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An integral approach to phase shifting interferometry using a super-resolution frequency estimation method

Abhijit Patil, Rajesh Langoju, and Pramod Rastogi

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Abstract

The objective of this paper is to describe an integral approach -based on the use of a super-resolution frequency estimation method - to phase shifting interferometry, starting from phase step estimation to phase evaluation at each point on the object surface. Denoising is also taken into consideration for the case of a signal contaminated with white Gaussian noise. The other significant features of the proposal are that it caters to the presence of multiple PZTs in an optical configuration, is capable of determining the harmonic content in the signal and effectively eliminating their influence on measurement, is insensitive to errors arising from PZT miscalibration, is applicable to spherical beams, and is a robust performer even in the presence of white Gaussian intensity noise.

©2004 Optical Society of America

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Figures (7)
Fig. 1.
Fig. 1. Plot showing the magnitude of diagonal values in matrix S versus N data points. From the plot the number of harmonics κ in the signal can be computed using M = 2 Hκ+1. In figure (o) and (+) represent the diagonal values for noiseless and noisy (SNR=10 dB) signals, respectively.

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Fig. 2.
Fig. 2. Holographic moiré for κ = 1 and for noise levels (a) 0 dB (b) 10 dB , and (c) 60 dB additive white Gaussian noise.

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Fig. 3.
Fig. 3. Holographic moiré for κ=2 and for noise levels (a) 0 dB (b) 10 dB , and (c) 60 dB additive white Gaussian noise.

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Fig. 4.
Fig. 4. Plot of phase steps α and β versus noise when computation is done over (a) eighteen (b) twenty seven, and (c) thirty six frames. Plots in. (b) and (c) are obtained after applying the denoising procedure. Large number of frames are due to the presence of two PZTs (H = 2) and two harmonics κ = 2, which in turn impose lower limit on data samples (4Hκ+ 2) as eighteen for phase step estimation.

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Fig. 5.
Fig. 5. Plot of error in the computation of phase φ 1 when SNR = 30 dB obtained a) without and b) with the application of the denoising procedure.

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Fig. 6.
Fig. 6. Plot of error in the computation of phase φ 2 when SNR = 30 dB obtained a) without and b) with the application of the denoising procedure.

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Fig. 7.
Fig. 7. Wrapped phase distributions φ 1 (solid line) and φ 2 (broken line) as functions of pixel position when SNR = 30 dB. Denoising procedure has been applied for obtaining these results.

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Equations (38)

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I ( x , y ; m ) = I dc + ∑ k = − 1 − κ a k exp ⁡ [ ik ( φ 1 + mα ) ] + ∑ k = 1 κ a k exp ⁡ [ ik ( φ 1 + mα ) ] +
∑ k = − 1 − κ b k exp ⁡ [ ik ( φ 2 + mβ ) ] + ∑ k = 1 κ b k exp ⁡ [ ik ( φ 2 + mβ ) ] ; for m = 1,2 , …… , N
Φ = φ 1 − φ 2
Ψ = φ 1 + φ 2
I n ( x , y ; m ) = I dc + ∑ k = 1 κ ℓ k u k m + ∑ k = 1 κ ℓ k * ( u k * ) m + ∑ k = 1 κ ℘ k v k m + ∑ k = 1 κ ℘ k * ( v k * ) m ,
for m = n + 1 = 1,2 , …… , N
I ( z ) = ∑ n = 0 N − 1 I n z − n = ∑ n = 0 N − 1 I dc z − n + ∑ n = 0 N − 1 ∑ k = 1 κ ℓ k u k ( n + 1 ) z − n + ∑ n = 0 N − 1 ∑ k = 1 κ ℓ k * ( u k * ) ( n + 1 ) z − n +
∑ n = 0 N − 1 ∑ k = 1 κ ℘ k v k ( n + 1 ) z − n + ∑ n = 0 N − 1 ∑ k = 1 κ ℘ k * ( v k * ) ( n + 1 ) z − n
I ( z ) = I dc ( 1 − z − N 1 − z − 1 ) + ∑ k = 1 κ ℓ k e iαk ( 1 − e iαkN z − N ) [ D 1 ( z ) P 1 ( z ) ] +
∑ k = 1 κ ℓ k * e − iαk ( 1 − e −iαkN z − N ) [ D 2 ( z ) P 2 ( z ) ] +
∑ k = 1 κ ℘ k e iβk ( 1 − e iβkN z − N ) [ D 3 ( z ) P 3 ( z ) ] +
∑ k = 1 κ ℘ k * e − iβk ( 1 − e iβkN z − N ) [ D 4 ( z ) P 4 ( z ) ]
D 1 ( z ) = ∏ j = 1 , j ≠ k κ ( 1 − e iαj z − 1 ) D 2 ( z ) = ∏ j = 1 , j ≠ k κ ( 1 − e − iαj z − 1 ) , , = ∑ j = 0 κ − 1 d 1 j z − j = ∑ j = 0 κ − 1 d 2 j z − j D 3 ( z ) = ∏ j = 1 , j ≠ k κ ( 1 − e iβj z − 1 ) D 4 ( z ) = ∏ j = 1 , j ≠ k κ ( 1 − e − iβj z − 1 ) , , = ∑ j = 0 κ − 1 d 3 j z − j = ∑ j = 0 κ − 1 d 4 j z − j
P 1 ( z ) = ∏ k = 1 κ ( 1 − e iαk z − 1 ) , P 2 ( z ) = ∏ k = 1 κ ( 1 − e − iαk z − 1 ) , P 3 ( z ) = ∏ k = 1 κ ( 1 − e iβk z − 1 )
and P 4 ( z ) = ∏ k = 1 κ ( 1 − e − iβk z − 1 ) .
I ( z ) = I dc ( 1 − z − N 1 − z − 1 ) + { ∑ k = 1 κ ℓ k e iαk [ ∑ j = 0 κ − 1 d 1 j z − j − ∑ j = 0 κ − 1 d 1 j e iαkN z − ( j + N ) ] ∏ k = 1 κ ( 1 − e iαk z − 1 ) } +
{ ∑ k = 1 κ ℓ k * e − iαk [ ∑ j = 0 κ − 1 d 2 j z − j − ∑ j = 0 κ − 1 d 2 j e − iαkN z − ( j + N ) ] ∏ k = 1 κ ( 1 − e − iαk z − 1 ) } +
{ ∑ k = 1 κ ℘ k e iβk [ ∑ j = 0 κ − 1 d 3 j z − j − ∑ j = 0 κ − 1 d 3 j e − iβkN z − ( j + N ) ] ∏ k = 1 κ ( 1 − e iβk z − 1 ) } +
{ ∑ k = 1 κ ℘ k * e − iβk [ ∑ j = 0 κ − 1 d 4 j z − j − ∑ j = 0 κ − 1 d 4 j e − iβkN z − ( j + N ) ] ∏ k = 1 κ ( 1 − e − iβk z − 1 ) }
I ( z ) P ( z ) = D ( z )
P ( z ) = ( 1 − z − 1 ) ∏ k = 1 4 κ ( 1 − e iαk z − 1 ) ( 1 − e − iαk z − 1 ) ( 1 − e iβk z − 1 ) ( 1 − e − iβk z − 1 )
= ∑ k = 0 4 κ + 1 p k z − k
C 1 ( z ) = ∑ k = 1 κ I dc ( 1 − z − N ) P 1 ( z ) P 2 ( z ) P 3 ( z ) P 4 ( z )
= ∑ j = 0 4 κ C 1 j z − j − ∑ j = 0 4 κ C 1 j + N z − ( j + N )
I n ⊗ p n = D n
p n = p 0 δ ( n ) + p 1 δ ( n − 1 ) + p 2 δ ( n − 2 ) + ……. + p 4 κ + 1 δ ( n − 4 κ − 1 )
∑ k = 0 4 κ + 1 I n − k p k = D n ; for n = 0,1 , …… , N , ….. , N + 4 κ
I n ( n + 1 ) = { 0 I n ( n + 1 ) 0 for n < 0 0 ≤ n ≤ N − 1 , n ≥ N
and δ ( n ) = { 0 1 for n = 0 else where
[ I 0 ( 1 ) 0 0 . . 0 I 1 ( 2 ) I 0 ( 1 ) 0 . . 0 . . . . . . . . . . . . -------------- -------------- -------------- -------------- -------------- -------------- I 4 κ + 1 ( 4 κ + 2 ) I 4 κ ( 4 κ + 1 ) . . . I 0 ( 1 ) I 4 κ + 2 ( 4 κ + 3 ) I 4 κ + 1 ( 4 κ + 2 ) . . . I 1 ( 2 ) . . . . . . . . . . . . I N − 2 ( N − 1 ) I N − 3 ( N − 2 ) . . . I N − 4 κ − 3 ( N − 4 κ − 2 ) -------------- -------------- -------------- -------------- -------------- -------------- I N − 1 ( N ) I N − 2 ( N − 1 ) . . . I N − 4 κ − 2 ( N − 4 κ − 1 ) 0 I N − 1 ( N ) I N − 2 ( N − 1 ) . . . 0 0 . . . . 0 . . . . . 0 . . . I N − 1 ( N ) I N − 2 ( N − 1 ) 0 0 0 0 0 I N − 1 ( N ) ] [ p 0 p 1 p 2 . . . p 4 κ + 1 ] = [ D 0 D 1 . D 4 κ ------ 0 0 . . 0 ------ D N . . . . D N + 4 κ ]
[ I 4 κ + 1 ( 4 κ + 2 ) I 4 κ ( 4 κ + 1 ) . . . I 0 ( 1 ) I 4 κ + 2 ( 4 κ + 3 ) I 4 κ + 1 ( 4 κ + 2 ) . . . I 1 ( 2 ) . . . . . . I N − 3 ( N − 2 ) I N − 4 ( N − 3 ) . . . I N − 4 κ − 4 ( N − 4 κ − 3 ) I N − 2 ( N − 1 ) I N − 3 ( N − 2 ) . . . I N − 4 κ − 3 ( N − 4 κ − 2 ) ] ︸ R [ p 0 p 1 . . p 4 κ + 1 ] ︸ P = [ 0 0 0 . 0 ]
I ( x , y ; m ) = I dc + a − 1 exp ⁡ [ − i ( φ 1 + mα ) ] + a 1 exp ⁡ [ i ( φ 1 + mα ) ] ︸ T 1 +
b − 1 exp ⁡ [ − i ( φ 2 + mβ ) ] + b 1 exp ⁡ [ i ( φ 2 + mβ ) ] ︸ ; for m T 2 = 1,2 , …… , N .
Ī n = 1 r − q + 1 ∑ j = 0 N − 2 Z M ( n − j + 1 , j + 1 ) ; for n = 0 , 1,2 , ….. N − 2 ;
I n ( x , y ; m ) = ∑ k = − 2 2 a k exp ⁡ ( i { 2 π [ ( x − x 0 ) 2 + ( y − y 0 ) 2 ] λ 1 ξ + mα } ) +
∑ k = − 2 2 b k exp ⁡ ( i { 2 π [ ( x − p 0 ) 2 + ( y − y 0 ) 2 ] λ 2 ξ + mβ } )
; for m = n + 1 = 1,2 , …… N
[ e iκ α 1 e − iκ α 1 e iκ β 1 e iκ β 1 e i ( κ − 1 ) α 1 . . 1 e iκ α 2 e − iκ α 2 e iκ β 2 e iκ β 2 e i ( κ − 1 ) α 2 . . 1 . . . . . . . . . . . . . . . . e iκ α N e − iκ α N e iκ β N e − iκ β N e i ( κ − 1 ) α N . . 1 ] [ ℓ κ ℓ κ * ℘ κ . I dc ] = [ I 0 I 1 I 2 . I N − 1 ]
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