Abstract

Statistical analysis of 2-D image data or data gathered from a scanning radiometer requires that both the non-Gaussian nature and finite sample size of the process be considered. To aid the statistical analysis of this data, a higher moment description density function has been defined, and parameters have been identified with the estimated moments of the data. It is shown that the first two moments may be computed from a knowledge of the Weiner spectrum, whereas all higher moments require the complex spatial frequency spectrum. Parameter identification is carried out for a three-parameter density function and applied to a scene in the IR region, 8–14 μm. Results indicate that a three-parameter distribution density generally provides different probabilities than does a two-parameter Gaussian description if maximum entropy (minimum bias) forms are sought.

© 1980 Optical Society of America

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References

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  1. R. Roberts, “Signal Processing Techniques,” Interstate Electronics Corp., Anaheim, Calif. (1977).
  2. W. D. Montgomery, P. W. Broome, J. Opt. Soc. Am. 52, 1259 (1962).
    [CrossRef]
  3. Y. S. Itakura, S. Tsutsumi, T. Takagi, Infrared Phys. 14, 17 (1974).
    [CrossRef]
  4. J. R. Maxwell, L. Wilkins, IRIS Proc. 22, 101 (1978).
  5. D. Middleton, An Introduction to Statistical Communications Theory (McGraw-Hill, New York, 1960).
  6. A. J. Devaney, R. Chidlaw, J. Opt. Soc. Am. 68, 1352 (1978).
    [CrossRef]
  7. A. H. Greenaway, Opt. Lett. 1, 10 (1977).
    [CrossRef] [PubMed]
  8. C. Stein, “Approximation of Improper Prior Measures by Prior Probability Measures,” Department of Statistics, Stanford University Tech. Rept. 12 (1964).
  9. S. J. Dunning, S. R. Robinson, Appl. Opt. 18, 1507 (1979).
    [CrossRef]
  10. A. K. Majumdar, J. Opt. Soc. Am. 69, 199 (1979).
    [CrossRef]
  11. P. M. Lewis, Inf. Control 2, 214 (1959).
    [CrossRef]
  12. W. D. Montgomery, IEEE Trans. Inf. Theory 10, 2 (1964).
    [CrossRef]
  13. M. Abramowitz, I. R. Stegun, Eds., Handbook of Mathematical Functions (Dover, New York, 1968, Chap. 14.

1979 (2)

S. J. Dunning, S. R. Robinson, Appl. Opt. 18, 1507 (1979).
[CrossRef]

A. K. Majumdar, J. Opt. Soc. Am. 69, 199 (1979).
[CrossRef]

1978 (2)

J. R. Maxwell, L. Wilkins, IRIS Proc. 22, 101 (1978).

A. J. Devaney, R. Chidlaw, J. Opt. Soc. Am. 68, 1352 (1978).
[CrossRef]

1977 (1)

1974 (1)

Y. S. Itakura, S. Tsutsumi, T. Takagi, Infrared Phys. 14, 17 (1974).
[CrossRef]

1964 (1)

W. D. Montgomery, IEEE Trans. Inf. Theory 10, 2 (1964).
[CrossRef]

1962 (1)

1959 (1)

P. M. Lewis, Inf. Control 2, 214 (1959).
[CrossRef]

Broome, P. W.

Chidlaw, R.

Devaney, A. J.

Dunning, S. J.

S. J. Dunning, S. R. Robinson, Appl. Opt. 18, 1507 (1979).
[CrossRef]

Greenaway, A. H.

Itakura, Y. S.

Y. S. Itakura, S. Tsutsumi, T. Takagi, Infrared Phys. 14, 17 (1974).
[CrossRef]

Lewis, P. M.

P. M. Lewis, Inf. Control 2, 214 (1959).
[CrossRef]

Majumdar, A. K.

Maxwell, J. R.

J. R. Maxwell, L. Wilkins, IRIS Proc. 22, 101 (1978).

Middleton, D.

D. Middleton, An Introduction to Statistical Communications Theory (McGraw-Hill, New York, 1960).

Montgomery, W. D.

W. D. Montgomery, IEEE Trans. Inf. Theory 10, 2 (1964).
[CrossRef]

W. D. Montgomery, P. W. Broome, J. Opt. Soc. Am. 52, 1259 (1962).
[CrossRef]

Roberts, R.

R. Roberts, “Signal Processing Techniques,” Interstate Electronics Corp., Anaheim, Calif. (1977).

Robinson, S. R.

S. J. Dunning, S. R. Robinson, Appl. Opt. 18, 1507 (1979).
[CrossRef]

Stein, C.

C. Stein, “Approximation of Improper Prior Measures by Prior Probability Measures,” Department of Statistics, Stanford University Tech. Rept. 12 (1964).

Takagi, T.

Y. S. Itakura, S. Tsutsumi, T. Takagi, Infrared Phys. 14, 17 (1974).
[CrossRef]

Tsutsumi, S.

Y. S. Itakura, S. Tsutsumi, T. Takagi, Infrared Phys. 14, 17 (1974).
[CrossRef]

Wilkins, L.

J. R. Maxwell, L. Wilkins, IRIS Proc. 22, 101 (1978).

Appl. Opt. (1)

S. J. Dunning, S. R. Robinson, Appl. Opt. 18, 1507 (1979).
[CrossRef]

IEEE Trans. Inf. Theory (1)

W. D. Montgomery, IEEE Trans. Inf. Theory 10, 2 (1964).
[CrossRef]

Inf. Control (1)

P. M. Lewis, Inf. Control 2, 214 (1959).
[CrossRef]

Infrared Phys. (1)

Y. S. Itakura, S. Tsutsumi, T. Takagi, Infrared Phys. 14, 17 (1974).
[CrossRef]

IRIS Proc. (1)

J. R. Maxwell, L. Wilkins, IRIS Proc. 22, 101 (1978).

J. Opt. Soc. Am. (3)

Opt. Lett. (1)

Other (4)

D. Middleton, An Introduction to Statistical Communications Theory (McGraw-Hill, New York, 1960).

R. Roberts, “Signal Processing Techniques,” Interstate Electronics Corp., Anaheim, Calif. (1977).

C. Stein, “Approximation of Improper Prior Measures by Prior Probability Measures,” Department of Statistics, Stanford University Tech. Rept. 12 (1964).

M. Abramowitz, I. R. Stegun, Eds., Handbook of Mathematical Functions (Dover, New York, 1968, Chap. 14.

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Figures (4)

Fig. 1
Fig. 1

Normalized distribution of radiometric intensity (measured in degrees) for a series of IR scenes (8–14 μm) contains approximately 105 data samples, 8 bits each (a). (b) Expanded scale of (a) showing existence of data points past 300 deg.

Fig. 2
Fig. 2

(a) Three-parameter distribution for Gaussian data and (b) Gaussian distribution. No difference is observable.

Fig. 3
Fig. 3

Analysis of data of Fig. 1. (a) and (b) Inputs in histogram form, which two- and three-moment descriptions are generated. Three-moment description in (a) is shown as (c). Skewness is less than 1% in the main lobe, while reaches >10% in the tails. Gaussian description of (a) is shown in (d). The three-parameter distribution provides a larger variance.

Fig. 4
Fig. 4

Analysis of Fig. 3(a) using only 20% of the samples considered in Fig. 3. (a) Three-moment description; (b) the Gaussian. Using this lower information description, it can be seen that the three-moment description of the smaller number of data points predicts much the same distribution as the Gaussian (two moments) description employing the larger number of samples. Though both provide the same information, the three-parameter distribution is able to do so with fewer samples, since it uses more information from the moments.

Equations (44)

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ϕ ( k ) = 0 exp ( i k I ) H ( I ) d I ,
ϕ ( k ) = 1 + i k I - k 2 2 ! + + ( i k ) n n ! I n + ,
ψ ( f , g ) = - I ( x , y ) exp [ 2 π i ( f x , g y ) ] d x d y ,
W ( f , g ) = ψ ( f , g ) ψ * ( f , g ) ,
I n = area I n ( x , y ) d x d y ,
I n = A { ψ ( f , g ) exp ( 2 π i ( f x + g y ) d f d g } n d x d y ,
I n = A { l = 1 n ψ ( f l g l ) × exp [ 2 π i ( f l x + g l y ) ] d f l d g l } d x d y ,
exp ( 2 π i l = 1 n f l x + g l y ) d x d y = δ ( l = 1 n f l ) δ ( l = 1 n g l ) .
I n = ψ ( f 1 g 1 ) ψ ( f n g n ) δ ( l = 1 n f l ) δ ( l = 1 n g l ) × d f 1 d g 1 d f n d g n ,
I n = [ l = 1 n - 1 ψ ( f l g l ) ] × ψ * ( l = 1 n - 1 f l , l = 1 n - 1 g l ) l = 1 n = 1 d f l d g l
I = A I ( x , y ) d x d y = ψ ( 0 , 0 ) = [ W ( 0 , 0 ) ] 1 / 2 ,
I 2 = A I 2 ( x , y ) d x d y I 2 ( x , y ) d x d y = ψ ( f , g ) ψ * ( f , g ) d f d g = W ( f , g ) d f d g ,
I 3 = I 3 ( x , y ) d x d y = ψ ( f 1 g 1 ) ψ ( f 2 g 2 ) ψ * ( f 1 + f 2 , g 1 + g 2 ) × d f 1 d g 1 d f 2 d g 2 .
I 3 = ψ ( f 1 g 1 ) ψ ( f 2 g 2 ) ψ * ( f 1 + f 2 , g 1 + g 2 ) × d f 1 d g 1 d f 2 d g 2 ;
ψ ( f , g ) = X ( f , g ) + i Y ( f , g ) ,
X ( f , g ) = X ( - f , - g )             Y ( f , g ) = - Y ( - f , - g ) .
I 3 = d f 1 d g 1 d f 2 d g 2 { [ X ( f 1 , g 1 ) X ( f 2 , g 2 ) X ( f 1 + f 2 , g 1 + g 2 ) + X ( f 1 , g 1 ) Y ( f 2 , g 2 ) Y ( f 1 + f 2 , g 1 + g 2 ) + Y ( f 1 , g 1 ) X ( f 2 , g 2 ) Y ( f 1 + f 2 , g 1 + g 2 ) - Y ( f 1 , g 1 ) Y ( f 2 , g 2 ) X ( f 1 + f 2 , g 1 + g 2 ) ] + i [ X ( f 1 , g 1 ) X ( f 2 , g 2 ) Y ( f 1 + f 2 , g 1 + g 2 ) + X ( f 1 g 1 ) Y ( f 2 g 2 ) X ( f 1 + f 2 , g 1 + g 2 ) + Y ( f 1 g 1 ) X ( f 2 g 2 ) X ( f 1 + f 2 , g 1 + g 2 ) - Y ( f 1 g 1 ) Y ( f 2 g 2 ) Y ( f 1 + f 2 , g 1 + g 2 ) ] } .
X ( f 1 , g 1 ) X ( f 1 + f 2 , + g 1 + g 2 ) d f 1 d g 1 , Y ( f 1 , g 1 ) Y ( f 1 + f 2 , g 1 + g 2 ) d f 1 d g 1 ,
Y ( f 2 , g 2 ) [ X ( f 1 g 1 ) X ( f 1 + f 2 , g 1 + g 2 ) d f 1 d g 1 ] d f 2 d g 2 , Y ( f 2 , g 2 ) [ Y ( f 1 , g 1 ) Y ( f 1 + f 2 , g 1 + g 2 ) d f 1 d g 1 ] d f 2 d g 2 ,
I 3 = d f 1 d g 1 d f 2 d g 2 × X ( f 1 g 1 ) X ( f 2 g 2 ) X ( f 1 + f 2 , g 1 + g 2 ) + X ( f 1 g 1 ) Y ( f 2 g 2 ) Y ( f 1 + f 2 , g 1 + g 2 ) ] , + Y ( f 1 g 1 ) X ( f 2 g 2 ) Y ( f 1 + f 2 , g 1 + g 2 ) - Y ( f 1 g 1 ) Y ( f 2 g 2 ) X ( f 1 + f 2 , g 1 + g 2 ) .
ϕ ( k ) = 1 + i k I - k 2 2 ! I 2 + + ( i k ) n n ! I n + ,
P ( I ) = λ 0 exp j = 1 n λ j I j ,
P ( I ) = μ - 1 exp ( - I / μ )
P ( I ) = λ 0 exp ( - λ 1 I - λ 2 I 2 - λ 3 I 3 ) .
1 = λ 0 exp ( - λ 1 I - λ 2 I 2 - λ 3 I 3 ) d I ,
I = λ 0 exp ( - λ 1 I - λ 2 I 2 - λ 3 I 3 ) I d I ,
I 2 = λ 0 exp ( - λ 1 I - λ 2 I 2 - λ 3 I 3 ) I 2 d I ,
I 3 = λ 0 exp ( - λ 1 I - λ 2 I 2 - λ 3 I 3 ) I 3 d I .
I = - λ 1 exp ( - λ 1 I - λ 2 I 2 - λ 3 I 3 ) d I exp ( - λ 1 I - λ 2 I 2 - λ 3 I 3 ) d I I = - λ 1 ln [ exp ( - λ 1 I - λ 2 I 2 - λ 3 I 3 ) d I ] I = - λ 1 ln P ( I ) ; P ( I ) = exp ( - λ 1 I - λ 2 I 2 - λ 3 I 3 ) d I }
I 2 = - λ 2 ln P ( I ) ,
I 3 = - λ 3 ln P ( I ) .
P ( I ) 1 2 exp ( - λ 1 I 0 - λ 2 I 0 2 - λ 3 I 0 3 ) · [ ( - λ 2 λ 3 ) - ( - λ 2 λ 1 ) ] ,
I 0 = - λ 2 3 λ 3 + 1 ( 3 λ 3 ) 1 / 2 ( λ 2 2 3 λ 3 - λ 1 ) 1 / 2 .
I = λ 1 I 0 λ 2 + I 0 + 2 λ 2 I 0 I 0 λ 1 + 3 λ 3 I 0 2 I 0 λ 1 + [ λ 2 - 1 - λ 2 λ 3 - ( - λ 1 λ 2 ) ] ,
I 2 = λ 1 I 0 λ 2 + 2 λ 2 I 0 I 0 λ 2 + I 0 2 + 3 λ 3 I 0 2 I 0 λ 2 + [ - 1 λ 3 - ( - λ 1 λ 2 2 ) - λ 2 λ 3 - ( - λ 1 λ 2 ) ] ,
I 3 = λ 1 I 0 λ 3 + 2 λ 2 I 0 I 0 λ 3 + I 0 3 + 3 λ 3 I 0 2 I 0 λ 3 + λ 2 λ 3 - 2 [ - λ 2 λ 3 - ( - λ 1 λ 2 ) ] ,
I 0 λ 1 = - 1 6 λ 3 [ ( λ 2 3 λ 3 ) 2 - λ 1 3 λ 3 ] - 1 / 2 , I 0 λ 2 = - 1 3 λ 3 + λ 2 9 λ 3 2 [ ( λ 2 3 λ 3 ) - 2 - λ 1 3 λ 3 ] - 1 / 2 , I 0 λ 3 = λ 2 3 λ 3 2 + [ - λ 2 2 9 λ 3 3 + λ 1 6 λ 3 2 ] [ ( λ 2 3 λ 3 ) 2 - λ 1 3 λ 3 ] - 1 / 2 .
I = λ 1 2 λ 2 - 2 λ 2 3 λ 3 ,
I 2 = - 2 λ 1 3 λ 3 + 4 λ 2 2 9 λ 3 2 ,
I 3 = 2 λ 2 λ 1 3 λ 3 2 - 8 λ 2 3 27 λ 3 3 .
λ 3 = 4 λ 2 2 - 6 λ 2 I + 3 λ 1 ,
λ 2 = 2 λ 2 ( I 2 - I 2 ) 1 / 2 .
λ 2 = - [ ( I 3 - C Ω - 2 I ¯ Ω - I ¯ 3 - 3 K I ¯ 2 Ω ) ( C K 2 - K ) ] - 1 ,
C = - 2 ( I 2 - I 2 ) 1 / 2 , K = 4 - 6 I - 6 ( I 2 - I 2 ) 1 / 2 , I ¯ = - 1 3 K + [ ( 1 3 K ) 2 - C 3 K ] 1 / 2 , Ω = 1 3 K 2 + 1 2 [ ( 1 3 K ) 2 = C 3 K ] - 1 / 2 ( - 2 9 K 3 + C 3 K 2 ) .

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