Abstract

Methods in speckle imaging and adapative optics, as well as a new technique in digital image restoration, require the calculation of the Fourier phase spectrum from measurements of the differences on a two-dimensional grid of the phase spectrum. The calculation of phases from phase differences has been analyzed in the literature and relaxation mechanisms for computing the phase have been derived by least-squares analysis. In the following paper we formulate the phase reconstruction problem in terms of a vector-matrix multiplication, and we then show that previous solution methods are equivalent to this general description. We also analyze the errors in reconstruction and reconcile previously published error results based on simulations with an analytical error expression derived from Parseval’s theorem. Finally, we comment upon the rate of convergence of phase reconstructions, and discuss numerical analysis literature which indicates that the methods previously published for phase reconstruction can be made to converge much faster.

© 1979 Optical Society of America

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References

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  1. J. Wany, J. Opt. Soc. Am. 67, 383, (1977).
    [Crossref]
  2. M. Lee, J. Holmes, and J. Kerr, J. Opt. Soc. Am. 67, 1164, (1976).
    [Crossref]
  3. J. Hardy, J. Lefebvre, and C. Koliopoulos, J. Opt. Soc. Am. 66, 360, (1977).
    [Crossref]
  4. K. Knox, J. Opt. Soc. Am. 66, 1236, (1976).
    [Crossref]
  5. J. Morton, Semiannual Technical Report, University of Southern California, Report IPI-770, page 141; Los Angeles, (1977) (unpublished).
  6. D. Fried, J. Opt. Soc. Am. 67, 370, (1977).
    [Crossref]
  7. R. Hudgin, J. Opt. Soc. Am. 67, 375, (1977).
    [Crossref]
  8. R. Noll, J. Opt. Soc. Am. 68, 139, (1978).
    [Crossref]
  9. D. Young, Iterative Solution of Large Linear Systems (Academic, New York, 1971).

1978 (1)

1977 (4)

1976 (2)

K. Knox, J. Opt. Soc. Am. 66, 1236, (1976).
[Crossref]

M. Lee, J. Holmes, and J. Kerr, J. Opt. Soc. Am. 67, 1164, (1976).
[Crossref]

Fried, D.

Hardy, J.

J. Hardy, J. Lefebvre, and C. Koliopoulos, J. Opt. Soc. Am. 66, 360, (1977).
[Crossref]

Holmes, J.

M. Lee, J. Holmes, and J. Kerr, J. Opt. Soc. Am. 67, 1164, (1976).
[Crossref]

Hudgin, R.

Kerr, J.

M. Lee, J. Holmes, and J. Kerr, J. Opt. Soc. Am. 67, 1164, (1976).
[Crossref]

Knox, K.

Koliopoulos, C.

J. Hardy, J. Lefebvre, and C. Koliopoulos, J. Opt. Soc. Am. 66, 360, (1977).
[Crossref]

Lee, M.

M. Lee, J. Holmes, and J. Kerr, J. Opt. Soc. Am. 67, 1164, (1976).
[Crossref]

Lefebvre, J.

J. Hardy, J. Lefebvre, and C. Koliopoulos, J. Opt. Soc. Am. 66, 360, (1977).
[Crossref]

Morton, J.

J. Morton, Semiannual Technical Report, University of Southern California, Report IPI-770, page 141; Los Angeles, (1977) (unpublished).

Noll, R.

Wany, J.

Young, D.

D. Young, Iterative Solution of Large Linear Systems (Academic, New York, 1971).

J. Opt. Soc. Am. (7)

Other (2)

D. Young, Iterative Solution of Large Linear Systems (Academic, New York, 1971).

J. Morton, Semiannual Technical Report, University of Southern California, Report IPI-770, page 141; Los Angeles, (1977) (unpublished).

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Figures (2)

FIG. 1
FIG. 1

Phase array notation.

FIG. 2
FIG. 2

Error behavior, simulation results (Fried and Hudgin) vs Eq. (36)

Equations (41)

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s 11 1 = ϕ 11 ϕ 21 s 12 1 = ϕ 12 ϕ 22 s 13 1 = ϕ 13 ϕ 23 } s 11 2 = ϕ 11 ϕ 12 s 12 2 = ϕ 12 ϕ 13 s 13 2 = ϕ 13 ϕ 14 } etc .
ϕ = [ ϕ 11 ϕ 12 ϕ 13 ϕ 1 N ϕ 21 ϕ 22 ϕ N N ]
ϕ i = [ ϕ i 1 ϕ i 2 ϕ i 3 ϕ i N ]
ϕ = [ ϕ 1 ϕ 2 ϕ 3 ϕ N ]
[ s 11 2 s 11 2 s 13 2 · · · s 1 , N 1 2 ] = [ 1 1 0 0 0 1 1 0 0 0 0 1 1 0 0 · · · · · · · · · · 0 1 1 ] [ ϕ 11 ϕ 12 ϕ 13 ϕ 14 · · · ϕ N ] ,
s 1 2 = [ D 1 ] ϕ 1
s i 2 = [ D 1 ] ϕ i ,
s i 2 = [ s i 1 2 s i 2 2 s i 3 2 s i 2 , N 1 ] ,
s i 1 = ϕ i ϕ i + 1 ,
s i 1 = [ s i 1 1 s i 2 1 s i 3 1 s i N 1 ]
s = [ P ] ϕ ,
s = [ P ] ϕ + n .
ϕ = { [ P ] T [ P ] } 1 [ P ] T s ,
[ P ] T [ P ] ϕ = [ P ] T s .
[ P ] = [ P 1 P 2 ] ,
[ P ] T [ P ] = [ P 1 ] T [ P 1 ] + [ P 2 ] T [ P 2 ]
[ P 2 ] T [ P 2 ] = [ I I 0 0 . 0 I 2 I I 0 0 I 2 I I · · 0 I 2 I I 0 I I ] .
[ D 1 T D + I ] = [ 2 1 0 · · · · · · · 0 1 3 1 0 · 0 1 3 1 · · · · · · 0 · 1 3 1 0 · · · · · · · · 0 1 2 ] .
[ D 1 T D + 2 I ] = [ 3 1 0 · · · · · · · 0 1 4 1 0 · 0 1 4 1 · · · · 0 · 1 4 1 0 · · · · · · · · 0 1 3 ] .
rhs ( j ) = [ D 1 ] T s j 2 + s j 1 s j 1 1 .
component ( k ) rhs ( j ) = s j k 2 s j k 1 2 + s j k 1 s j 1 , k 1 .
( lhs ) ( j ) = [ I ] ϕ j 1 + [ D 1 T D + 2 I ] ϕ j [ I ] ϕ j + 1 .
component ( k ) lhs ( j ) = ϕ j 1 , k ϕ j , k 1 + 4 ϕ j k ϕ j , k + 1 ϕ j + 1 , k .
4 ϕ j k ( ϕ j 1 , k ϕ j , k 1 + ϕ j + 1 , k + ϕ j , k + 1 ) = s j k 2 s j , k 1 2 + s j k 1 s j 1 , k 1 .
[ P ] T [ P ] ϕ = [ P ] T s ,
E 2 = σ n 2 j , k , l , m , n ( b jkmn l ) 2 ,
ϕ ̂ = [ B ] s = { [ P ] T [ P ] } 1 [ P ] T s ,
ϕ ̂ j k = l , m , n b j m , k n l s m n l .
B r s l = j , k = 0 N 1 b j k l exp ( i 2 π N ( j r + k s ) ) ,
B r s 1 = 1 exp ( i 2 π r / N ) 2 cos ( 2 π r / N ) 2 cos ( 2 π s / N ) ,
B r s 2 = 1 exp ( i 2 π s / N ) 2 cos ( 2 π r / N ) 2 cos ( 2 π s / N ) .
j , k ( b j k l ) 2 = 1 N 2 r , s ( B r s l ) 2 .
E 2 = σ n 2 N 2 { r , s 4 sin 2 ( π r / N ) [ 4 2 cos ( 2 π r / N ) 2 cos ( 2 π s / N ) ] 2 + r , s 4 sin 2 ( π s / N ) [ 4 2 cos ( 2 π r / N ) 2 cos ( 2 π s / N ) ] 2 } = σ n 2 N 2 { r , s [ 2 sin ( π r / N ) 4 2 cos ( 2 π r / N ) 2 cos ( 2 π s / N ) ] 2 + r , s [ 2 sin ( π s / N ) 4 2 cos ( 2 π r / N ) 2 cos ( 2 π s / N ) ] 2 } .
e 2 ( N ) = a ln ( N ) + β ,
[ A ] ϕ = [ P ] s = s ,
[ A ] = [ L ] + [ D ] + [ R ] ,
ϕ m + 1 = [ G ] ϕ m + [ H ] s ,
[ G ] = [ D ] 1 ( [ R ] + [ L ] , [ H ] = [ D ] 1 .
ϕ j k ( n + 1 ) = 1 4 ( ϕ j 1 , k ( m ) + ϕ j + 1 , k ( m ) + ϕ j , k 1 ( m ) + ϕ j , k + 1 ( m ) ) + 1 4 ( s j k 1 s j 1 , 1 1 + s j k 2 s j , k 1 2 ) ,
[ G ] = ( [ D ] + [ L ] ) 1 [ R ] , [ H ] = ( [ D ] + [ L ] ) 1 ,
[ G ] = ( [ D ] + ω [ L ] ) 1 { ( 1 ω ) [ D ] ω [ R ] } , [ H ] = ω ( [ D ] + ω [ L ] ) 1 ,