Abstract

Principal-component decomposition is applied to the analysis of noise for infrared images. It provides a set of eigenimages, the principal components, that represents spatial patterns associated with different types of noise. We provide a method to classify the principal components into processes that explain a given amount of the variance of the images under analysis. Each process can reconstruct the set of data, thus allowing a calculation of the weight of the given process in the total noise. The method is successfully applied to an actual set of infrared images. The extension of the method to images in the visible spectrum is possible and would provide similar results.

© 2002 Optical Society of America

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References

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  1. J. M. Mooney, “Effect of spatial noise on the minimum resolvable temperature of a staring sensor,” Appl. Opt. 30, 3324–3332 (1991).
    [CrossRef] [PubMed]
  2. C. Webb, J. A. D’Agostino, “Manual reference of FLIR 92,” (U.S. Army Night Vision and Electronic Sensor Directorate, Washington, D.C., 1992).
  3. H. Rothe, A. Duparr, S. Jacobs, “Generic detrending of surface profiles,” Opt. Eng. 33, 3023–3030 (1994).
    [CrossRef]
  4. H. Rothe, M. Tuerschmann, P. Mager, R. Endter, “Improved accuracy in laser triangulation by variance-stabilizing transformations,” Opt. Eng. 31, 1538–1545 (1992).
    [CrossRef]
  5. H. Rothe, O. Ginter, C. Woldenga, “Assessment and robust reconstruction of laser radar signals,” Opt. Laser Technol. 25, 289–297 (1993).
    [CrossRef]
  6. G. Rasigni, F. Varnier, M. Rasigni, J. P. Palmari, “Autocovariance functions for polished optical surfaces,” J. Opt. Soc. Am. 73, 222–233 (1983).
    [CrossRef]
  7. D. F. Morrison, Multivariate Statistical Methods, 3rd ed. (McGraw-Hill, Singapore, 1990), Chap. 8.
  8. R. B. Cattell, “The scree test for the number of factors,” J. Multivar. Behav. Res. 1, 245–276 (1966).
    [CrossRef]
  9. G. R. North, T. L. Bell, R. F. Cahalan, F. J. Moeng, “Sampling errors in the estimation of empirical orthogonal functions,” Mon. Weather Rev. 19, 699–706 (1982).
    [CrossRef]
  10. C. L. Bennett, “The effect of jitter on an imaging FTIR spectrometer,” in Infrared Imaging Systems: Design, Analysis, Modeling and Testing VIII, C. Holst, ed., Proc. SPIE3063, 174–184 (1997).
    [CrossRef]
  11. H. Rothe, H. Truckenbrodt, “Discrimination of surface properties using BRDF-variance estimators as feature variables,” in Specification and Measurement of Optical Systems, L. R. Baker, ed., Proc. SPIE1781, 152–162 (1992).
    [CrossRef]
  12. S. Hare, “Low frequency climate variability and salmon production,” Ph.D. dissertation (University of Washington, Seattle, Washington, 1996), Chap. 1. See http://www.iphc.washington.edu/Staff/hare/html/diss/chapter1/chap1.html .
  13. C. M. Waternaux, “Asymptotic distribution of the sample roots for a nonnormal population,” Biometrika 63, 639–645 (1976).
    [CrossRef]
  14. A. W. Davis, “Asymptotic theory for principal component analysis: non-normal case,” Aust. J. Stat. 19, 206–212 (1977).
    [CrossRef]

1994

H. Rothe, A. Duparr, S. Jacobs, “Generic detrending of surface profiles,” Opt. Eng. 33, 3023–3030 (1994).
[CrossRef]

1993

H. Rothe, O. Ginter, C. Woldenga, “Assessment and robust reconstruction of laser radar signals,” Opt. Laser Technol. 25, 289–297 (1993).
[CrossRef]

1992

H. Rothe, M. Tuerschmann, P. Mager, R. Endter, “Improved accuracy in laser triangulation by variance-stabilizing transformations,” Opt. Eng. 31, 1538–1545 (1992).
[CrossRef]

1991

1983

1982

G. R. North, T. L. Bell, R. F. Cahalan, F. J. Moeng, “Sampling errors in the estimation of empirical orthogonal functions,” Mon. Weather Rev. 19, 699–706 (1982).
[CrossRef]

1977

A. W. Davis, “Asymptotic theory for principal component analysis: non-normal case,” Aust. J. Stat. 19, 206–212 (1977).
[CrossRef]

1976

C. M. Waternaux, “Asymptotic distribution of the sample roots for a nonnormal population,” Biometrika 63, 639–645 (1976).
[CrossRef]

1966

R. B. Cattell, “The scree test for the number of factors,” J. Multivar. Behav. Res. 1, 245–276 (1966).
[CrossRef]

Bell, T. L.

G. R. North, T. L. Bell, R. F. Cahalan, F. J. Moeng, “Sampling errors in the estimation of empirical orthogonal functions,” Mon. Weather Rev. 19, 699–706 (1982).
[CrossRef]

Bennett, C. L.

C. L. Bennett, “The effect of jitter on an imaging FTIR spectrometer,” in Infrared Imaging Systems: Design, Analysis, Modeling and Testing VIII, C. Holst, ed., Proc. SPIE3063, 174–184 (1997).
[CrossRef]

Cahalan, R. F.

G. R. North, T. L. Bell, R. F. Cahalan, F. J. Moeng, “Sampling errors in the estimation of empirical orthogonal functions,” Mon. Weather Rev. 19, 699–706 (1982).
[CrossRef]

Cattell, R. B.

R. B. Cattell, “The scree test for the number of factors,” J. Multivar. Behav. Res. 1, 245–276 (1966).
[CrossRef]

D’Agostino, J. A.

C. Webb, J. A. D’Agostino, “Manual reference of FLIR 92,” (U.S. Army Night Vision and Electronic Sensor Directorate, Washington, D.C., 1992).

Davis, A. W.

A. W. Davis, “Asymptotic theory for principal component analysis: non-normal case,” Aust. J. Stat. 19, 206–212 (1977).
[CrossRef]

Duparr, A.

H. Rothe, A. Duparr, S. Jacobs, “Generic detrending of surface profiles,” Opt. Eng. 33, 3023–3030 (1994).
[CrossRef]

Endter, R.

H. Rothe, M. Tuerschmann, P. Mager, R. Endter, “Improved accuracy in laser triangulation by variance-stabilizing transformations,” Opt. Eng. 31, 1538–1545 (1992).
[CrossRef]

Ginter, O.

H. Rothe, O. Ginter, C. Woldenga, “Assessment and robust reconstruction of laser radar signals,” Opt. Laser Technol. 25, 289–297 (1993).
[CrossRef]

Hare, S.

S. Hare, “Low frequency climate variability and salmon production,” Ph.D. dissertation (University of Washington, Seattle, Washington, 1996), Chap. 1. See http://www.iphc.washington.edu/Staff/hare/html/diss/chapter1/chap1.html .

Jacobs, S.

H. Rothe, A. Duparr, S. Jacobs, “Generic detrending of surface profiles,” Opt. Eng. 33, 3023–3030 (1994).
[CrossRef]

Mager, P.

H. Rothe, M. Tuerschmann, P. Mager, R. Endter, “Improved accuracy in laser triangulation by variance-stabilizing transformations,” Opt. Eng. 31, 1538–1545 (1992).
[CrossRef]

Moeng, F. J.

G. R. North, T. L. Bell, R. F. Cahalan, F. J. Moeng, “Sampling errors in the estimation of empirical orthogonal functions,” Mon. Weather Rev. 19, 699–706 (1982).
[CrossRef]

Mooney, J. M.

Morrison, D. F.

D. F. Morrison, Multivariate Statistical Methods, 3rd ed. (McGraw-Hill, Singapore, 1990), Chap. 8.

North, G. R.

G. R. North, T. L. Bell, R. F. Cahalan, F. J. Moeng, “Sampling errors in the estimation of empirical orthogonal functions,” Mon. Weather Rev. 19, 699–706 (1982).
[CrossRef]

Palmari, J. P.

Rasigni, G.

Rasigni, M.

Rothe, H.

H. Rothe, A. Duparr, S. Jacobs, “Generic detrending of surface profiles,” Opt. Eng. 33, 3023–3030 (1994).
[CrossRef]

H. Rothe, O. Ginter, C. Woldenga, “Assessment and robust reconstruction of laser radar signals,” Opt. Laser Technol. 25, 289–297 (1993).
[CrossRef]

H. Rothe, M. Tuerschmann, P. Mager, R. Endter, “Improved accuracy in laser triangulation by variance-stabilizing transformations,” Opt. Eng. 31, 1538–1545 (1992).
[CrossRef]

H. Rothe, H. Truckenbrodt, “Discrimination of surface properties using BRDF-variance estimators as feature variables,” in Specification and Measurement of Optical Systems, L. R. Baker, ed., Proc. SPIE1781, 152–162 (1992).
[CrossRef]

Truckenbrodt, H.

H. Rothe, H. Truckenbrodt, “Discrimination of surface properties using BRDF-variance estimators as feature variables,” in Specification and Measurement of Optical Systems, L. R. Baker, ed., Proc. SPIE1781, 152–162 (1992).
[CrossRef]

Tuerschmann, M.

H. Rothe, M. Tuerschmann, P. Mager, R. Endter, “Improved accuracy in laser triangulation by variance-stabilizing transformations,” Opt. Eng. 31, 1538–1545 (1992).
[CrossRef]

Varnier, F.

Waternaux, C. M.

C. M. Waternaux, “Asymptotic distribution of the sample roots for a nonnormal population,” Biometrika 63, 639–645 (1976).
[CrossRef]

Webb, C.

C. Webb, J. A. D’Agostino, “Manual reference of FLIR 92,” (U.S. Army Night Vision and Electronic Sensor Directorate, Washington, D.C., 1992).

Woldenga, C.

H. Rothe, O. Ginter, C. Woldenga, “Assessment and robust reconstruction of laser radar signals,” Opt. Laser Technol. 25, 289–297 (1993).
[CrossRef]

Appl. Opt.

Aust. J. Stat.

A. W. Davis, “Asymptotic theory for principal component analysis: non-normal case,” Aust. J. Stat. 19, 206–212 (1977).
[CrossRef]

Biometrika

C. M. Waternaux, “Asymptotic distribution of the sample roots for a nonnormal population,” Biometrika 63, 639–645 (1976).
[CrossRef]

J. Multivar. Behav. Res.

R. B. Cattell, “The scree test for the number of factors,” J. Multivar. Behav. Res. 1, 245–276 (1966).
[CrossRef]

J. Opt. Soc. Am.

Mon. Weather Rev.

G. R. North, T. L. Bell, R. F. Cahalan, F. J. Moeng, “Sampling errors in the estimation of empirical orthogonal functions,” Mon. Weather Rev. 19, 699–706 (1982).
[CrossRef]

Opt. Eng.

H. Rothe, A. Duparr, S. Jacobs, “Generic detrending of surface profiles,” Opt. Eng. 33, 3023–3030 (1994).
[CrossRef]

H. Rothe, M. Tuerschmann, P. Mager, R. Endter, “Improved accuracy in laser triangulation by variance-stabilizing transformations,” Opt. Eng. 31, 1538–1545 (1992).
[CrossRef]

Opt. Laser Technol.

H. Rothe, O. Ginter, C. Woldenga, “Assessment and robust reconstruction of laser radar signals,” Opt. Laser Technol. 25, 289–297 (1993).
[CrossRef]

Other

C. Webb, J. A. D’Agostino, “Manual reference of FLIR 92,” (U.S. Army Night Vision and Electronic Sensor Directorate, Washington, D.C., 1992).

C. L. Bennett, “The effect of jitter on an imaging FTIR spectrometer,” in Infrared Imaging Systems: Design, Analysis, Modeling and Testing VIII, C. Holst, ed., Proc. SPIE3063, 174–184 (1997).
[CrossRef]

H. Rothe, H. Truckenbrodt, “Discrimination of surface properties using BRDF-variance estimators as feature variables,” in Specification and Measurement of Optical Systems, L. R. Baker, ed., Proc. SPIE1781, 152–162 (1992).
[CrossRef]

S. Hare, “Low frequency climate variability and salmon production,” Ph.D. dissertation (University of Washington, Seattle, Washington, 1996), Chap. 1. See http://www.iphc.washington.edu/Staff/hare/html/diss/chapter1/chap1.html .

D. F. Morrison, Multivariate Statistical Methods, 3rd ed. (McGraw-Hill, Singapore, 1990), Chap. 8.

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Figures (8)

Fig. 1
Fig. 1

Scatter plot of the values of 4756 pixels in three frames. The principal components correspond with the main directions of the 3D ellipsoid that contains the points. These directions are represented as three orthogonal arrows with lengths proportional to the eigenvalue. For the sake of clarity, the scale for the second and third values has been modified. The actual values of the eigenvalues are, in decreasing percentage, 95.82%, 2.21%, and 1.96%.

Fig. 2
Fig. 2

The original frames (after subtraction of the mean) are related to the principal component through a rotation matrix. In this plot we show how the original frames ( 1, 2, 3), whose pixels’ values are represented in Fig. 1, are transformed into three principal components, (Y 1, Y 2, Y 3), that are presented as images. The 4756 pixels are actually arranged as a 58 × 82 array. Below each one of the principal components we plot the three frames generated by each principal component [see Eq. (14)]. To improve the clarity of the representation, we have maximized the range of the gray level to the maximum for each image (principal component, frames, and generated frames).

Fig. 3
Fig. 3

Plot of the 21 eigenvectors for a set of 21 experimental frames analyzed by the principal-component method. The scree test clearly identifies three groups of eigenvalues. The method to group eigenvalues described in Section 3 also produces the same grouping. The values are also plotted with the error bars obtained after calculation by use of the operational method described in Appendix A. The grouping appears when the eigenvalues overlap their error bars continuously.

Fig. 4
Fig. 4

Eigenimages corresponding to the principal components obtained at two different temperatures. The first principal component can be associated with spatial noise. The second and the third correspond with a process clearly different from the other two. It represents a fringe pattern crossing the scene periodically. The rest of eigenimages can be associated with temporal noise.

Fig. 5
Fig. 5

Plot of the 21 eigenvalues at several temperatures of the blackbody. The same three processes shown in Fig. 3 are found. The first process, associated with spatial noise, changes in accordance with the expected evolution of spatial noise with temperature, reaching a minimum around the calibration point.

Fig. 6
Fig. 6

Mean standard deviation and uncertainty for the frames generated for the three processes at different temperatures.

Fig. 7
Fig. 7

Values of the noise for the three processes within the 3D model of noise. (a) is for a blackbody temperature of 20 °C, and (b) is for a blackbody temperature of 40 °C. The original set of data is analyzed along three directions: horizontal (h), vertical (v), and temporal (t) (see Ref. 2). Each subset of columns represents the rms value of the data along the labeled directions. TN is the total noise of the data obtained by adding in the quadrature all the other contributions.

Fig. 8
Fig. 8

Plot of the variance matrix of the whole set of frames obtained at T = 40 °C, and its decomposition into the three processes derived by the principal-component expansion.

Equations (36)

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ρt1, t2=1Mi=1Mfi,t1-ft1fi,t2-ft2σsf,t12,
ρt1,t2=Ft1  Ft2-ft1ft2Ft1  Ft1-ft1,
F=F1, F2,  , Ft,  , FN,
k=c-1R+r,
S=1M-1 F¯TF¯,
F¯=F-1M UMUMTF,
S-λαIEα=0,
Y=F¯E.
Yα=tN et,αF¯t.
F¯=YET,
F¯t=αN et,αYα.
SY=1M-1 YTY=ETSE.
P=EET,
F=F¯P+1M UMUMTF.
Ho : λk1=λk2==λkn; kiK,
e1e2=cos θ1,2sin θ1,2-sin θ1,2cos θ1,2e1e2.
Y1=t e1,tFt=Y1 cos θ1,2+Y2 sin θ1,2,Y2=t e2,tFt=-Y1 sin θ1,2+Y2 cos θ1,2,
l1=l1 cos2 θ1,2+l2 sin2 θ1,2,l2=l1 sin2 θ1,2+l2 cos2 θ1,2,
covY1, Y2l2-l12sin 2θ1,2,
ΔYr+s-1=ΔYr+s sin θr+s, r+s-1, ΔYr+s=ΔYr+s cos θr+s, r+s-1.
Ci,j=1M2λi2δi,j+c4iδi,j+c2,2i,j,
c2,2i,j=1Mk=1MYik2Yjk2-1Mk=1MYik2×1Mk=1MYjk2,
Yik21r-1j=0r-1Yi+jk2=sk2.
c2,2i,j=1Mk=1Msk22-1Mk=1M sk22.
sk2=χr-12r-1 λ,
c2,2i,j=c2,2=2λ2r-1,
c4i=1Mk=1MYik4-3 1Mk=1MYik22.
c4i1Mk=1MYik4-31Mk=1MYik22.
Yik4=3σk43sk4,
μy=1Mk=1M σk21Mk=1M sk2,
σy2=2M2k=1M σk42M2k=1M sk4.
-31Mk=1MYik22=-32M2k=1M sk4+1Mk=1M sk22.
c4i=c4=3-6Mc2,2-6M λ2.
c43c2,2.
μ=1ri=1r li=λ,
σ2=1M1r2λ2+c4+r-1c2,2=λ2M4r+6rr-1.

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