Abstract

Requirements for adaptive optics and compensating imaging systems lead to wave front reconstruction problems which we formulate as generalized least-squares problems. For a given array of phase-difference measurements, we construct explicit and exact solutions for the least-squares wave front error. Of particular interest are solutions with minimum norm. Two different discretizations for the gradient are used and the reasons for the different results are given.

© 1980 Optical Society of America

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References

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  1. H. Margenau and G. M. Murphy, Inc., The Mathematics of Physics and Chemistry (Van Nostrand, New York, 1956), Chap. 6.
  2. G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1961), Chap. 15.
  3. R. H. Hudgin, “Wave-front reconstruction for compensated imaging,” J. Opt. Soc. Am. 67, 374–378 (1977).
    [CrossRef]
  4. D. L. Fried, “Least-squares fitting a wave-front distortion estimate to an array of phase-difference measurements,” J. Opt. Soc. Am. 67, 370–375 (1977).
    [CrossRef]
  5. A. Ben-Israel and N. E. Greville, Generalized Inverses: Theory and Applications (Wiley, New York, 1974), Chaps. 3 and 5.
  6. C. R. Rao and S. K. Mitra, Generalized Inverse of Matrices and its Applications (Wiley, New York, 1971), Chap. 11.
  7. C. L. Lawson and R. J. Hanson, Solving Least Squares Problems (Prentice Hall, Englewood Cliffs, New Jersey, 1974), Chap. 25.
  8. International Mathematical & Statistical Libraries, Inc., Sixth Floor GNB Building, 7500 Bellaire, Houston, Texas 77036.
  9. N. Balabanian and T. A. Bickart, Electrical Network Theory (Wiley, New York, 1969).
  10. W. H. Kim and H. E. Meadows, Modern Network Analysis (Wiley, New York, 1971), Chap. 3.

1977 (2)

Balabanian, N.

N. Balabanian and T. A. Bickart, Electrical Network Theory (Wiley, New York, 1969).

Ben-Israel, A.

A. Ben-Israel and N. E. Greville, Generalized Inverses: Theory and Applications (Wiley, New York, 1974), Chaps. 3 and 5.

Bickart, T. A.

N. Balabanian and T. A. Bickart, Electrical Network Theory (Wiley, New York, 1969).

Fried, D. L.

Greville, N. E.

A. Ben-Israel and N. E. Greville, Generalized Inverses: Theory and Applications (Wiley, New York, 1974), Chaps. 3 and 5.

Hanson, R. J.

C. L. Lawson and R. J. Hanson, Solving Least Squares Problems (Prentice Hall, Englewood Cliffs, New Jersey, 1974), Chap. 25.

Hudgin, R. H.

R. H. Hudgin, “Wave-front reconstruction for compensated imaging,” J. Opt. Soc. Am. 67, 374–378 (1977).
[CrossRef]

Kim, W. H.

W. H. Kim and H. E. Meadows, Modern Network Analysis (Wiley, New York, 1971), Chap. 3.

Korn, G. A.

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1961), Chap. 15.

Korn, T. M.

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1961), Chap. 15.

Lawson, C. L.

C. L. Lawson and R. J. Hanson, Solving Least Squares Problems (Prentice Hall, Englewood Cliffs, New Jersey, 1974), Chap. 25.

Margenau, H.

H. Margenau and G. M. Murphy, Inc., The Mathematics of Physics and Chemistry (Van Nostrand, New York, 1956), Chap. 6.

Meadows, H. E.

W. H. Kim and H. E. Meadows, Modern Network Analysis (Wiley, New York, 1971), Chap. 3.

Mitra, S. K.

C. R. Rao and S. K. Mitra, Generalized Inverse of Matrices and its Applications (Wiley, New York, 1971), Chap. 11.

Murphy, G. M.

H. Margenau and G. M. Murphy, Inc., The Mathematics of Physics and Chemistry (Van Nostrand, New York, 1956), Chap. 6.

Rao, C. R.

C. R. Rao and S. K. Mitra, Generalized Inverse of Matrices and its Applications (Wiley, New York, 1971), Chap. 11.

J. Opt. Soc. Am. (2)

Other (8)

H. Margenau and G. M. Murphy, Inc., The Mathematics of Physics and Chemistry (Van Nostrand, New York, 1956), Chap. 6.

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1961), Chap. 15.

A. Ben-Israel and N. E. Greville, Generalized Inverses: Theory and Applications (Wiley, New York, 1974), Chaps. 3 and 5.

C. R. Rao and S. K. Mitra, Generalized Inverse of Matrices and its Applications (Wiley, New York, 1971), Chap. 11.

C. L. Lawson and R. J. Hanson, Solving Least Squares Problems (Prentice Hall, Englewood Cliffs, New Jersey, 1974), Chap. 25.

International Mathematical & Statistical Libraries, Inc., Sixth Floor GNB Building, 7500 Bellaire, Houston, Texas 77036.

N. Balabanian and T. A. Bickart, Electrical Network Theory (Wiley, New York, 1969).

W. H. Kim and H. E. Meadows, Modern Network Analysis (Wiley, New York, 1971), Chap. 3.

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Figures (7)

FIG. 1
FIG. 1

Hudgin’s discretization for N = 3, M = 12, and the corresponding matrix A, the reduced matrix Ar for x9 = 0, and the extended matrix Ae.

FIG. 2
FIG. 2

Fried’s discretization for N = 3, M = 8, and the corresponding matrix A, the reduced matrix Ar for x8 = x9 = 0, and the extended matrix Ae.

FIG. 3
FIG. 3

Error propagator for Hudgin’s discretization as function of number of points N2 for a variety of conditions.

FIG. 4
FIG. 4

Error propagator for Fried’s and Hudgin’s discretization for the minimum-norm solutions and the values from Ref. 4.

FIG. 5
FIG. 5

Example showing the difference between least-squares phase fit and least-squares gradient fit.

FIG. 6
FIG. 6

Special 9 × 9 array with corners cut off. The outer circle contains 120 gradients and 69 phases. The Inner circle contains 45 phases. A central obstruction removes four gradients and one phase.

FIG. 7
FIG. 7

Curl operator and its generalized inverse for N = 3.

Tables (2)

Tables Icon

TABLE I Wave-front error for the 69 phase-point system.

Tables Icon

TABLE II Matrix for least-squares phase fit of tilt (× 104).Tilt coefficients: [ c x c y ] = [ ( P + A + ) x x ( P + A + ) x y ( P + A + ) y x ( P + A + ) y y ] [ g x g y ] .

Equations (72)

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ϕ ( r ) = C ϕ · d s + ϕ ( r 0 ) ,
ϕ + n = g ,
( ϕ ˆ g ) 2 d x d y = minimum .
2 ϕ ˆ = · g ,
W ( x , y ) ( ϕ ˆ g ) 2 d x d y = minimum ,
· ( W ϕ ˆ ) = · ( W g )
= ϕ ˆ ϕ ,
2 = · n .
A x = g ,
A [ q + ( p 1 ) ( N 1 ) , q + ( p 1 ) N ] = 1 A [ q + ( p 1 ) ( N 1 ) , 1 + q + ( p 1 ) N ] = 1 } p = 1 , N q = 1 , N 1 A [ q + M / 2 , q ] = 1 A [ q + M / 2 , q + N ] = 1 } q = 1 , M / 2
M = 2 N ( N 1 ) , K = N 2 ,
R ( A ) = N 2 1.
A [ m , n ] = 1 A [ m , n + M ] = 1 A [ m , n + N M ] = 1 A [ m , n + N M + M ] = 1 A [ m + 1 , n ] = 1 A [ m + 1 , n + M ] = 1 A [ m + 1 , n + N M ] = 1 A [ m + 1 , n + N M + M ] = 1 m = 1 + 2 ( q 1 ) + 2 ( p 1 ) ( N 1 ) n = [ q + ( p 1 ) N ] M p = 1 , N 1 q = 1 , N 1
M = 2 ( N 1 ) 2 , K = N 2 ,
R ( A ) = N 2 2.
x = A + g ,
A = [ A 1 A 2 A 3 A 4 ] ,
A + = [ A 1 A 2 ] T U T [ A 1 A 3 ] T ,
U = ( [ A 1 A 2 ] A T [ A 1 A 3 ] ) 1 ,
2 = σ 2 S / N 2 ,
A = U S V T ,
A + = V S 1 U T .
B r = [ ( A r T A r ) 1 A r T 0 ] ,
A r T A r x = A r T g ,
B = [ B r C C ] ,
C m = ( 1 / N 2 ) n = 1 k 1 B n m , m = 1 , M .
f = 1 h ( f n + 1 f n ) + O ( h 2 ) ,
f = 1 h ( 1 24 f n + 2 + 9 8 f n + 1 9 8 f n + 1 24 f n 2 ) + O ( h 4 ) .
A e x = g e .
A s ( q ) = 1 , q = 1 , K
A e = [ A A s ] ,
A s [ 1 , 1 + 2 p ] = 1 p = 0 , ( N 2 1 ) / 2 A s [ 2 , 2 + 2 p ] = 1 q = 0 , ( N 2 3 ) / 2 for odd N , A s [ 1 , 1 + 2 ( q 1 ) + N ( p 1 ) + mod ( p + 1 , 2 ) ] = 1 A s [ 2 , 1 + 2 ( q 1 ) + N ( p 1 ) + mod ( p , 2 ) ] = 1 for even N .
A e T A e x = A e T g e = A T g ,
x = ( A e T A e ) 1 A T g ,
ϕ ( x , y ) = Σ c i P i ( x , y ) .
ϕ = P c ,
g = A ϕ ,
ϕ = P c .
ϕ = A + g ,
c = P + ϕ .
c = P + A + g .
g = A P c ,
c g = ( A P ) + g ,
ϕ g = P ( A P ) + g .
c g = ( A P ) + g ,
c = P + A + g .
[ c x g c y g ] = ( 2 M ) [ 1 1 1 0 0 0 0 0 0 1 1 0 ] [ g x g y ] ,
Q [ p , q ] = 1 Q [ p , q + N 1 ] = 1 } p = 1 , ( N 1 ) 2 q = 1 , ( N 1 ) 2 Q [ q + ( p 1 ) ( N 1 ) , M / 2 + q + ( p 1 ) N ] = 1 Q [ q + ( p 1 ) ( N 1 ) , M / 2 + 1 + q + ( p 1 ) N ] = 1 } P = 1 , N 1 q = 1 , N 1.
g c = g g s .
Q g s = Q g ,
g s = Q + Q g .
g c = ( I Q + Q ) g = R g ,
A ϕ = g c .
A r d ϕ r = g r d ,
ϕ r = A r d 1 g r d = S L g r d ,
S = [ C S L C ] ,
C j = 1 N 2 i ( S L ) i j .
ϕ = S R r d g ,
ϕ = A + g .
q = Q g
σ q 2 = ( N 1 ) 2 | Q | 2 σ g 2 = 4 σ g 2 .
σ g s 2 = 1 M | Q + Q | 2 σ g 2 = N 1 2 N σ g 2 ,
A x = b
x = A 1 b .
x = ( A T A ) 1 A T b
A x b = min .
x = A T ( A A T ) 1 b
x = A + b + ( I A + A ) y ,
A x b = min ,
A x b = min ,
x = min .
A A + A = A , A + A A + = A + , ( A A + ) T = A A + , ( A + A ) T = A + A .