Abstract

The statistical behavior of images is inherently nonstationary. Unfortunately, most image processing algorithms assume stationary image models. Spatially adaptive algorithms have been developed which take into account local image statistics. In this paper we derive radiometric and geometric transforms which generate nearly stationary (block stationary) images in the first and second moments. We show that true stationarity is impossible to realize. The aim of these transformations is to enhance the performance of nonadaptive processing techniques, in particular data compression.

© 1983 Optical Society of America

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References

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  1. C. W. Helstrom, J. Opt. Soc. Am. 57, 297 (1967).
    [CrossRef]
  2. G. L. Anderson, A. N. Netravali, IEEE Trans. Syst. Man Cybern. SMC-6, 845 (1976).
    [CrossRef]
  3. H. J. Trussel, B. R. Hunt, IEEE Trans. Acoust. Speech Signal Process. ASSP-26, 157 (1978).
    [CrossRef]
  4. D. T. Kuan, A. A. Sawchuk, J. Opt. Soc. Am. 71, 1641A (1981).
  5. R. Kasturi, “Adaptive Image Restoration in Signal-Dependent Noise,” Ph.D. Dissertation, Institute for Electronic Science, Texas Tech U., Lubbock (1982).
  6. W. K. Pratt, Digital Image Processing (Wiley, New York, 1978), pp. 729 and 21.
  7. A. Habibi, IEEE Trans. Commun. COM-25, 1315 (1977).
  8. B. R. Hunt, Comput. Graphics Image Process. 12, 173 (1980).
    [CrossRef]
  9. T. S. Huang, IEEE Spectrum 2, (12), 57 (1965).
  10. H. G. Musmann, in Image Transmission Techniques, W. K. Pratt, Ed. (Academic, New York, 1979), pp. 92–97.
  11. A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, New York, 1965), p. 302.
  12. R. Wallis, “An Approach to the Space-Variant Restoration and Enhancement of Images,” in Proceedings, Symposium on Current Mathematical Problems in Image Science, Naval Postgraduate School, Monterey, Calif., Nov. 1976.
  13. J. S. Lee, IEEE Trans. Pattern Anal. Mach. Intell. PAMI-2, 165 (1980).
    [CrossRef]
  14. A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975), pp. 539–541.
  15. G. S. Robinson, Comput. Graphics Image Process. 6, 492 (1977).
    [CrossRef]
  16. R. Bernstein, IBM J. Res. Dev. 20, 40 (1976).
    [CrossRef]
  17. R. N. Strickland, J. J. Burke, Proc. Soc. Photo-Opt. Instrum. Eng. 249, 47 (1980).

1981

D. T. Kuan, A. A. Sawchuk, J. Opt. Soc. Am. 71, 1641A (1981).

1980

B. R. Hunt, Comput. Graphics Image Process. 12, 173 (1980).
[CrossRef]

J. S. Lee, IEEE Trans. Pattern Anal. Mach. Intell. PAMI-2, 165 (1980).
[CrossRef]

R. N. Strickland, J. J. Burke, Proc. Soc. Photo-Opt. Instrum. Eng. 249, 47 (1980).

1978

H. J. Trussel, B. R. Hunt, IEEE Trans. Acoust. Speech Signal Process. ASSP-26, 157 (1978).
[CrossRef]

1977

A. Habibi, IEEE Trans. Commun. COM-25, 1315 (1977).

G. S. Robinson, Comput. Graphics Image Process. 6, 492 (1977).
[CrossRef]

1976

R. Bernstein, IBM J. Res. Dev. 20, 40 (1976).
[CrossRef]

G. L. Anderson, A. N. Netravali, IEEE Trans. Syst. Man Cybern. SMC-6, 845 (1976).
[CrossRef]

1967

1965

T. S. Huang, IEEE Spectrum 2, (12), 57 (1965).

Anderson, G. L.

G. L. Anderson, A. N. Netravali, IEEE Trans. Syst. Man Cybern. SMC-6, 845 (1976).
[CrossRef]

Bernstein, R.

R. Bernstein, IBM J. Res. Dev. 20, 40 (1976).
[CrossRef]

Burke, J. J.

R. N. Strickland, J. J. Burke, Proc. Soc. Photo-Opt. Instrum. Eng. 249, 47 (1980).

Habibi, A.

A. Habibi, IEEE Trans. Commun. COM-25, 1315 (1977).

Helstrom, C. W.

Huang, T. S.

T. S. Huang, IEEE Spectrum 2, (12), 57 (1965).

Hunt, B. R.

B. R. Hunt, Comput. Graphics Image Process. 12, 173 (1980).
[CrossRef]

H. J. Trussel, B. R. Hunt, IEEE Trans. Acoust. Speech Signal Process. ASSP-26, 157 (1978).
[CrossRef]

Kasturi, R.

R. Kasturi, “Adaptive Image Restoration in Signal-Dependent Noise,” Ph.D. Dissertation, Institute for Electronic Science, Texas Tech U., Lubbock (1982).

Kuan, D. T.

D. T. Kuan, A. A. Sawchuk, J. Opt. Soc. Am. 71, 1641A (1981).

Lee, J. S.

J. S. Lee, IEEE Trans. Pattern Anal. Mach. Intell. PAMI-2, 165 (1980).
[CrossRef]

Musmann, H. G.

H. G. Musmann, in Image Transmission Techniques, W. K. Pratt, Ed. (Academic, New York, 1979), pp. 92–97.

Netravali, A. N.

G. L. Anderson, A. N. Netravali, IEEE Trans. Syst. Man Cybern. SMC-6, 845 (1976).
[CrossRef]

Oppenheim, A. V.

A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975), pp. 539–541.

Papoulis, A.

A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, New York, 1965), p. 302.

Pratt, W. K.

W. K. Pratt, Digital Image Processing (Wiley, New York, 1978), pp. 729 and 21.

Robinson, G. S.

G. S. Robinson, Comput. Graphics Image Process. 6, 492 (1977).
[CrossRef]

Sawchuk, A. A.

D. T. Kuan, A. A. Sawchuk, J. Opt. Soc. Am. 71, 1641A (1981).

Schafer, R. W.

A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975), pp. 539–541.

Strickland, R. N.

R. N. Strickland, J. J. Burke, Proc. Soc. Photo-Opt. Instrum. Eng. 249, 47 (1980).

Trussel, H. J.

H. J. Trussel, B. R. Hunt, IEEE Trans. Acoust. Speech Signal Process. ASSP-26, 157 (1978).
[CrossRef]

Wallis, R.

R. Wallis, “An Approach to the Space-Variant Restoration and Enhancement of Images,” in Proceedings, Symposium on Current Mathematical Problems in Image Science, Naval Postgraduate School, Monterey, Calif., Nov. 1976.

Comput. Graphics Image Process.

B. R. Hunt, Comput. Graphics Image Process. 12, 173 (1980).
[CrossRef]

G. S. Robinson, Comput. Graphics Image Process. 6, 492 (1977).
[CrossRef]

IBM J. Res. Dev.

R. Bernstein, IBM J. Res. Dev. 20, 40 (1976).
[CrossRef]

IEEE Spectrum

T. S. Huang, IEEE Spectrum 2, (12), 57 (1965).

IEEE Trans. Acoust. Speech Signal Process.

H. J. Trussel, B. R. Hunt, IEEE Trans. Acoust. Speech Signal Process. ASSP-26, 157 (1978).
[CrossRef]

IEEE Trans. Commun.

A. Habibi, IEEE Trans. Commun. COM-25, 1315 (1977).

IEEE Trans. Pattern Anal. Mach. Intell.

J. S. Lee, IEEE Trans. Pattern Anal. Mach. Intell. PAMI-2, 165 (1980).
[CrossRef]

IEEE Trans. Syst. Man Cybern.

G. L. Anderson, A. N. Netravali, IEEE Trans. Syst. Man Cybern. SMC-6, 845 (1976).
[CrossRef]

J. Opt. Soc. Am.

C. W. Helstrom, J. Opt. Soc. Am. 57, 297 (1967).
[CrossRef]

D. T. Kuan, A. A. Sawchuk, J. Opt. Soc. Am. 71, 1641A (1981).

Proc. Soc. Photo-Opt. Instrum. Eng.

R. N. Strickland, J. J. Burke, Proc. Soc. Photo-Opt. Instrum. Eng. 249, 47 (1980).

Other

A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975), pp. 539–541.

R. Kasturi, “Adaptive Image Restoration in Signal-Dependent Noise,” Ph.D. Dissertation, Institute for Electronic Science, Texas Tech U., Lubbock (1982).

W. K. Pratt, Digital Image Processing (Wiley, New York, 1978), pp. 729 and 21.

H. G. Musmann, in Image Transmission Techniques, W. K. Pratt, Ed. (Academic, New York, 1979), pp. 92–97.

A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, New York, 1965), p. 302.

R. Wallis, “An Approach to the Space-Variant Restoration and Enhancement of Images,” in Proceedings, Symposium on Current Mathematical Problems in Image Science, Naval Postgraduate School, Monterey, Calif., Nov. 1976.

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Figures (14)

Fig. 1
Fig. 1

Autocorrelation function profile designations.

Fig. 2
Fig. 2

Original test image.

Fig. 3
Fig. 3

Radiometric transform of Fig 2 (a) blockwise transformation; (b) transformation with interpolated coefficients.

Fig. 4
Fig. 4

Expansion ratios map: circular symmetric autocorrelation model.

Fig. 5
Fig. 5

Expansion ratios map: elliptically symmetric autocorrelation model: (a) NS data; (b) EW data.

Fig. 6
Fig. 6

Principles of 1-D and 2-D control point mapping.

Fig. 7
Fig. 7

Control point grid warping by Eqs. (37) and (38) (two-component expansion) ENS = 1.5, EEW = 2.0.

Fig. 8
Fig. 8

Quadrilateral neighborhood geometry.

Fig. 9
Fig. 9

Iterative warping: control point grid after four iterations: (a) single expansion ENS = 1.5, EEW = 2.0; (b) expansions of Fig. 5.

Fig. 10
Fig. 10

Iterative warping using circular expansion; rms expansion (area) error <0.01% after 100 iterations:

Fig. 11
Fig. 11

Comparison of three- and four-point transformations: three-point, p = a0 + a1x + a2y; four-point, p = a0 + a1x + a2y + a3xy.

Fig. 12
Fig. 12

Radiometric and geometric transforms of Fig. 2.

Fig. 13
Fig. 13

Hybrid optical–digital implementation for parameter measurements.

Fig. 14
Fig. 14

Radiometric and geometric transforms.

Equations (43)

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μ = E [ f ( x , y ) ] ,
R ( ξ , η ) = E [ f ( x , y ) f ( x + ξ , y + η ) ] ,
μ N = N f ( x , y ) d x d y ,
R N ( ξ , η ) = N f ( x , y ) f ( x + ξ , y + η ) d x d y .
μ N = constant R N ( ξ , η ) = R s ( ξ , η ) } all N .
μ N = 1 M 2 j = 0 M - 1 k = 0 M - 1 f ( j , k ) ,
R N ( r , s ) = 1 M 2 j = 0 M - r - 1 k = 0 M - s - 1 f ( j , k ) f ( j + r , k + s ) ,
R N ( 0 , 0 ) = 1 M 2 j = 0 M - 1 k = 0 M - 1 f ( j , k ) 2 .
σ N 2 = R N ( 0 , 0 ) - μ N 2 .
g ( j , k ) = A f ( j , k ) + B .
1 M 2 j = 0 M - 1 k = 0 M - 1 g ( j , k ) = μ s ,
1 M 2 j = 0 M - 1 k = 0 M - 1 g ( j , k ) 2 = R s ( 0 , 0 ) ,
A = [ μ s 2 - R s ( 0 , 0 ) μ N 2 - R N ( 0 , 0 ) ] 1 / 2 = σ s σ N = stationary standard deviation nonstationary standard deviation , B = μ s - A μ N .
g ( j , k ) = σ s σ N [ f ( j , k ) - μ N ] + μ s .
g = K ( variations about mean of f ) + new mean .
R N ( ξ , η ) = σ N 2 exp [ - ρ N ( ξ 2 + η 2 ) 1 / 2 ] + μ N 2 .
R N S = R N ( ξ , 0 ) = σ N 2 exp ( - ρ N S ξ ) + μ N 2 ,
R E W = R N ( 0 , η ) = σ N 2 exp ( - ρ E W η ) + μ N 2 ,
ρ N S = M ln σ N 2 - r = 0 M - 1 ln [ R ( r , 0 ) - μ N 2 ] r = 0 M - 1 r .
E N = ρ N ρ s .
R g N ( r , s ) = ( σ s / σ N ) 2 [ R N ( r , s ) - μ N 2 ] + μ s 2 .
R g N ( r , s ) - μ s 2 = ( σ s / σ N ) 2 [ R N ( r , s ) - μ N 2 ] ,
C g N ( r , s ) = ( σ s / σ N ) 2 C N ( r , s ) ,
f ( j , k ) = f ( j , k ) + n ( j , k ) ,
g ( j , k ) = σ s σ N [ f ( j , k ) + n ( j , k ) - μ N ] + μ s .
g ( j , k ) = K [ f ( j , k ) - μ N ] + μ s ,
K = σ s σ N ; σ N > T , K = 1 ; σ N T .
σ N 2 = [ f ( j , k ) - μ N ] 2 ¯ .
E N S = ρ N S ρ N S min , E N E = ρ N E ρ N E min , E E W = ρ E W ρ E W min , E S E = ρ S E ρ S E min
p = i = 0 m j = 0 m - i a i j x i y i , q = i = 0 m j = 0 m - i b i j x i y j ,
p = x + a 1 x + a 2 y + a 3 x 2 + a 4 x y + a 5 y 2 q = y + b 1 x + b 2 y + b 3 x 2 + b 4 x y + b 5 y 2 . identity term             perturbation terms
E ( x , y ) = p x q y - p y q x .
E ( x , y ) = 1 + L ( x , y ) + Q ( x , y ) . linear             quadratic terms             terms
p = ( E - 1 ) d 2 + x ,
p = d 2 cos θ · E E W ,
q = d 2 sin θ · E N S .
p = d 2 cos θ · E E W + ( x - d 2 cos θ ) , p = d 2 cos θ · ( E E W - 1 ) + x .
q = d 2 sin θ · ( E N S - 1 ) + y .
E N S = l N S d ,             E E W = l E W d ,
corrected expanasion = ( E 2 × original area measured area ) 1 / 2 .
p = a 0 + a 1 x + a 2 y + a 3 x y , q = b 0 + b 1 x + b 2 y + b 3 x y .
p = a 0 + a 1 x + a 2 y , q = b 0 + b 1 x + b 2 y .
t = 2 ( open / close ) × shutter time of mask × CCD integration and read - out time × no . of subblocks + CCD computing time .

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