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Estimation of multiple phases in holographic moiré in presence of harmonics and noise using minimum-norm algorithm

Abhijit Patil and Pramod Rastogi

Open Access Open Access

Abstract

The paper proposes a novel approach for estimating multiple phases in holographic moiré. The need to design such an algorithm is necessitated by the development of optical configurations containing two phase stepping devices, e.g. PZTs, with a view to measure simultaneously two phase distributions. The approach consists of first applying minimum-norm algorithm to extract phase steps imparted to the PZTs. Salient feature of the algorithm lies in its ability to handle nonsinusoidal waveforms and noise. This approach also provides the flexibility of using arbitrary phase steps, a feature most commonly attributed to generalized phase shifting interferometry. Once the phase steps are estimated for each PZT, the Vandermonde system of equations is designed to estimate the phase distributions.

©2005 Optical Society of America

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Figures (6)
Fig. 1.
Fig. 1. Schematic of the optical setup in holographic moiré.

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Fig. 2.
Fig. 2. Fringe map corresponding to a) κ=1 and pure signal, a) κ=1 and 10 SNR, a) κ=2 and pure signal, a) κ=2 and 10 SNR.

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Fig. 3.
Fig. 3. Plot for phase step values α and β (in degrees) obtained at an arbitrary pixel location on a data frame for different values of N and m using the forward approach. During the simulation the phase steps are assumed to be α=45° and β=70°.

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Fig. 4.
Fig. 4. Plot for phase step values α and β (in degrees) obtained at an arbitrary pixel location on a data frame for different values of N and m using the forward-backward approach. During the simulation the phase steps are assumed to be α=45° and β=70°.

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Fig. 5.
Fig. 5. Plots show typical error in computation of phase distribution a) φ 1 (in radians), and b) φ 2 (in radians), for phase step obtained from Fig. 4(c) for 30dB noise.

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Fig. 6.
Fig. 6. Plot shows wrapped phase for a) φ 1 and b) φ 2 for phase step obtained from Fig. 4(c) for 30dB noise.

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Equations (39)

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I ( t ) = I dc + ∑ k = − 1 − κ a k exp [ ik ( φ 1 + t α ) ] + ∑ k = 1 κ a k exp [ ik ( φ 1 + t α ) ] +
∑ k = − 1 − κ b k exp [ ik ( φ 2 + t β ) ] + ∑ k = 1 κ b k exp [ ik ( φ 2 + t β ) ] ;
for t = 0 , 1 , 2 , ... , m , ... , N − 1
I ( t ) = I dc + ∑ k = 1 κ ℓ k u k t + ∑ k = 1 κ ℓ k * ( u k * ) t + ∑ k = 1 κ ℘ k v k t + ∑ k = 1 κ ℘ k * ( v k * ) t + η ( t ) ;
for t = 0 , 1 , ... , m , ... , N − 1
r ( p ) = E [ I ( t ) I * ( t − p ) ]
I ( t ) = I dc + a 1 e i φ 1 e i α t + a 1 e − i φ 1 e − e α t + b 1 e i φ 2 e i β t + b 1 e − i φ 2 e − i β t + η ( t )
I * ( t − p ) = I dc + a 1 e i φ 1 e i α ( t − p ) + a 1 e − φ 1 e − i α ( t − p ) +
b 1 e i φ 2 e i β ( t − p ) + b 1 e − i φ 2 e − i β ( t − p ) + η * ( t − p )
r ( p ) = E { I dc 2 + c 1 + e i α p ( a 1 2 + c 2 ) + e − i α p ( a 1 2 + c 3 ) + e i β p ( b 1 2 + c 4 ) + e − i β p ( b 1 2 + c 5 ) + η ( t ) η * ( t − p ) }
r ( p ) = A 0 2 + A 1 2 e i α p + A 2 2 e − i α p + A 3 2 e i β p + A 4 2 e − i β p + σ 2 δ p , 0
E [ η ( k ) η * ( j ) ] = σ 2 δ k , j
r ( p ) = E [ I ( t ) I * ( t − p ) ] = ∑ n = 0 4 κ A n 2 e i ω n p + σ 2 δ p , 0
R I = E [ I * ( t ) I ( t ) ] = [ r ( 0 ) r * ( 1 ) r * ( 2 ) . r * ( m − 1 ) r ( 1 ) r ( 0 ) . . . r ( 2 ) . . . . . . . . r * ( 1 ) r ( m − 1 ) . . . r ( 0 ) ]
R I = A P A c ︸ R s + σ 2 I ︸ R ε
P = [ A 0 2 0 . 0 0 A 1 2 . . . . . . 0 . . A m 2 ]
R I G = G [ A 4 κ + 1 2 0 . . 0 0 A 4 κ + 2 2 . . . . . . . . . . . . 0 0 0 . . A m 2 ] = σ 2 G = AP A c G + σ 2 G
A c G = 0
R ( G ) = N ( A c )
S c G = 0
R ( S ) = R ( A )
a T ( z − 1 ) G ̂ G ̂ c a ( z ) = 0
a T ( z − 1 ) [ 1 g ̂ ] = 0
S ̂ = [ χ c S ¯ ] } m − 1 } 1
S ̂ c [ 1 g ̂ ] = 0
S ¯ c g ̂ = − χ
g ̂ = − S ¯ ( S ¯ c S ¯ ) − 1 χ
I = S ̂ c S ̂ = χ χ c + S ¯ c S ¯
∥ χ ∥ 2 ≠ 1
rank ( S ¯ ) = n
I ( x , y ; t ) = I dc + a − 1 exp [ − i ( φ 1 + t α ) ] + a 1 exp [ i ( φ 1 + t α ) ] +
a − 2 exp [ − 2 i ( φ 1 + t α ) ] + a 2 exp [ 2 i ( φ 1 + t α ) ] +
b − 1 exp [ − i ( φ 2 + t β ) ] + b 1 exp [ i ( φ 2 + t β ) ] +
b − 2 exp [ − 2 i ( φ 2 + t β ) ] + b 2 exp [ 2 i ( φ 2 + t β ) ] + η ( t )
φ 1 ( x , y ) = 2 π λ ( p ′ − x ) 2 + ( p ′ − y ) 2 + φ R 1
φ 2 ( x , y ) = 2 π λ ( p ″ − x ) 2 + ( p ′ − y ) 2 + φ R 2 .
R ̂ I = 1 N ∑ t = m N [ I * ( t − 1 ) I * ( t − 2 ) . . I * ( t − m ) ] [ I ( t − 1 ) I ( t − 2 ) . . I ( t − m ) ]
R ̂ I = 1 2 N ∑ t = m N { [ I * ( t − 1 ) I * ( t − 2 ) . I * ( t − m ) ] [ I ( t − 1 ) I ( t − 2 ) . . I ( t − m ) ] + [ I * ( t − m ) . I * ( t − 2 ) I * ( t − 1 ) ] [ I ( t − m ) . . I ( t − 2 ) I ( t − 1 ) ] }
[ e i κ α 0 e − i κ α 0 e i κ β 0 . e i ( κ − 1 ) α 0 . . 1 e i κ α 1 e − i κ α 1 e i κ β 1 . e i ( κ − 1 ) α 1 . . 1 . . . . . . . . . . . . . . . . e i κ α ( N − 1 ) e − i κ α ( N − 1 ) e i κ β ( N − 1 ) . e i ( κ − 1 ) α ( N − 1 ) . . 1 ] [ ℓ κ ℓ κ * ℘ κ . I dc ] = [ I 0 I 1 I 2 . I N − 1 ]
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