Abstract

A novel statistical approach is undertaken for the adaptive estimation of the gain and bias nonuniformity in infrared focal-plane array sensors from scene data. The gain and the bias of each detector are regarded as random state variables modeled by a discrete-time Gauss–Markov process. The proposed Gauss–Markov framework provides a mechanism for capturing the slow and random drift in the fixed-pattern noise as the operational conditions of the sensor vary in time. With a temporal stochastic model for each detector’s gain and bias at hand, a Kalman filter is derived that uses scene data, comprising the detector’s readout values sampled over a short period of time, to optimally update the detector’s gain and bias estimates as these parameters drift. The proposed technique relies on a certain spatiotemporal diversity condition in the data, which is satisfied when all detectors see approximately the same range of temperatures within the periods between successive estimation epochs. The performance of the proposed technique is thoroughly studied, and its utility in mitigating fixed-pattern noise is demonstrated with both real infrared and simulated imagery.

© 2003 Optical Society of America

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References

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  1. G. C. Holst, CCD Arrays, Cameras and Displays (SPIE Optical Engineering Press, Bellingham, Wash., 1996).
  2. D. L. Perry, E. L. Dereniak, “Linear theory of nonuni-formity 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 Integrated Circ. Signal Process. 12, 231–238 (1997).
  8. D. A. Scribner, K. A. Sarkay, J. T. Caldfield, M. R. Kruer, G. Katz, C. J. Gridley, “Nonuniformity correction for staring focal plane arrays using scene-based techniques,” in Infrared Detectors and Focal Plane Arrays, E. L. Dereniak, R. E. Sampson, eds., Proc. SPIE1308, 24–233 (1990).
  9. D. Scribner, K. Sarkady, M. Kruer, J. Caldfield, J. Hunt, M. Colbert, M. Descour, “Adaptive nonuniformity correction for IR focal plane arrays using neural networks,” in Infrared Sensors: Detectors, Electronics, and Signal Processing, T. S. Jayadev, ed., Proc. SPIE1541, 100–109 (1991).
    [CrossRef]
  10. D. Scribner, K. Sarkady, M. Kruer, J. Caldfield, J. Hunt, M. Colbert, M. Descour, “Adaptive retina-like preprocessing for imaging detector arrays,” in Proceedings of the IEEE International Conference on Neural Networks (Institute of Electrical and Electronics Engineers, New York, 1993), pp. 1955–1960.
  11. W. F. O’Neil, “Experimental verification of dithered scan non-uniformity correction,” in Proceedings of the 1996 International Meeting of the Infrared Information Symposium Specialty Group on Passive Sensors (Infrared Information Analysis Center, Ann Arbor, Mich., 1997), Vol. 1, pp. 329–339.
  12. 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]
  13. K. C. Hepfer, S. R. Horman, B. Horsch, “Method and device for improved IR detection with compensations for individual detector response,” U.S. patent5,276,319 (1994).
  14. M. M. Hayat, S. N. Torres, E. E. Armstrong, S. C. Cain, B. Yasuda, “Statistical algorithm for non-uniformity correction in focal-plane arrays,” Appl. Opt. 38, 772–780 (1999).
    [CrossRef]
  15. B. M. Ratliff, M. M. Hayat, R. C. Hardie, “An algebraic algorithm for nonuniformity correction in focal plane arrays,” J. Opt. Soc. Am. A 19, 1737–1747 (2002).
    [CrossRef]
  16. R. E. Kalman, R. S. Bucy, “New results in linear filtering and prediction: theory,” ASME J. Basic Eng. 83, 95–107 (1961).
    [CrossRef]
  17. H. V. Poor, Introduction to Signal Detection and Estimation (Springer-Verlag, New York, 1988).
  18. G. Minkler, J. Minkler, Theory and Applications of Kalman Filtering (Magellan, Palm Bay, Fla., 1993).
  19. R. C. Gonzalez, R. E. Woods, Digital Image Processing (Addison-Wesley, New York, 1993).
  20. M. Schulz, L. Caldwell, “Nonuniformity correction and correctability of infrared focal plane arrays,” Infrared Phys. Technol. 36, 763–777 (1995).
    [CrossRef]

2002 (1)

2000 (1)

1999 (1)

1997 (1)

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

1995 (1)

M. Schulz, L. Caldwell, “Nonuniformity correction and correctability of infrared focal plane arrays,” Infrared Phys. Technol. 36, 763–777 (1995).
[CrossRef]

1993 (1)

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

1961 (1)

R. E. Kalman, R. S. Bucy, “New results in linear filtering and prediction: theory,” ASME J. Basic Eng. 83, 95–107 (1961).
[CrossRef]

Armstrong, E. E.

Bucy, R. S.

R. E. Kalman, R. S. Bucy, “New results in linear filtering and prediction: theory,” ASME J. Basic Eng. 83, 95–107 (1961).
[CrossRef]

Cain, S. C.

Caldfield, J.

D. Scribner, K. Sarkady, M. Kruer, J. Caldfield, J. Hunt, M. Colbert, M. Descour, “Adaptive retina-like preprocessing for imaging detector arrays,” in Proceedings of the IEEE International Conference on Neural Networks (Institute of Electrical and Electronics Engineers, New York, 1993), pp. 1955–1960.

D. Scribner, K. Sarkady, M. Kruer, J. Caldfield, J. Hunt, M. Colbert, M. Descour, “Adaptive nonuniformity correction for IR focal plane arrays using neural networks,” in Infrared Sensors: Detectors, Electronics, and Signal Processing, T. S. Jayadev, ed., Proc. SPIE1541, 100–109 (1991).
[CrossRef]

Caldfield, J. T.

D. A. Scribner, K. A. Sarkay, J. T. Caldfield, M. R. Kruer, G. Katz, C. J. Gridley, “Nonuniformity correction for staring focal plane arrays using scene-based techniques,” in Infrared Detectors and Focal Plane Arrays, E. L. Dereniak, R. E. Sampson, eds., Proc. SPIE1308, 24–233 (1990).

Caldwell, L.

M. Schulz, L. Caldwell, “Nonuniformity correction and correctability of infrared focal plane arrays,” Infrared Phys. Technol. 36, 763–777 (1995).
[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 Integrated Circ. Signal Process. 12, 231–238 (1997).

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]

Colbert, M.

D. Scribner, K. Sarkady, M. Kruer, J. Caldfield, J. Hunt, M. Colbert, M. Descour, “Adaptive retina-like preprocessing for imaging detector arrays,” in Proceedings of the IEEE International Conference on Neural Networks (Institute of Electrical and Electronics Engineers, New York, 1993), pp. 1955–1960.

D. Scribner, K. Sarkady, M. Kruer, J. Caldfield, J. Hunt, M. Colbert, M. Descour, “Adaptive nonuniformity correction for IR focal plane arrays using neural networks,” in Infrared Sensors: Detectors, Electronics, and Signal Processing, T. S. Jayadev, ed., Proc. SPIE1541, 100–109 (1991).
[CrossRef]

Dereniak, E. L.

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

Descour, M.

D. Scribner, K. Sarkady, M. Kruer, J. Caldfield, J. Hunt, M. Colbert, M. Descour, “Adaptive retina-like preprocessing for imaging detector arrays,” in Proceedings of the IEEE International Conference on Neural Networks (Institute of Electrical and Electronics Engineers, New York, 1993), pp. 1955–1960.

D. Scribner, K. Sarkady, M. Kruer, J. Caldfield, J. Hunt, M. Colbert, M. Descour, “Adaptive nonuniformity correction for IR focal plane arrays using neural networks,” in Infrared Sensors: Detectors, Electronics, and Signal Processing, T. S. Jayadev, ed., Proc. SPIE1541, 100–109 (1991).
[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]

Gonzalez, R. C.

R. C. Gonzalez, R. E. Woods, Digital Image Processing (Addison-Wesley, New York, 1993).

Gridley, C. J.

D. A. Scribner, K. A. Sarkay, J. T. Caldfield, M. R. Kruer, G. Katz, C. J. Gridley, “Nonuniformity correction for staring focal plane arrays using scene-based techniques,” in Infrared Detectors and Focal Plane Arrays, E. L. Dereniak, R. E. Sampson, eds., Proc. SPIE1308, 24–233 (1990).

Hardie, R. C.

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 Integrated Circ. Signal Process. 12, 231–238 (1997).

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]

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]

Hayat, M. M.

Hepfer, K. C.

K. C. Hepfer, S. R. Horman, B. Horsch, “Method and device for improved IR detection with compensations for individual detector response,” U.S. patent5,276,319 (1994).

Holst, G. C.

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

Horman, S. R.

K. C. Hepfer, S. R. Horman, B. Horsch, “Method and device for improved IR detection with compensations for individual detector response,” U.S. patent5,276,319 (1994).

Horsch, B.

K. C. Hepfer, S. R. Horman, B. Horsch, “Method and device for improved IR detection with compensations for individual detector response,” U.S. patent5,276,319 (1994).

Hunt, J.

D. Scribner, K. Sarkady, M. Kruer, J. Caldfield, J. Hunt, M. Colbert, M. Descour, “Adaptive nonuniformity correction for IR focal plane arrays using neural networks,” in Infrared Sensors: Detectors, Electronics, and Signal Processing, T. S. Jayadev, ed., Proc. SPIE1541, 100–109 (1991).
[CrossRef]

D. Scribner, K. Sarkady, M. Kruer, J. Caldfield, J. Hunt, M. Colbert, M. Descour, “Adaptive retina-like preprocessing for imaging detector arrays,” in Proceedings of the IEEE International Conference on Neural Networks (Institute of Electrical and Electronics Engineers, New York, 1993), pp. 1955–1960.

Kalman, R. E.

R. E. Kalman, R. S. Bucy, “New results in linear filtering and prediction: theory,” ASME J. Basic Eng. 83, 95–107 (1961).
[CrossRef]

Katz, G.

D. A. Scribner, K. A. Sarkay, J. T. Caldfield, M. R. Kruer, G. Katz, C. J. Gridley, “Nonuniformity correction for staring focal plane arrays using scene-based techniques,” in Infrared Detectors and Focal Plane Arrays, E. L. Dereniak, R. E. Sampson, eds., Proc. SPIE1308, 24–233 (1990).

Kruer, M.

D. Scribner, K. Sarkady, M. Kruer, J. Caldfield, J. Hunt, M. Colbert, M. Descour, “Adaptive retina-like preprocessing for imaging detector arrays,” in Proceedings of the IEEE International Conference on Neural Networks (Institute of Electrical and Electronics Engineers, New York, 1993), pp. 1955–1960.

D. Scribner, K. Sarkady, M. Kruer, J. Caldfield, J. Hunt, M. Colbert, M. Descour, “Adaptive nonuniformity correction for IR focal plane arrays using neural networks,” in Infrared Sensors: Detectors, Electronics, and Signal Processing, T. S. Jayadev, ed., Proc. SPIE1541, 100–109 (1991).
[CrossRef]

Kruer, M. R.

D. A. Scribner, K. A. Sarkay, J. T. Caldfield, M. R. Kruer, G. Katz, C. J. Gridley, “Nonuniformity correction for staring focal plane arrays using scene-based techniques,” in Infrared Detectors and Focal Plane Arrays, E. L. Dereniak, R. E. Sampson, eds., Proc. SPIE1308, 24–233 (1990).

Minkler, G.

G. Minkler, J. Minkler, Theory and Applications of Kalman Filtering (Magellan, Palm Bay, Fla., 1993).

Minkler, J.

G. Minkler, J. Minkler, Theory and Applications of Kalman Filtering (Magellan, Palm Bay, Fla., 1993).

Narendra, P. M.

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]

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]

O’Neil, W. F.

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

Perry, D. L.

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

Poor, H. V.

H. V. Poor, Introduction to Signal Detection and Estimation (Springer-Verlag, New York, 1988).

Ratliff, B. M.

Sarkady, K.

D. Scribner, K. Sarkady, M. Kruer, J. Caldfield, J. Hunt, M. Colbert, M. Descour, “Adaptive retina-like preprocessing for imaging detector arrays,” in Proceedings of the IEEE International Conference on Neural Networks (Institute of Electrical and Electronics Engineers, New York, 1993), pp. 1955–1960.

D. Scribner, K. Sarkady, M. Kruer, J. Caldfield, J. Hunt, M. Colbert, M. Descour, “Adaptive nonuniformity correction for IR focal plane arrays using neural networks,” in Infrared Sensors: Detectors, Electronics, and Signal Processing, T. S. Jayadev, ed., Proc. SPIE1541, 100–109 (1991).
[CrossRef]

Sarkay, K. A.

D. A. Scribner, K. A. Sarkay, J. T. Caldfield, M. R. Kruer, G. Katz, C. J. Gridley, “Nonuniformity correction for staring focal plane arrays using scene-based techniques,” in Infrared Detectors and Focal Plane Arrays, E. L. Dereniak, R. E. Sampson, eds., Proc. SPIE1308, 24–233 (1990).

Schulz, M.

M. Schulz, L. Caldwell, “Nonuniformity correction and correctability of infrared focal plane arrays,” Infrared Phys. Technol. 36, 763–777 (1995).
[CrossRef]

Scribner, D.

D. Scribner, K. Sarkady, M. Kruer, J. Caldfield, J. Hunt, M. Colbert, M. Descour, “Adaptive nonuniformity correction for IR focal plane arrays using neural networks,” in Infrared Sensors: Detectors, Electronics, and Signal Processing, T. S. Jayadev, ed., Proc. SPIE1541, 100–109 (1991).
[CrossRef]

D. Scribner, K. Sarkady, M. Kruer, J. Caldfield, J. Hunt, M. Colbert, M. Descour, “Adaptive retina-like preprocessing for imaging detector arrays,” in Proceedings of the IEEE International Conference on Neural Networks (Institute of Electrical and Electronics Engineers, New York, 1993), pp. 1955–1960.

Scribner, D. A.

D. A. Scribner, K. A. Sarkay, J. T. Caldfield, M. R. Kruer, G. Katz, C. J. Gridley, “Nonuniformity correction for staring focal plane arrays using scene-based techniques,” in Infrared Detectors and Focal Plane Arrays, E. L. Dereniak, R. E. Sampson, eds., Proc. SPIE1308, 24–233 (1990).

Torres, S. N.

Woods, R. E.

R. C. Gonzalez, R. E. Woods, Digital Image Processing (Addison-Wesley, New York, 1993).

Yasuda, B.

Appl. Opt. (2)

ASME J. Basic Eng. (1)

R. E. Kalman, R. S. Bucy, “New results in linear filtering and prediction: theory,” ASME J. Basic Eng. 83, 95–107 (1961).
[CrossRef]

Infrared Phys. Technol. (1)

M. Schulz, L. Caldwell, “Nonuniformity correction and correctability of infrared focal plane arrays,” Infrared Phys. Technol. 36, 763–777 (1995).
[CrossRef]

J. Analog Integrated Circ. Signal Process. (1)

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

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

Opt. Eng. (1)

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

Other (13)

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]

D. A. Scribner, K. A. Sarkay, J. T. Caldfield, M. R. Kruer, G. Katz, C. J. Gridley, “Nonuniformity correction for staring focal plane arrays using scene-based techniques,” in Infrared Detectors and Focal Plane Arrays, E. L. Dereniak, R. E. Sampson, eds., Proc. SPIE1308, 24–233 (1990).

D. Scribner, K. Sarkady, M. Kruer, J. Caldfield, J. Hunt, M. Colbert, M. Descour, “Adaptive nonuniformity correction for IR focal plane arrays using neural networks,” in Infrared Sensors: Detectors, Electronics, and Signal Processing, T. S. Jayadev, ed., Proc. SPIE1541, 100–109 (1991).
[CrossRef]

D. Scribner, K. Sarkady, M. Kruer, J. Caldfield, J. Hunt, M. Colbert, M. Descour, “Adaptive retina-like preprocessing for imaging detector arrays,” in Proceedings of the IEEE International Conference on Neural Networks (Institute of Electrical and Electronics Engineers, New York, 1993), pp. 1955–1960.

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

K. C. Hepfer, S. R. Horman, B. Horsch, “Method and device for improved IR detection with compensations for individual detector response,” U.S. patent5,276,319 (1994).

H. V. Poor, Introduction to Signal Detection and Estimation (Springer-Verlag, New York, 1988).

G. Minkler, J. Minkler, Theory and Applications of Kalman Filtering (Magellan, Palm Bay, Fla., 1993).

R. C. Gonzalez, R. E. Woods, Digital Image Processing (Addison-Wesley, New York, 1993).

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

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Figures (15)

Fig. 1
Fig. 1

True 8-bit image from the fifth data set (k=5).

Fig. 2
Fig. 2

Image of Fig. 1 corrupted with simulated nonuniformity. The nonuniformity is generated with standard deviations for the gain and the offset of 0.15 and 5, respectively.

Fig. 3
Fig. 3

Corrected version of the image in Fig. 2.

Fig. 4
Fig. 4

Image of Fig. 1 with a high level of simulated bias nonuniformity. The gain and offset standard deviations are 0.01 and 100, respectively.

Fig. 5
Fig. 5

Corrected version of the image in Fig. 4.

Fig. 6
Fig. 6

Nonuniformity correction (NUC) with 50 consecutive frames. Note the artifacts at the bottom of the image, which result from the lack of statistical gray-value diversity within the 50 frames. This lack of diversity violates the constant-range assumption.

Fig. 7
Fig. 7

NUC with 150 consecutive frames. Note the improvement in comparison with Fig. 6; however, some residual artifacts remain, again as a result of a minor violation of the constant-range assumption.

Fig. 8
Fig. 8

NUC with 1000 consecutive frames. Note the improvement in comparison with Fig. 7.

Fig. 9
Fig. 9

Root mean square error (RMSE) as a function of the error in the drift parameters α and β. The symbols* and + correspond to the case with 150 and 300 consecutive frames, respectively.

Fig. 10
Fig. 10

Raw frame from the second real infrared (IR) data set (k=2).

Fig. 11
Fig. 11

Corrected version of the image in Fig. 10 using 300 sampled frames. The drift parameters are assumed as α=β=0.95.

Fig. 12
Fig. 12

Example of a raw frame from the second real IR data set (k=2). The dots at the bottom of the image correspond to branches of trees.

Fig. 13
Fig. 13

Corrected version of the image in Fig. 12 using 300 consecutive frames. The drift parameters are assumed as α=β=0.95, which corresponds to weak drift.

Fig. 14
Fig. 14

Corrected version of the image in Fig. 12 using 300 consecutive frames. The drift parameters are assumed as α=β=0.05, which corresponds to strong drift.

Fig. 15
Fig. 15

Corrected version of the image in Fig. 10 using the gain and the bias corresponding to data set 1. Note that less NUC is achieved in comparison with that in Fig. 11 because of the drift in the gain and the bias. Also note that there is no compensation for the dark spots (dead pixels) that appear in Fig. 10.

Tables (4)

Tables Icon

Table 1 Empirical Mean Square Error (MSE) for the Gain and Bias Estimates for Low and High Drift Conditions

Tables Icon

Table 2 Root Mean Square Error (RMSE) of a Corrected Image for Several Numbers of Sampled Frames per Block

Tables Icon

Table 3 Correctability Parameter c as a Function of the Temporal-Noise Variance for Six Flat-Field Levels

Tables Icon

Table 4 Performance Parameter for a Corrected Frame for Different Values of the Drift Parametersa

Equations (38)

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

Ykij(n)=AkijTkij(n)+Bkij+Vkij(n),
E[(Xk-Xˆk)Yl]=0,l=1,, k.
Xˆk=E[Xk|Y1,, Yk].
Xk+1=ΦkXk+Wk,
Φk=αk00βk
Qk=σWk(1)200σWk(2)2,
MkE[Wk]=X¯01-α001-β,k0,
X¯0=E[X0][A¯0, B¯0].
σWk(1)2=(1-α2)σA02,σWk(2)2=(1-β2)σB02,
Yk=HkXk+Vk,
Hk=Tk(1)1Tk(lk)1,
Rk=IlkσVk2,
Xˆk=Xˆk-+Kk(Yk-H¯kXˆk-),
Xˆk-=Φk-1Xˆk-1+Mk-1,
Kk=Pk-H¯k[H¯kPk-H¯k+Rk+σT2(σA02+A¯0)Ilk]-1,
σT2=112(Tkmin-Tkmax)2,
H¯k=0.5(Tkmin+Tkmax)10.5(Tkmin+Tkmax)1.
Pk-E[(Xk-Xˆk-)(Xk-Xˆk-)],
Pk-=Φk-1Pk-1Φk-1+Qk-1,
PkE[(Xk-Xˆk)(Xk-Xˆk)],
Pk=(I2-KkH¯k)Pk-,
Xˆ0=X¯0,
P0=σA0200σB02,
Xˆ1-=Φ0X¯0+M0,
MSEAk=1pmi=1pj=1m(Aˆkij-Akij)2,
ρ(f)h1*f1+h2*f1f1.
RMSE=1pmi=1pj=1m(Tˆij-Tij)21/2,
c=σtotal2σV2-11/2,
σtotal2=i=1pj=1m(Yij-Y¯)2pm-1,
Kk=E[(Xk-Xˆk-)(Yk-Yˆk-)]×{E[(Yk-Yˆk-)(Yk-Yˆk-)]}-1,
Yˆk-E[Yk|Y1,, Yk-1],
Yˆk-=E[HkXk|Y1,, Yk-1]+E[Vk|Y1,, Yk-1].
E[HkXk|Y1,, Yk-1]=E[Hk|Y1,, Yk-1]E[Xk|Y1,, Yk-1].
Yˆk-=E[Hk]E[Xk|Y1,, Yk-1]=H¯kXˆk-,
E[(Yk-Yˆk-)(Yk-Yˆk-)]=H¯kPk-H¯k+Rk+E[HkXkXkHk]-H¯kE[XkXk]H¯k,
E[HkXkXkHk]-H¯kE[XkXk]H¯k=σT2[σA02+A¯0Ik,
E[(Xk-Xˆk-)(Yk-Yˆk-)]=Pk-H¯k,
Kk=Pk-H¯k{H¯kPk-H¯k+Rk+σT2[σA02+A¯0]Ik}-1.

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