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Fast wave-front reconstruction by solving the Sylvester equation with the alternating direction implicit method

Hongwu Ren and Richard Dekany

Open Access Open Access

Abstract

Large degree-of-freedom real-time adaptive optics (AO) control requires reconstruction algorithms that are computationally efficient and readily parallelized for hardware implementation. In particular, we find the wave-front reconstruction for the Hudgin and Fried geometry can be cast into a form of the well-known Sylvester equation using the Kronecker product properties of matrices. We derive the filters and inverse filtering formulas for wave-front reconstruction in two-dimensional (2-D) Discrete Cosine Transform (DCT) domain for these two geometries using the Hadamard product concept of matrices and the principle of separable variables. We introduce a recursive filtering (RF) method for the wave-front reconstruction on an annular aperture, in which, an imbedding step is used to convert an annular-aperture wave-front reconstruction into a square-aperture wave-front reconstruction, and then solving the Hudgin geometry problem on the square aperture. We apply the Alternating Direction Implicit (ADI) method to this imbedding step of the RF algorithm, to efficiently solve the annular-aperture wave-front reconstruction problem at cost of order of the number of degrees of freedom, O(n). Moreover, the ADI method is better suited for parallel implementation and we describe a practical real-time implementation for AO systems of order 3,000 actuators.

©2004 Optical Society of America

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Figures (6)
Fig. 1.
Fig. 1. Flowchart for the RF in which the preconditioning solution step is solved through the Sylvester equation to accelerate the convergence rate of the iterative process.

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Fig. 2.
Fig. 2. Annular wave-front phase screen embedded in a 255×255 size square-aperture sampling grid. The colormap is shown in radians.

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Fig. 3.
Fig. 3. Comparison of the rTRMS performance of the RFCG algorithm for the Fried geometry, when the imbedding step is solved using the ADI and BS methods respectively. The subaperture slope SNR is equal to 2, 8, 32 and 128, respectively, for both methods from top curves to bottom ones.

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Fig. 4.
Fig. 4. RPE (in radians) after 50 iterations when solving the wave-front reconstruction for the Fried geometry using the RFCG algorithm, and the imbedding step is solved using the ADI method. The wave-front reconstruction is done in a 255×255 sized sampling grids and the subaperture slope SNR is equal to 2, 8, 32 and 128 for the RPE in image (a), (b), (c) and (d), respectively.

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Fig. 5.
Fig. 5. Comparison of the rTRMS performance of the RFD algorithm for the Fried geometry, when the imbedding step is solved using the ADI and BS methods respectively. The subaperture slope SNR is equal to 2, 8, 32 and 128, respectively, for both methods from top curves to bottom ones.

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Fig. 6.
Fig. 6. RPE (in radians) after 50 iterations when solving the wave-front reconstruction for the Fried geometry using the RFD algorithm, and the imbedding step is solved using the ADI method. The wave-front reconstruction is done in a 255×255 sized sampling grids and the subaperture slope SNR is equal to 2, 8, 32 and 128 for the RPE in image (a), (b), (c) and (d), respectively.

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Tables (1)

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Table 1. The ρj parameter for the ADI method when ε=5×10-4 and N=255

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Equations (73)

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s = P ϕ + η ,
P T P ϕ = P T s .
P = [ P 1 P 2 ] ,
r = P T s = P 1 T s x + P 2 T s y ,
P 1 = I ⊗ D 1 ,
P 2 = D 1 ⊗ I ,
Y ⊗ Z = [ y 11 Z y 12 Z … y 1 n Z y 21 Z y 22 Z … y 2 n Z ⋮ ⋮ ⋮ ⋮ y m 1 Z y m 2 Z … y mn Z ] ,
D 1 = [ 1 − 1 1 − 1 ⋱ ⋱ 1 − 1 1 − 1 ] .
P T P = ( I ⊗ D 1 ) T ( I ⊗ D 1 ) + ( D 1 ⊗ I ) T ( D 1 ⊗ I ) .
( Y ⊗ Z ) T = Y T ⊗ Z T ,
( Y ⊗ Z ) ( W ⊗ X ) = ( YW ) ⊗ ( ZX ) ,
P T P = I ⊗ A + A ⊗ I ,
A = [ 1 − 1 − 1 2 − 1 ⋱ ⋱ ⋱ − 1 2 − 1 − 1 1 ] .
( I ⊗ A + A ⊗ I ) ϕ = r .
vec A = [ a 1 a 2 ⋮ a N ] ,
( I ⊗ A + A ⊗ I ) vec ( Φ T ) = vec ( R T ) ,
vec ( AYB ) = ( B T ⊗ A ) vec Y ,
A Φ + Φ A = R .
F = [ 0.5 0.5 0.5 0.5 ⋱ ⋱ 0.5 0.5 0.5 0.5 ] .
P 1 = F ⊗ D 1 ,
P 2 = D 1 ⊗ F ,
P T P = ( F ⊗ D 1 ) T ( F ⊗ D 1 ) + ( D 1 ⊗ F ) T ( D 1 ⊗ F ) .
P T P = H ⊗ A + A ⊗ H ,
H = 1 4 [ 1 1 1 2 1 ⋱ ⋱ ⋱ 1 2 1 1 1 ] ,
( H ⊗ A + A ⊗ H ) ϕ = r .
A Φ H + H Φ A = R ,
A = M Λ A M T .
Λ A M T Φ M + M T Φ M Λ A = M T R M .
( λ u T + u λ T ) ∘ ( M T Φ M ) = M T R M ,
Φ = M ( M T R M λ u T + u λ T ) M T = M ( M T R M T H ) M T .
M mn = 2 N κ m cos ⁡ [ m ( 2 n + 1 ) π 2 N ] , ( m , n = 0 , 1 , … , N − 1 ) ,
λ m = 4 sin 2 ⁡ ( m π 2 N ) , ( m = 0 , 1 , … , N − 1 ) ,
T mn H = 4 [ sin 2 ⁡ ( m π 2 N ) + sin 2 ⁡ ( nπ 2 N ) ] , ( m , n = 0 , 1 , … , N − 1 ) .
S mn = 2 N κ m sin ⁡ [ ( m + 1 ) ( 2 n + 1 ) π 2 N ] , ( m , n = 0 , 1 , … , N − 1 ) ,
σ m = cos 2 ⁡ [ ⁡ ( m + 1 ) π 2 N ] , ( m = 0 , 1 , … , N − 1 ) .
τ m = cos 2 ( m π 2 N ) , ( m = 0 , 1 , … , N − 1 ) .
H = M Λ H M T .
Φ = M ( M T RM λ τ T + τ λ T ) M T .
T mn F = 4 [ sin 2 ( m π 2 N ) cos 2 ( n π 2 N ) + sin 2 ( n π 2 N ) cos 2 ( m π 2 N ) ] ,
( m , n = 0 , 1 , … , N − 1 ) .
A Φ k + Φ k A = R k .
r k = b k − C x k ,
( A + ρ j I ) Φ k j − 1 ⁄ 2 = R k − Φ k j − 1 ( A − ρ j I ) ,
Φ k j ( A + ρ j I ) = R k − ( A − ρ j I ) Φ k j − 1 ⁄ 2 .
ϖ a = 4 sin 2 ( π 2 N ) ,
ϖ b = 4 sin 2 [ ( N − 1 ) π 2 N ] .
h = 1 2 ( ϖ a ϖ b + ϖ b ϖ a ) ,
g ′ = 1 h + h 2 − 1 ,
g = 1 − g ′ 2 .
J = ln 4 ε ln 4 g ′ π 2 .
ρ j = ϖ a ϖ b g ′ dn ( υ j , g ) , ( j = 1 , 2 , … , J ) ,
dn ( υ j , g ) = 2 q υ j ⁄ 2 ( 1 + q 1 − υ j + q 1 + υ j ) ( 1 + 2 q ) ( 1 + q ) , ( υ j ≥ 0.5 ) ,
dn ( υ j , g ) = g ′ dn ( υ J − j + 1 , g ) , ( υ j < 0.5 ) ,
υ j = 2 j − 1 2 J ,
q = ( g ′ 4 ) 2 ( 1 + g ′ 2 2 ) .
Q x = d ,
Q = [ a 1 f 1 c 2 a 2 f 2 ⋱ ⋱ ⋱ c N − 1 a N − 1 f N − 1 c N a N ] .
L = [ 1 l 2 1 ⋱ ⋱ l N − 1 1 l N 1 ] ,
U = [ u 1 f 1 u 2 f 2 ⋱ ⋱ u N − 1 f N − 1 u N ] .
u 1 = a 1 ,
l i = c i ⁄ u i ,
u i = a i − l i f i − 1 ,
( i = 2 , 3 , … , N ) .
y 1 = d 1 ,
y i = d i − l i y i − 1 ,
( i = 2 , 3 , … , N ) ,
x N = y N ⁄ u N ,
x i = ( y i − x i + 1 f i ) ⁄ u i ,
( i = N − 1 , N − 2 , … , 1 ) .
e r = w 0 · * [ e − ( v 1 T e ) v 1 ] ,
RMS = ( e r · * e r ) 1 ⁄ 2 ,
TRMS = [ 1 n e r T e r ] 1 ⁄ 2 ,
rTRMS = [ e r T e r x T x ] 1 ⁄ 2 .
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