Abstract

An alternative method for statistical interpolation is formalized. A new theorem is proved, providing theoretical basis for optimizing statistical accuracy in successively conditioned rendering applications. The theorem is empirically validated by two simulations, each comparing two different statistical interpolators. The interpolators are used to model high- resolution phase fluctuations over finite apertures. The theorem correctly predicts which interpolator is more optimal, based on empirical trials with greater than 99.9% certainty. The theorem is suitable as a quick alternative to the Monte Carlo optimization techniques used previously.

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References

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  1. M. C. Roggemann and B. Welsh, Imaging Through Turbulence (CRC Press, 1996).
  2. R. G. Lane, A. Glindemann, and J. C. Dainty, "Simulation of a Kolmogorov phase screen," Waves Random Media 2, 209-224 (1992).
    [CrossRef]
  3. C. M. Harding, R. A. Johnston, and R. G. Lane, "Fast simulation of a Kolmogorov phase screen," Appl. Opt. 28, 2161-2170 (1999).
    [CrossRef]
  4. A. Papoulis and S. Unnikrishna Pillai, Probability, Random Variables and Stochastic Processes, 4th ed. (McGraw-Hill, 2002).
  5. G. Strang, Linear Algebra and Its Applications, 3rd ed. (Harcourt Brace Jovanovich College Publishers, 1988).
  6. F. Cucker and A. Gonzalez Corbalan, "An alternate proof of the continuity of the roots of a polynomial," Am. Math. Monthly 96, 342-345 (1989).
    [CrossRef]

1999 (1)

C. M. Harding, R. A. Johnston, and R. G. Lane, "Fast simulation of a Kolmogorov phase screen," Appl. Opt. 28, 2161-2170 (1999).
[CrossRef]

1992 (1)

R. G. Lane, A. Glindemann, and J. C. Dainty, "Simulation of a Kolmogorov phase screen," Waves Random Media 2, 209-224 (1992).
[CrossRef]

1989 (1)

F. Cucker and A. Gonzalez Corbalan, "An alternate proof of the continuity of the roots of a polynomial," Am. Math. Monthly 96, 342-345 (1989).
[CrossRef]

Corbalan, A. Gonzalez

F. Cucker and A. Gonzalez Corbalan, "An alternate proof of the continuity of the roots of a polynomial," Am. Math. Monthly 96, 342-345 (1989).
[CrossRef]

Cucker, F.

F. Cucker and A. Gonzalez Corbalan, "An alternate proof of the continuity of the roots of a polynomial," Am. Math. Monthly 96, 342-345 (1989).
[CrossRef]

Dainty, J. C.

R. G. Lane, A. Glindemann, and J. C. Dainty, "Simulation of a Kolmogorov phase screen," Waves Random Media 2, 209-224 (1992).
[CrossRef]

Glindemann, A.

R. G. Lane, A. Glindemann, and J. C. Dainty, "Simulation of a Kolmogorov phase screen," Waves Random Media 2, 209-224 (1992).
[CrossRef]

Harding, C. M.

C. M. Harding, R. A. Johnston, and R. G. Lane, "Fast simulation of a Kolmogorov phase screen," Appl. Opt. 28, 2161-2170 (1999).
[CrossRef]

Johnston, R. A.

C. M. Harding, R. A. Johnston, and R. G. Lane, "Fast simulation of a Kolmogorov phase screen," Appl. Opt. 28, 2161-2170 (1999).
[CrossRef]

Lane, R. G.

C. M. Harding, R. A. Johnston, and R. G. Lane, "Fast simulation of a Kolmogorov phase screen," Appl. Opt. 28, 2161-2170 (1999).
[CrossRef]

R. G. Lane, A. Glindemann, and J. C. Dainty, "Simulation of a Kolmogorov phase screen," Waves Random Media 2, 209-224 (1992).
[CrossRef]

Papoulis, A.

A. Papoulis and S. Unnikrishna Pillai, Probability, Random Variables and Stochastic Processes, 4th ed. (McGraw-Hill, 2002).

Pillai, S. Unnikrishna

A. Papoulis and S. Unnikrishna Pillai, Probability, Random Variables and Stochastic Processes, 4th ed. (McGraw-Hill, 2002).

Roggemann, M. C.

M. C. Roggemann and B. Welsh, Imaging Through Turbulence (CRC Press, 1996).

Strang, G.

G. Strang, Linear Algebra and Its Applications, 3rd ed. (Harcourt Brace Jovanovich College Publishers, 1988).

Welsh, B.

M. C. Roggemann and B. Welsh, Imaging Through Turbulence (CRC Press, 1996).

Am. Math. Monthly (1)

F. Cucker and A. Gonzalez Corbalan, "An alternate proof of the continuity of the roots of a polynomial," Am. Math. Monthly 96, 342-345 (1989).
[CrossRef]

Appl. Opt. (1)

C. M. Harding, R. A. Johnston, and R. G. Lane, "Fast simulation of a Kolmogorov phase screen," Appl. Opt. 28, 2161-2170 (1999).
[CrossRef]

Waves Random Media (1)

R. G. Lane, A. Glindemann, and J. C. Dainty, "Simulation of a Kolmogorov phase screen," Waves Random Media 2, 209-224 (1992).
[CrossRef]

Other (3)

M. C. Roggemann and B. Welsh, Imaging Through Turbulence (CRC Press, 1996).

A. Papoulis and S. Unnikrishna Pillai, Probability, Random Variables and Stochastic Processes, 4th ed. (McGraw-Hill, 2002).

G. Strang, Linear Algebra and Its Applications, 3rd ed. (Harcourt Brace Jovanovich College Publishers, 1988).

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Figures (6)

Fig. 1
Fig. 1

(Color online) Four-nearest-neighbors statistical interpolator, which illustrates the three generations rendered during each level of recursion by the 4-nearest-neighbors recursive interpolator. Curved arrows indicate the cycle of generations, while the black arrows indicate which prior data (tails) are used to render additional data (arrowheads). Step 0 represents the initial coarse grid of phase samples.

Fig. 2
Fig. 2

(Color online) Two-nearest-neighbors statistical interpolator, which illustrates the three generations rendered during each level of recursion by the 2-nearest-neighbors recursive interpolator, using the same graphic conventions as in Fig. 1.

Fig. 3
Fig. 3

Statistical error estimates for level 1, generation A. 7.5 × 106 simulations were performed for each interpolator to obtain maximum-likelihood estimates. Solid regions indicate 99.9% confidence intervals.

Fig. 4
Fig. 4

Statistical error attributed to each level. This plot indicates the degree of statistical error per level of statistical interpolation. The data represent 4.6 × 106 phase screen simulations per interpolator. The original phase screens were 2 × 2, resulting in 33 × 33 phase screens after five levels of recursion. Dotted lines indicate 99.9% confidence intervals.

Fig. 5
Fig. 5

Histogram of multilevel statistical error compares the distribution of statistical error for each interpolator, without regard to any particular level or generation. The 99.9% confidence interval has a mean width of ±1.56% of the estimated SERE values shown. This histogram reflects the same data shown in Fig. 4.

Fig. 6
Fig. 6

Multilevel phase covariance estimates. This plot compares the ideal theoretical phase covariance function to the actual covariance resulting from each interpolator, and reflects the same data shown in Figs. 4 and 5. The covariance at some separations appears noisy due to the low frequency of samples that possess these separations.

Equations (57)

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p w ( v ) [ ( 2 π ) M | C | ] 1 / 2 exp { v T C 1 v } ,
p w 0 ( v 0 ) [ ( 2 π ) M + 1 | C 0 | ] 1 / 2 exp { v 0 T C 1 v 0 } ,
p w 0 | w ( v 0 ) = p w 0 ( v 0 ) p w ( v ) = [ ( 2 π ) M + 1 | C 0 | ] 1 / 2 exp { v 0 T C 0 - 1 v 0 } [ ( 2 π ) M | C 0 | ] 1 / 2 exp { v T C 1 v } = K 1 ( v ) exp { v 0 T C 0 - 1 v 0 } ,
K 1 ( v ) [ | C | 2 π | C 0 | ] 1 / 2 exp { v T C 1 v }
v 0 T C 0 - 1 v 0 = i = 0 M w i j = 0 M v j s i , j = i = 0 M j = 0 M v i v j s i , j = v 0 2 s 0 , 0 + i = 1 M v i v 0 s i , 0 j = 1 M v 0 v j s 0 , j + K 2 ( v ) ,
v 0 T C 0 - 1 v 0 = v 0 2 s 0 , 0 + 2 i = 1 M v i v 0 s i , 0 + K 2 ( v ) .
a x 2 + b x + c = ( a 1 / 2 x + b 2 a 1 / 2 ) 2 + c b 2 4 a ,
v 0 T C 0 - 1 v 0 = ( s 0 , 0 1 / 2 v 0 + s 0 , 0 - 1 / 2 i = 1 M v i s i , 0 ) 2 + K 3 ( v ) = s 0 , 0 2 ( v 0 + s 0 , 0 - 1 i = 1 M v i s i , 0 ) 2 + K 3 ( v ) .
p w 0 | w ( v 0 ) = K 4 ( v ) exp { s 0 , 0 ( v 0 + s 0 , 0 - 1 i = 1 M v i s i , 0 ) 2 } ,
μ 0 = s 0 , 0 - 1 i = 1 M v i s i , 0 , σ = s 0 , 0 - 1 / 2 .
ι [ E [ ( w 0 E [ w 0 ] ) 2 ] i = 0 M E [ ( w i E [ w i ] ) 2 ] ] 1 / 2 × [ i = 0 M | ( E [ w 0 w i ] E [ w 0 ] E [ w i ] ) ( E [ w ^ 0 ι w i ] E [ w ^ 0 ι ] E [ w i ] ) | 2 ] 1 / 2 = [ E [ ( w 0 E [ w 0 ] ) 2 ] i = 0 M E [ ( w i E [ w i ] ) 2 ] ] 1 / 2 × [ i = 0 M | E [ w 0 w i ] + E [ w i ] ( E [ w ^ 0 ι ] E [ w 0 ] ) E [ w ^ 0 ι w i ] | 2 ] 1 / 2 .
ι 2 [ 1 ϒ ι ] 1 / 2 ,
ϒ ι ( j = 1 m 1 x j c 0 , j ) 2 c 0 , 0 i = 1 m 1 j = 1 m 1 x i c i , j x j
x T C x 0 , x .
x T C x = i = 0 m 1 j = 0 m 1 x i E [ ( w i ι E [ w i ι ] ) ( w j ι E [ w j ι ] ) ] x j = E [ i = 0 m 1 j = 0 m 1 x i ( w i ι E [ w i ι ] ) ( w j ι E [ w j ι ] ) x j ] = E [ i = 0 m 1 x i ( w i ι E [ w i ι ] ) j = 0 m 1 x j ( w j ι E [ w j ι ] ) ] = E [ ( i = 0 m 1 x i ( w i ι E [ w i ι ] ) ) 2 ] 0.
x T Q x = x T ( C + Δ ) x = x T C x + x T Δ x = x T C x + i = 0 m j = 0 m x i Δ i , j x j = x T C x + 2 x 0 x m Δ 0 , m .
Δ 0 , m = x T C x 2 x 0 x m = 1 2 x 0 x m i = 0 m j = 0 m x i c i , j x j = 1 2 x 0 x m i = 0 m 1 j = 0 m 1 x i c i , j x j 1 x 0 x m i = 0 m 1 x i c i , m x m x m 2 c m , m 2 x 0 x m = x T C x 2 x 0 x m 1 x 0 i = 0 m 1 x i c i , m x m c m , m 2 x 0 = λ 2 x 0 x m 1 x 0 i = 0 m 1 x i c i , m x m c m , m 2 x 0 ,
Δ 0 , m x m = λ 2 x 0 x m       2 c m , m 2 x 0 ,
x m = ± [ λ c m , m ] 1 / 2 ,
2 Δ 0 , m ( x m ) x m 2 = λ x 0 x m 3 ,
Δ 0 , m ( x m ) Δ 0 , m ( x m a ) > 0 , x m <    0 ,
Δ 0 , m ( x m ) Δ 0 , m ( x m b ) < 0 , x m > 0,
Δ 0 , m ( x m ) Δ 0 , m ( x m a ) <    0 , x m < 0 ,
Δ 0 , m ( x m ) Δ 0 , m ( x m b ) > 0 , x m > 0.
λ | λ | = x T C x | x T C x |
= ( a x ^ ) T C ( a x ^ ) | ( a x ^ ) T C ( a x ^ ) |
= a 2 x ^ T C x ^ | a 2 x ^ T C x ^ |
= x ^ T C x ^ | x ^ T C x ^ | ,
Δ + = | Δ 0 , m ( x m b ) Δ 0 , m ( x m a ) | = | ( [ λ 0 c m , m ] 1 / 2 e ^ 0 1 e ^ 0 i = 0 m 1 e ^ i c i , m ) ( [ λ 0 c m , m ] 1 / 2 e ^ 0 1 e ^ 0 i = 0 m 1 e ^ i c i , m ) | = 2 [ λ 0 c m , m ] 1 / 2 | e ^ 0 | .
Δ + = 2 [ x T C x c m , m ] 1 / 2 | x 0 | = 2 | x 0 | [ c m , m i = 0 m 1 j = 0 m 1 x i x j c i , j ] 1 / 2 = 2 | x 0 | [ c m , m ( i = 1 m 1 j = 1 m 1 x i x j c i , j + 2 x 0 j = 1 m 1 x j c 0 , j + x 0 2 c 0 , 0 ) ] 1 / 2 = 2 | x 0 | [ c m , m ( λ + 2 x 0 j = 1 m 1 x j c 0 , j + x 0 2 c 0 , 0 ) ] 1 / 2 ,
f ( x 0 ) = λ + 2 x 0 j = 1 m 1 x j c 0 , j + x 0 2 c 0 , 0 x 0 2 ,
Δ + = 2 [ c m , m f ( x 0 ) ] 1 / 2
f x 0 = 2 λ x 0 3 2 x 0 2 j = 1 m 1 x j c 0 , j ,
x 0 a = λ j = 1 m 1 x j c 0 , j .
2 f x 0 2 = 6 λ x 0 4 + 4 x 0 3 j = 1 m 1 x j c 0 , j .
2 f x 0 2 | x 0 = x 0 a = 2 λ [ j = 1 m 1 x j c 0 , j ] 4 0 ,
    f ( x 0 ) | x 0 = x 0 a = λ [ λ j = 1 m 1 x j c 0 , j ] 2 + 2 j = 1 m 1 x j c 0 , j λ j = 1 m 1 x j c 0 , j + c 0 , 0 = c 0 , 0 1 λ [ j = 1 m 1 x j c 0 , j ] 2 ,
Δ + = 2 [ c m , m c 0 , 0 ( 1 1 λ c 0 , 0 [ j = 1 m 1 x j c 0 , j ] 2 ) ] 1 / 2 .
ϒ ( x ˜ ) = 1 λ c 0 , 0 [ j = 1 m 1 x j c 0 , j ] 2 ,
ϒ ( x ˜ ) = ( j = 1 m 1 x j c 0 , j ) 2 c 0 , 0 i = 1 m 1 j = 1 m 1 x i x j c i , j .
ϒ ( x k ) = c 0 , 0           1 [ x k 2 c k , k + 2 x k j = 1 m 1 ( 1 δ j , k ) x j c j , k + i = 1 m 1 j = 1 m 1 ( 1 δ j , k ) ( 1 δ i , k ) x i x j c i , j ] 1 × [ x k c 0 , k + j = 1 m 1 ( 1 δ j , k ) x j c 0 , j ] 2 = ( c 0 , k 2 c 0 , 0 c k , k ) g 1 ( x k ) g 2 ( x k ) ,
g 1 ( x k ) = ( x k + 1 c 0 , k j = 1 m 1 ( 1 δ j , k ) x j c 0 , j ) 2
g 2 ( x k ) = x k 2 + 2 x k c k , k j = 1 m 1 ( 1 δ j , k ) x j c j , k + 1 c k , k i = 1 m 1 j = 1 m 1 ( 1 δ j , k ) ( 1 δ i , k ) × x i x j c i , j .
g 1 ( x k ) = ( x k b 1 ) 2 ,
g 2 ( x k ) = ( x k b 2 ) 2 + d 2 ,
b 1 = 1 c 0 , k    j = 1 m 1 ( 1 δ j , k ) x j c 0 , j ,
b 2 = 1 c k , k j = 1 m 1 ( 1 δ j , k ) x j c j , k ,
d 2 = b 2 2 + 1 c k , k    i = 1 m 1 j = 1 m 1 ( 1 δ j , k ) × ( 1 δ i , k ) x i x j c i , j .
ϒ x k = ( c 0 , k 2 c 0 , 0 c k , k ) g 2 g 1 g 1 g 2 g 2 2 = ( c 0 , k 2 c 0 , 0 c k , k ) { [ ( x k b 2 ) 2 + d 2 ] ( 2 ) ( x k b 1 ) [ ( x k b 2 ) 2 + d 2 ] 2 ( x k b 1 ) 2 ( 2 ) ( x k b 2 ) [ ( x k b 2 ) 2 + d 2 ] 2 } = ( 2 c 0 , k 2 c 0 , 0 c k , k ) { x k 2 ( b 1 b 2 ) + x k ( b 2 2 b 1 2 + d 2 ) [ ( x k b 2 ) 2 + d 2 ] 2 + b 1 2 b 2 b 1 b 2 2 b 1 d 2 [ ( x k b 2 ) 2 + d 2 ] 2 } ,
x k = [ 2 ( b 1 b 2 ) ] 1 [ b 1 2 b 2 2 d 2 ± [ ( b 1 2 b 2 2 d 2 ) 2 4 ( b 1 b 2 ) ( b 1 2 b 2 b 1 b 2 2 b 1 d 2 ) ] 1 / 2 ] = [ 2 ( b 1 b 2 ) ] 1 { b 1 2 b 2 2 d 2 ± [ b 1 4 + b 2 4 + d 4 + 6 b 1 2 b 2 2 4 ( b 1 3 b 2 + b 1 b 2 3 + b 1 b 2 d 2 ) + 2 ( b 1 2 b 2 2 + b 1 2 d 2 + b 2 2 d 2 ) ] 1 / 2 } = b 1 2 b 2 2 d 2 ± [ ( b 1 b 2 ) 2 + d 2 ] 2 ( b 1 b 2 ) = { b 1 ,   b 2 d 2 b 1 b 2 } .
ϒ x k | x k = b 1 + ϵ = ( 2 c 0 , k 2 c 0 , 0 c k , k ) [ ( b 1 b 2 ) 2 + d 2 ] 2 × [ ( 2 b 1 ϵ + ϵ 2 ) ( b 1 b 2 ) + ϵ ( b 2 2 b 2 2 + d 2 ) ] = ( 2 c 0 , k 2 c 0 , 0 c k , k ) [ ( b 1 b 2 ) 2 + d 2 ] 2 × [ ( b 1 2 2 b 1 b 2 + b 2 2 + d 2 ) ϵ + ( b 1 b 2 ) ϵ 2 ] = ( 2 c 0 , k 2 c 0 , 0 c k , k ) [ ( b 1 b 2 ) 2 + d 2 ] 2 × [ ( ( b 1 b 2 ) 2 + d 2 ) ϵ + ( b 1 b 2 ) ϵ 2 ] > 0 ,
ϒ x k | x k = b 1 ϵ = ( 2 c 0 , k             2 c 0 , 0 c k , k ) [ ( b 2 b 1 ) 2 + d 2 ] 2 × [ ( ( b 2 b 1 ) 2 + d 2 ) ϵ ( b 2 b 1 ) ϵ 2 ] < 0 ,
Δ min             + = 2 [ c 0 , 0 c m , m ( 1 ϒ ι ) ] 1 / 2 .
Δ i , min + = 2 [ E [ ( w 0 E [ w 0 ] ) 2 ] E [ ( w i E [ w i ] ) 2 ] ×   ( 1 ϒ ι ) ] 1 / 2 .
ι [ i = 0 M | Δ i , min + | 2 E [ ( w 0 E [ w 0 ] ) 2 ] i = 0 M E [ ( w i E [ w i ] ) 2 ] ] 1 / 2 = [ 4 E [ ( w 0 E [ w 0 ] ) 2 ] i = 0 M E [ ( w i E [ w i ] ) 2 ] ( 1 ϒ ι ) E [ ( w 0 E [ w 0 ] ) 2 ] i = 0 M E [ ( w i E [ w i ] ) 2 ] ] 1 / 2
= 2 [ 1 ϒ ι ] 1 / 2 ,
B ψ ( R ) = 0.033 ( 2 π ) 2 k 2 C n 2 Δ z × 0 κ J 0 ( κ R ) ( κ 2 + 4 π 2 / L 0 2 ) 11 / 16 d κ ,

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