Abstract

Gaussian assumptions for the irradiance probability density in a digital image are often employed but rarely justified. We provide a mathematical justification for these assumptions and indicate the limitations of their use. Beginning with the context-dependent image ensemble considerations introduced by Hunt and Cannon [IEEE Trans. Syst. Man Cybern. SMC-6, 876 (1976)], such an ensemble is found to be accurately modeled as a Gaussian random process with nonstationary mean and nonstationary variance. An ensemble transformation is deduced and confirmed empirically that yields a Gaussian, zero-mean, unit-variance, ergodic random process. This conclusion leads to an image model that predicts that the distribution of image irradiance values after local mean removal is determined only by the distribution of local standard deviation values in the image. An analytic expression is derived for the probability density function of these irradiance values and is validated experimentally. This expression indicates that the distribution of image irradiance values after local mean removal may be assumed to be Gaussian only when the local standard deviation in an image is a nearly stationary quantity.

© 1998 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. C. Gonzalez, R. E. Woods, Digital Image Processing (Addison-Wesley, Reading, Mass., 1992).
  2. B. R. Hunt, T. M. Cannon, “Nonstationary assumptions of Gaussian models of images,” IEEE Trans. Syst. Man Cybern. SMC-6, 876–882 (1976).
  3. A. Margalit, I. S. Reed, R. M. Gagliardi, “Adaptive Optical Target Detection Using Correlated Images,” IEEE Trans. Aerosp. Electron. Syst. AES-21, 394–405 (1985).
  4. J. Y. Chen, I. S. Reed, “A detection algorithm for optical targets in clutter,” IEEE Trans. Aerosp. Electron. Syst. AES-23, 46–59 (1987).
  5. I. S. Reed, X. Yu, “Adaptive multiple-band CFAR detection of an optical pattern with unknown spectral distribution,” IEEE Trans. Acoust. Speech Signal Process. 38, 1760–1770 (1990).
    [CrossRef]
  6. A. D. Stocker, I. S. Reed, X. Yu, “Multi-dimensional signal processing for electro-optical target detection,” in Signal and Data Processing of Small Targets, O. E. Drummond, ed., Proc. SPIE1305, 218–231 (1990).
    [CrossRef]
  7. X. Yu, I. S. Reed, A. D. Stocker, “Comparative performance analysis of adaptive multispectral detectors,” IEEE Trans. Signal Process. 41, 2639–2655 (1993).
    [CrossRef]
  8. G. H. Watson, S. K. Watson, “Detection of unusual events in intermittent nonGaussian images using multiresolution background models,” Opt. Eng. 35, 3159–3171 (1996).
    [CrossRef]
  9. T. M. Apostol, Mathematical Analysis, 2nd ed. (Addison-Wesley, Reading, Mass., 1974).
  10. R. V. Hogg, A. T. Craig, Introduction to Mathematical Statistics, 4th ed. (Macmillan, New York, 1978).
  11. A. Papoulis, Probability, Random Variables, and Stochastic Processes, 3rd ed. (McGraw-Hill, New York, 1991).
  12. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

1996

G. H. Watson, S. K. Watson, “Detection of unusual events in intermittent nonGaussian images using multiresolution background models,” Opt. Eng. 35, 3159–3171 (1996).
[CrossRef]

1993

X. Yu, I. S. Reed, A. D. Stocker, “Comparative performance analysis of adaptive multispectral detectors,” IEEE Trans. Signal Process. 41, 2639–2655 (1993).
[CrossRef]

1990

I. S. Reed, X. Yu, “Adaptive multiple-band CFAR detection of an optical pattern with unknown spectral distribution,” IEEE Trans. Acoust. Speech Signal Process. 38, 1760–1770 (1990).
[CrossRef]

1987

J. Y. Chen, I. S. Reed, “A detection algorithm for optical targets in clutter,” IEEE Trans. Aerosp. Electron. Syst. AES-23, 46–59 (1987).

1985

A. Margalit, I. S. Reed, R. M. Gagliardi, “Adaptive Optical Target Detection Using Correlated Images,” IEEE Trans. Aerosp. Electron. Syst. AES-21, 394–405 (1985).

1976

B. R. Hunt, T. M. Cannon, “Nonstationary assumptions of Gaussian models of images,” IEEE Trans. Syst. Man Cybern. SMC-6, 876–882 (1976).

Apostol, T. M.

T. M. Apostol, Mathematical Analysis, 2nd ed. (Addison-Wesley, Reading, Mass., 1974).

Cannon, T. M.

B. R. Hunt, T. M. Cannon, “Nonstationary assumptions of Gaussian models of images,” IEEE Trans. Syst. Man Cybern. SMC-6, 876–882 (1976).

Chen, J. Y.

J. Y. Chen, I. S. Reed, “A detection algorithm for optical targets in clutter,” IEEE Trans. Aerosp. Electron. Syst. AES-23, 46–59 (1987).

Craig, A. T.

R. V. Hogg, A. T. Craig, Introduction to Mathematical Statistics, 4th ed. (Macmillan, New York, 1978).

Gagliardi, R. M.

A. Margalit, I. S. Reed, R. M. Gagliardi, “Adaptive Optical Target Detection Using Correlated Images,” IEEE Trans. Aerosp. Electron. Syst. AES-21, 394–405 (1985).

Gonzalez, R. C.

R. C. Gonzalez, R. E. Woods, Digital Image Processing (Addison-Wesley, Reading, Mass., 1992).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Hogg, R. V.

R. V. Hogg, A. T. Craig, Introduction to Mathematical Statistics, 4th ed. (Macmillan, New York, 1978).

Hunt, B. R.

B. R. Hunt, T. M. Cannon, “Nonstationary assumptions of Gaussian models of images,” IEEE Trans. Syst. Man Cybern. SMC-6, 876–882 (1976).

Margalit, A.

A. Margalit, I. S. Reed, R. M. Gagliardi, “Adaptive Optical Target Detection Using Correlated Images,” IEEE Trans. Aerosp. Electron. Syst. AES-21, 394–405 (1985).

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes, 3rd ed. (McGraw-Hill, New York, 1991).

Reed, I. S.

X. Yu, I. S. Reed, A. D. Stocker, “Comparative performance analysis of adaptive multispectral detectors,” IEEE Trans. Signal Process. 41, 2639–2655 (1993).
[CrossRef]

I. S. Reed, X. Yu, “Adaptive multiple-band CFAR detection of an optical pattern with unknown spectral distribution,” IEEE Trans. Acoust. Speech Signal Process. 38, 1760–1770 (1990).
[CrossRef]

J. Y. Chen, I. S. Reed, “A detection algorithm for optical targets in clutter,” IEEE Trans. Aerosp. Electron. Syst. AES-23, 46–59 (1987).

A. Margalit, I. S. Reed, R. M. Gagliardi, “Adaptive Optical Target Detection Using Correlated Images,” IEEE Trans. Aerosp. Electron. Syst. AES-21, 394–405 (1985).

A. D. Stocker, I. S. Reed, X. Yu, “Multi-dimensional signal processing for electro-optical target detection,” in Signal and Data Processing of Small Targets, O. E. Drummond, ed., Proc. SPIE1305, 218–231 (1990).
[CrossRef]

Stocker, A. D.

X. Yu, I. S. Reed, A. D. Stocker, “Comparative performance analysis of adaptive multispectral detectors,” IEEE Trans. Signal Process. 41, 2639–2655 (1993).
[CrossRef]

A. D. Stocker, I. S. Reed, X. Yu, “Multi-dimensional signal processing for electro-optical target detection,” in Signal and Data Processing of Small Targets, O. E. Drummond, ed., Proc. SPIE1305, 218–231 (1990).
[CrossRef]

Watson, G. H.

G. H. Watson, S. K. Watson, “Detection of unusual events in intermittent nonGaussian images using multiresolution background models,” Opt. Eng. 35, 3159–3171 (1996).
[CrossRef]

Watson, S. K.

G. H. Watson, S. K. Watson, “Detection of unusual events in intermittent nonGaussian images using multiresolution background models,” Opt. Eng. 35, 3159–3171 (1996).
[CrossRef]

Woods, R. E.

R. C. Gonzalez, R. E. Woods, Digital Image Processing (Addison-Wesley, Reading, Mass., 1992).

Yu, X.

X. Yu, I. S. Reed, A. D. Stocker, “Comparative performance analysis of adaptive multispectral detectors,” IEEE Trans. Signal Process. 41, 2639–2655 (1993).
[CrossRef]

I. S. Reed, X. Yu, “Adaptive multiple-band CFAR detection of an optical pattern with unknown spectral distribution,” IEEE Trans. Acoust. Speech Signal Process. 38, 1760–1770 (1990).
[CrossRef]

A. D. Stocker, I. S. Reed, X. Yu, “Multi-dimensional signal processing for electro-optical target detection,” in Signal and Data Processing of Small Targets, O. E. Drummond, ed., Proc. SPIE1305, 218–231 (1990).
[CrossRef]

IEEE Trans. Acoust. Speech Signal Process.

I. S. Reed, X. Yu, “Adaptive multiple-band CFAR detection of an optical pattern with unknown spectral distribution,” IEEE Trans. Acoust. Speech Signal Process. 38, 1760–1770 (1990).
[CrossRef]

IEEE Trans. Aerosp. Electron. Syst.

A. Margalit, I. S. Reed, R. M. Gagliardi, “Adaptive Optical Target Detection Using Correlated Images,” IEEE Trans. Aerosp. Electron. Syst. AES-21, 394–405 (1985).

J. Y. Chen, I. S. Reed, “A detection algorithm for optical targets in clutter,” IEEE Trans. Aerosp. Electron. Syst. AES-23, 46–59 (1987).

IEEE Trans. Signal Process.

X. Yu, I. S. Reed, A. D. Stocker, “Comparative performance analysis of adaptive multispectral detectors,” IEEE Trans. Signal Process. 41, 2639–2655 (1993).
[CrossRef]

IEEE Trans. Syst. Man Cybern.

B. R. Hunt, T. M. Cannon, “Nonstationary assumptions of Gaussian models of images,” IEEE Trans. Syst. Man Cybern. SMC-6, 876–882 (1976).

Opt. Eng.

G. H. Watson, S. K. Watson, “Detection of unusual events in intermittent nonGaussian images using multiresolution background models,” Opt. Eng. 35, 3159–3171 (1996).
[CrossRef]

Other

T. M. Apostol, Mathematical Analysis, 2nd ed. (Addison-Wesley, Reading, Mass., 1974).

R. V. Hogg, A. T. Craig, Introduction to Mathematical Statistics, 4th ed. (Macmillan, New York, 1978).

A. Papoulis, Probability, Random Variables, and Stochastic Processes, 3rd ed. (McGraw-Hill, New York, 1991).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

R. C. Gonzalez, R. E. Woods, Digital Image Processing (Addison-Wesley, Reading, Mass., 1992).

A. D. Stocker, I. S. Reed, X. Yu, “Multi-dimensional signal processing for electro-optical target detection,” in Signal and Data Processing of Small Targets, O. E. Drummond, ed., Proc. SPIE1305, 218–231 (1990).
[CrossRef]

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 (8)

Fig. 1
Fig. 1

(a) Typical long-wave infrared image of targets and background with natural clutter obtained with sensor in a dynamic flight environment. (b) Histogram of the irradiance values of the image in (a).

Fig. 2
Fig. 2

Three typical members of the image ensemble obtained by sequential time observation of a bright star as imaged by a ground-based telescope in the presence of turbulent atmosphere. These images are displayed in negative for clarity.

Fig. 3
Fig. 3

(a) Mean PSF obtained from the ensemble of 100 images and (b) the PSF variance of the ensemble. These images are displayed in negative for clarity.

Fig. 4
Fig. 4

(a) Ensemble correlation after transformation as a function of radial pixel separation. (b) Histogram of image irradiance values after ensemble transformation. As seen in this figure, theory accurately models the observed image ensemble behavior.

Fig. 5
Fig. 5

(a) Local mean and (b) local variance of the long-wave infrared image of targets and background in Fig. 1(a) computed with use of a 29×29 pixel processing window.

Fig. 6
Fig. 6

(a) Result of the single image transformation on the long-wave infrared image of targets and background in Fig. 1(a). (b) The histogram of image irradiance values after this transformation is shown by the solid curve. The dashed curve indicates the PDF predicted by the current theory.

Fig. 7
Fig. 7

Histogram of the local standard deviation values of the infrared image shown in Fig. 1(a), taken to be an estimate of pV(v).

Fig. 8
Fig. 8

Histogram of the irradiance values of the infrared image of Fig. 1(a) after local mean removal shown by the solid curve. The dashed curve shows the histogram predicted by the proposed image model. For comparison, a zero-mean normal distribution with variance predicted by the current theory is shown by the dotted curve.

Equations (29)

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

(M, d)={f, gA|d(f, g)<r, rR},
d(f, g)=f-g=i=1mn(fi-gi)21/2,
X=F(ξ=a, η=b).
h(x)ln[pX(x)],
dh(x)dxx=x*=h(x*)=0.
h(x)=h(x*)+(x-x*)h(x*)+12h(x*)(x-x*)2+h(x*)+12h(x*)(x-x*)2.
ln[pX(x)]ln[pX(x*)]-12(x-x*)2σ2,
σ2-d2dx2ln[pX(x)]x=x*-1.
pX(x)N exp-(x-x*)22σ2,
pX(x)N exp-(x-μ)22σ2.
F(ξ, η)N(μξη, σξη2),
W(ξ, η)F(ξ, η)-μξησξη.
W(ξ, η)N(0, 1).
w(ξ, η)=f(ξ, η)-fμ(ξ, η)fσ(ξ, η).
f(ξ, η)=fμ(ξ, η)+t(ξ, η),
f(ξ, η)=fμ(ξ, η)+w(ξ, η)fσ(ξ, η).
f(ξ, η)-fμ(ξ, η)=w(ξ, η)fσ(ξ, η).
T=WV.
pWV(w, v)=12πexp-w22pV(v),
t=wv,
u=v,
pTU(t, u)=pWVtu, u|J|=12πpV(u)uexp-t22u2.
pT(t)=pTU(t, u)du.
pT(t)=12π0 pV(u)uexp-t22u2du.
var(T)=T2-T2=T2-0=-t2pT(t)dt=-dt t2 12π0du pV(u)uexp-t22u2=0dupV(u)-dt t2 12πuexp-t22u2=0duu2pV(u)=V2.
var(T)=var(V)+V2;
pV(v)i=1Nαiδ(v-vi),
pT(t)0dui=1Nαiδ(u-vi) 12πuexp-t22u2=i=1Nαi0duδ(u-vi) 12πuexp-t22u2,
pT(t)i=1Nαi 12πviexp-t22vi2.

Metrics