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A self-calibrating phase-shifting algorithm based on the natural demodulation of two-dimensional fringe patterns

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Abstract

A new method of estimating the phase-shift between interferograms is introduced. The method is based on a recently introduced two-dimensional Fourier-Hilbert demodulation technique. Three or more interferogram frames in an arbitrary sequence are required. The first stage of the algorithm calculates frame differences to remove the fringe pattern offset; allowing increased fringe modulation. The second stage is spatial demodulation to estimate the analytic image for each frame difference. The third stage robustly estimates the inter-frame phase-shifts and then uses the generalised phase-shifting algorithm of Lai and Yatagai to extract the offset, the modulation and the phase exactly. Initial simulations of the method indicate that high accuracy phase estimates are obtainable even in the presence of closed or discontinuous fringe patterns.

©2001 Optical Society of America

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Figures (6)

Fig. 1.
Fig. 1. The left plot shows sequential inter-frame differences (dotted connecting lines) on a unit circle representing phase angles from 0 to 2π. The right plot shows all possible interframe differences as dotted lines
Fig. 2.
Fig. 2. (a). Flowchart for automatic phase-step calibration method.
Fig. 2.
Fig. 2. b). Continuation of flowchart for automatic calibration method.
Fig. 3.
Fig. 3. Simple weight function calculated from the estimated errors in the contingent analytic function (relative to the output from the PSA). Black regions indicate zero weight, white regions have a weight of 1. Note how the weight removes the contribution from regions with large fringe curvature, from regions with discontinuous phase, and from regions with stationary phase.
Fig. 4.
Fig. 4. Fringe pattern used for testing the phase-shift calibration algorithm. Note the closed fringes, the rapid fringe spacing variation and the fringe discontinuity – factors which defeat many other techniques. The sampled intensity varies between 0 and 255 grey levels.
Fig. 5.
Fig. 5. Phase error showing the classic second harmonic structure, and the disappearance of the vertical half-period discontinuity. Note that the phase error is just ±0.0068 radian, but is shown here normalised.

Tables (1)

Tables Icon

Table 1. Performance of phase-shift calibrating algorithm applied to fringe pattern of Fig. 4

Equations (32)

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f n ( x , y ) = a ( x , y ) + b ( x , y ) cos [ ψ ( x , y ) + δ n ] .
N M 3 M + N 1 or N 3 M 1 M 1 or M N 1 N 3 .
b ( x , y ) exp [ i ψ ( x , y ) + i δ n ] ,
g n m = f n f m = g m = b { cos [ ψ + δ n ] cos [ ψ + δ m ] }
$ { f ( x , y ) } = F 1 { exp [ i ϕ ] F { f ( x , y ) } } .
F { f ( x , y ) } = F ( u , v ) = + + f ( x , y ) exp [ 2 π i ( u x + v y ) ] d x d y F 1 { F ( u , v ) } = f ( x , y ) = F ( u , v ) [ + 2 π i ( u x + v y ) ] d u d v } .
u = q cos ϕ v = q sin ϕ } .
$ { g n m } i exp [ i β ] b [ sin ( ψ + δ n ) sin ( ψ + δ m ) ]
ψ ( x , y ) = ψ 00 + 2 π u 0 ( x x 0 ) + 2 π v 0 ( y y 0 ) +
ψ ( x , y ) x x = x 0 y = y 0 2 π u 0 ψ ( x , y ) y x = x 0 y = y 0 2 π v 0 } .
g n m x b ( sin ( ψ + δ n ) + sin ( ψ + δ m ) ) ψ x g n m y b ( sin ( ψ + δ n ) + sin ( ψ + δ m ) ) ψ y }
g n m y g n m x v 0 u 0 = tan β n m
exp [ i β e ] = cos β e + i sin β e = ± u 0 + i v 0 u 0 2 + v 0 2 .
g n m = 2 b sin [ ψ + ( δ n + δ m ) 2 ] sin [ ( δ n δ m ) 2 ]
V { g n m } = i exp [ i β e ] $ { g n m } 2 b exp [ i ( β β e ) ] cos [ ψ + ( δ n + δ m ) 2 ] sin [ ( δ n δ m ) 2 ]
h ( x , y ) = exp [ i ( β β e ) ] = { + exp [ i ε ] , β e = β ε exp [ i ε ] , β e = β ε + π
g ˜ n m = V { g n m } + i g n m
g ˜ n m = 2 b sin [ ( δ n δ m ) 2 ] ( h cos [ ψ + ( δ n + δ m ) 2 ] + i sin [ ψ + ( δ n + δ m ) 2 ] )
= 2 b sin [ ( δ n δ m ) 2 ] exp { i h cos [ ψ + ( δ n + δ m ) 2 ] + i π 2 } .
α n m = Arg ( g ˜ n m ) [ ψ + ( δ n + δ m ) 2 ] h + π 2
α n m α m k = [ ψ + ( δ n + δ m ) 2 ] h [ ψ + ( δ m + δ k ) 2 ] h = [ ( δ n δ k ) 2 ] h .
δ n δ k mean = 2 S α n m α m k d x d y S d x d y
α n m α m k α n m α m k = h ( x , y ) δ n δ k δ n δ k = h ( x , y ) sgn ( δ n δ k ) .
h g ( x , y ) = h n ̂ k ̂ ( x , y ) = sgn ( α n ̂ m ̂ α m ̂ k ̂ ) ,
p n k ( x , y ) = exp [ 2 i h { Arg ( g ˜ n m ) Arg ( g ˜ m k ) } ] exp [ i ( δ n δ k ) ] .
( δ n δ k ) ¯ phase mean = Arg [ p n k ¯ ] = Arg [ S p n k ( x , y ) d x d y S d x d y ] = Arg [ S p n k ( x , y ) d x d y ]
γ n k = ( δ n δ k ) ¯ phase weighted mean = Arg [ w p n k ¯ ] = Arg [ S p n k ( x , y ) w 2 ( x , y ) d x d y S w 2 ( x , y ) d x d y ]
N ! 2 ( N 2 ) !
{ γ 12 , γ 23 , γ 34 , γ 45 , γ 13 , γ 24 , γ 35 , γ 14 , γ 25 , γ 15 ,
δ 1 = 0 , δ 2 = δ 1 + γ 21 , δ 3 = δ 2 + γ 32 , δ 4 = δ 3 + γ 43 , δ 5 = δ 4 + γ 54
σ n k 2 = S p n k w p ¯ n k 2 w 2 ( x , y ) d x d y S w 2 ( x , y ) d x d y .
σ n k 2 = 1 w p ¯ n k 2 .
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