise ratio and dynamic range with respect to the analog gain,” Appl. Opt. 51, 7103–7114 (2012).
[Crossref] [PubMed]

Zhao, W.

Zhou, D.

Zhu, W.

Zuo, C.

Appl. Opt. (7)

IEEE Photon. J. (3)

L. Geng, Q. Chen, and W. Qian, “An adjacent differential statistics method for IRFPA nonuniformity correction,” IEEE Photon. J. 5, 6801615 (2013).
[Crossref]

L. Liu and T. Zhang, “Optics temperature-dependent nonuniformity correction via ℓ0-regularized prior for airborne infrared imaging systems,” IEEE Photon. J. 7, 6803016 (2016).

J. Zeng, X. Sui, and H. Gao, “Adaptive image-registration-based nonuniformity correction algorithm with ghost artifacts eliminating for infrared focal plane arrays,” IEEE Photon. J. 7, 1–16 (2015).

IEEE Trans. Pattern Anal. Mach. Intell. (1)

G. E. Healey and R. Kondepudy, “Radiometric CCD camera calibration and noise estimation,” IEEE Trans. Pattern Anal. Mach. Intell. 16, 267–276 (1994).
[Crossref]

J. Comput. Syst. Sci. (1)

M. Blum, R. W. Floyd, V. Pratt, R. L. Rivest, and R. E. Tarjan, “Time bounds for selection,” J. Comput. Syst. Sci. 7, 448–461 (1973).
[Crossref]

J. Opt. Soc. Am. A (3)

Opt. Eng. (2)

D. L. Perry and E. L. Dereniak, “Linear theory of nonuniformity correction in infrared staring sensors,” Opt. Eng. 32, 1854–1859 (1993).
[Crossref]

P. W. Nugent, J. A. Shaw, and N. J. Pust, “Radiometric calibration of infrared imagers using an internal shutter as an equivalent external blackbody,” Opt. Eng. 53, 123106 (2014).
[Crossref]

Opt. Express (2)

Opt. Lett. (2)

Proc. SPIE (2)

L. Fortunato, A. Ondini, C. Quaranta, and C. Giunti, “SKYWARD: the next generation airborne infrared search and track,” Proc. SPIE 9819, 98190K (2016).

G. A. Page, B. D. Carroll, A. E. Pratt, and P. N. Randall, “Long-range target detection algorithms for infrared search and track,” Proc. SPIE 3698, 48–57 (1999).
[Crossref]

Other (2)

A. Rogalski, Infrared Detectors (CRC, 2011), 2nd ed.

Texas Instruments, TMS320C645x DSP External Memory Interface (EMIF) User’s Guide (2005).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (17)

Fig. 1
Fig. 1 Relevance with four adjacent pixels through (a) Eq. (4) and (b) Eq. (5).
Fig. 2
Fig. 2 Diagram of the CSAR method.
Fig. 3
Fig. 3 Error propagation when (a) αi,j = 1.00025 and (b) αi,j = 1.00050.
Fig. 4
Fig. 4 Experimental equipment: (a) schematic diagram, (b) practical calibration platform.
Fig. 5
Fig. 5 Comparison of global STD with different models and the CSAR method.
Fig. 6
Fig. 6 Random noise and global STD with the proposed method as a function of gray value.
Fig. 7
Fig. 7 Proof-of-concept IRST system. On the right was the screen shot of the GNS viewer, where a Boeing B738 (airliner 1) was detected.
Fig. 8
Fig. 8 Comparison of the coefficients. (a) k in TP. (b) b in TP. (c) b in ADS. (d) k in CSAR.
Fig. 9
Fig. 9 Comparison of the corrected images. (a) Raw image. (b) BBNUC. (c) ADS. (d) CSAR. The smaller images below the main images are magnifications of the content within the red squares.
Fig. 10
Fig. 10 Comparison of (a) PDF and (b) CDF of the local STD.
Fig. 11
Fig. 11 Local STD with the percentage fixed to 50% in the CDF.
Fig. 12
Fig. 12 The cropped target of a Boeing B738 from 59.37 km away. (a) The raw image, and the images corrected by (b) BBNUC, (c) ADS, and (d) CSAR.
Fig. 13
Fig. 13 Comparison of the corrected images. (a) Raw image. (b) BBNUC. (c) ADS. (d) Proposed.
Fig. 14
Fig. 14 Plot of (a) the ratio and (b) the sorted ratio for two typical pixels.
Fig. 15
Fig. 15 Roughness (×10−2) as a function of frame index.
Fig. 16
Fig. 16 Corrected images when (a) F = 100 (b) F = 300 (c) F = 500.
Fig. 17
Fig. 17 Ratios for the pixel (194, 201) with different F.

Tables (2)

Tables Icon

Table 1 Detailed features of the tested IR systems.

Tables Icon

Table 2 Global STD of the images calibrated with different models and the CSAR method.

Equations (19)

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

J i , j ( f ) = k i , j I i , j ( f ) + b i , j + n i , j ,
J i , j ( f ) = k i , j I i , j ( f ) .
mid F [ J i , j ( f ) J i 1 ,   j ( f ) J i + 1 , j ( f ) J i , j 1 ( f ) J i , j + 1 ( f ) 4 ] = 1 .
k i , j k i 1 , j k i + 1 , j k i , j 1 k i , j + 1 4 mid F [ I i , j ( f ) I i 1 , j ( f ) I i + 1 , j ( f ) I i , j 1 ( f ) I i , j + 1 ( f ) 4 ] = 1 .
k i , j k i 1 , j k i , j 1 mid F [ I i , j ( f ) I i 1 , j ( f ) I i , j 1 ( f ) ] = 1 .
r i , j ( f ) = I i , j ( f ) I i 1 , j ( f ) I i , j 1 ( f ) .
k i , j = k i 1 , j k i , j 1 r ˜ i , j .
{ k i , j k i , j k i , j 1 mid F [ I i , j ( f ) I i , j ( f ) I i , j 1 ( f ) ] = 1 i = 1 , j [ 2 , N ] k i , j k i , j k i 1 , j mid F [ I i , j ( f ) I i , j ( f ) I i 1 , j ( f ) ] = 1 i [ 2 , M ] , j = 1 .
r i , j ( f ) = { I i , j ( f ) I i , j 1 ( f ) i = 1 , j [ 2 , N ] I i , j ( f ) I i 1 , j ( f ) i [ 2 , M ] , j = 1 .
k i , j = { k i , j 1 r ˜ i , j 2 i = 1 , j [ 2 , N ] k i 1 , j r ˜ i , j 2 i [ 2 , M ] , j = 1 .
g Δ i , Δ j = { α i , j 2 Δ j Δ i = 0 α i , j 2 ( Δ i + Δ j ) s = 1 Δ i s + Δ i s Δ i [ 1 , M i ] .
E Δ i , Δ j = J i + Δ i , j + Δ j ( f ) J i + Δ i , j + Δ j ( f ) = ( g Δ i , Δ j 1 ) J i + Δ i , j + Δ j ( f ) .
ρ ( J ) = h 1 J 1 + h 2 J 1 J 1 ,
J i , j ( f ) = I i , j ( f ) + b i , j .
I ^ i , j ( f ) = I i , j ( f ) + Δ I i , j ( f ) .
{ J i , j ( f ) = I ¯ ( f ) + Δ I i , j ( f ) J i , j ( f ) = I ¯ ( f ) + k i , j Δ I i , j ( f ) J i , j ( f ) = I ¯ ( f ) + k i , j Δ I i , j ( f ) .
P Δ i , Δ j = { 2 ( 1 + Δ j ) 1 + Δ j 1 Δ i = 1 2 ( 2 + Δ j ) ( 1 + Δ j ) ( 2 + Δ j ) 1 × 2 Δ i = 2 2 ( 3 + Δ j ) ( 1 + Δ j ) ( 2 + Δ j ) ( 3 + Δ j ) 1 × 2 × 3 Δ i = 3 2 ( Δ i + Δ j ) ( 1 + Δ j ) ( 2 + Δ j ) ( Δ i + Δ j ) 1 × 2 × × Δ i Δ i = Δ i .
P Δ i , Δ j = { 2 Δ j Δ i = 0 2 ( Δ i + Δ j ) s = 1 Δ i s + Δ i s Δ i [ 1 , M i ] .
P Δ j , Δ i = 2 ( Δ i + Δ j ) ( 1 + Δ i ) ( 2 + Δ i ) ( Δ j + Δ i ) 1 × 2 × × Δ j = 2 ( Δ i + Δ j ) [ ( 1 + Δ i ) ( 2 + Δ i ) Δ j ] [ ( 1 + Δ j ) ( 2 + Δ j ) ( Δ j + Δ i ) ] [ 1 × 2 × × Δ i ] [ ( 1 + Δ i ) ( 2 + Δ i ) Δ j ] . = 2 ( Δ i + Δ j ) ( 1 + Δ j ) ( 2 + Δ j ) ( Δ i + Δ j ) 1 × 2 × × Δ i

Metrics