Abstract

The Fourier analysis of two-stage phase-shifting (TSPS) algorithms is growing in interest as a research topic, specifically, the algorithm’s insensitivity properties to various error sources. The main motivation of this paper is to propose TSPS algorithms that perform well in the face of detuning and harmonics for each of the two sets of interferograms with different or equal reference frequencies. TSPS algorithms based on the development of generalized equations consider both the frequency sampling functions that represent them and nonconstant phase shifts.

© 2013 Optical Society of America

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  1. M. Miranda, V. Álvarez-Valado, B. V. Dorrío, and H. González-Jorge, “Error propagation in differential phase evaluation,” Opt. Express 18, 3199–3209 (2010).
    [CrossRef]
  2. M. Miranda and B. V. Dorrío, “Monte Carlo based techniques of two-stage phase shifting algorithms,” Opt. Lasers Eng. 49, 439–444 (2011).
    [CrossRef]
  3. M. Miranda and B. V. Dorrío, “Fourier analysis of two-stage phase-shifting algorithms,” J. Opt. Soc. Am. A 27, 276–285 (2010).
    [CrossRef]
  4. M. Miranda, B. V. Dorrío, J. Blanco, J. Diz-Bugarin, and F. Ribas, “Characteristic polynomial theory of two-stage phase shifting algorithms,” Opt. Lasers Eng. 50, 522–528 (2012).
    [CrossRef]
  5. M. Miranda, B. V. Dorrío, J. Blanci, and J. Diz-Bugarin, “Linear error analysis of differential phase shifting algorithms,” Optik 124, 710–717 (2013).
    [CrossRef]
  6. Q. Kemao, H. S. Seah, and A. K. Asundi, “Algorithm for directly retrieving the phase difference: a generalization,” Opt. Eng. 42, 1721–1724 (2003).
    [CrossRef]
  7. M. Novak, J. Millerd, N. Brock, M. North-Morris, J. Hayes, and J. Wyant, “Analysis of a micropolarizer array-based simultaneous phase-shifting interferometer,” Appl. Opt. 44, 6861–6868 (2005).
    [CrossRef]
  8. N. I. Toto-Arellano, G. Rodriguez-Zurita, C. Meneses-Fabian, and J. F. Vazquez-Castillo, “Phase shifts in the Fourier spectra of phase gratings and phase grids: an application for one-shot phase-shifting interferometry,” Opt. Express 16, 19330–19341 (2008).
    [CrossRef]
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    [CrossRef]
  10. D. Malacara-Doblado and B. V. Dorrío, “Family of detuning-insensitive phase-shifting algorithms,” J. Opt. Soc. Am. A 17, 1857–1863 (2000).
    [CrossRef]
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2013 (1)

M. Miranda, B. V. Dorrío, J. Blanci, and J. Diz-Bugarin, “Linear error analysis of differential phase shifting algorithms,” Optik 124, 710–717 (2013).
[CrossRef]

2012 (3)

2011 (2)

2010 (2)

2009 (1)

2008 (1)

2005 (1)

2004 (1)

2003 (1)

Q. Kemao, H. S. Seah, and A. K. Asundi, “Algorithm for directly retrieving the phase difference: a generalization,” Opt. Eng. 42, 1721–1724 (2003).
[CrossRef]

2000 (1)

1999 (1)

1998 (1)

1997 (2)

K. Hibino, B. F. Oreb, D. I. Farrant, and K. G. Larkin, “Phase-shifting algorithms for nonlinear and spatially nonuniform phase shifts,” J. Opt. Soc. Am. A 14, 918–930 (1997).
[CrossRef]

M. Servín, D. Malacara, J. L. Marroquín, and F. J. Cuevas, “Complex linear filters for phase shifting with very low detuning sensitivity,” J. Mod. Opt. 44, 1269–1278 (1997).
[CrossRef]

1996 (1)

1995 (1)

1991 (1)

Álvarez-Valado, V.

Asundi, A. K.

Q. Kemao, H. S. Seah, and A. K. Asundi, “Algorithm for directly retrieving the phase difference: a generalization,” Opt. Eng. 42, 1721–1724 (2003).
[CrossRef]

Blanci, J.

M. Miranda, B. V. Dorrío, J. Blanci, and J. Diz-Bugarin, “Linear error analysis of differential phase shifting algorithms,” Optik 124, 710–717 (2013).
[CrossRef]

Blanco, J.

M. Miranda, B. V. Dorrío, J. Blanco, J. Diz-Bugarin, and F. Ribas, “Characteristic polynomial theory of two-stage phase shifting algorithms,” Opt. Lasers Eng. 50, 522–528 (2012).
[CrossRef]

Brock, N.

Cai, L. Z.

Creath, K.

Cuevas, F. J.

M. Servín, D. Malacara, J. L. Marroquín, and F. J. Cuevas, “Complex linear filters for phase shifting with very low detuning sensitivity,” J. Mod. Opt. 44, 1269–1278 (1997).
[CrossRef]

Diz-Bugarin, J.

M. Miranda, B. V. Dorrío, J. Blanci, and J. Diz-Bugarin, “Linear error analysis of differential phase shifting algorithms,” Optik 124, 710–717 (2013).
[CrossRef]

M. Miranda, B. V. Dorrío, J. Blanco, J. Diz-Bugarin, and F. Ribas, “Characteristic polynomial theory of two-stage phase shifting algorithms,” Opt. Lasers Eng. 50, 522–528 (2012).
[CrossRef]

Dorrío, B. V.

M. Miranda, B. V. Dorrío, J. Blanci, and J. Diz-Bugarin, “Linear error analysis of differential phase shifting algorithms,” Optik 124, 710–717 (2013).
[CrossRef]

M. Miranda, B. V. Dorrío, J. Blanco, J. Diz-Bugarin, and F. Ribas, “Characteristic polynomial theory of two-stage phase shifting algorithms,” Opt. Lasers Eng. 50, 522–528 (2012).
[CrossRef]

M. Miranda and B. V. Dorrío, “Monte Carlo based techniques of two-stage phase shifting algorithms,” Opt. Lasers Eng. 49, 439–444 (2011).
[CrossRef]

M. Miranda, V. Álvarez-Valado, B. V. Dorrío, and H. González-Jorge, “Error propagation in differential phase evaluation,” Opt. Express 18, 3199–3209 (2010).
[CrossRef]

M. Miranda and B. V. Dorrío, “Fourier analysis of two-stage phase-shifting algorithms,” J. Opt. Soc. Am. A 27, 276–285 (2010).
[CrossRef]

D. Malacara-Doblado and B. V. Dorrío, “Family of detuning-insensitive phase-shifting algorithms,” J. Opt. Soc. Am. A 17, 1857–1863 (2000).
[CrossRef]

Farrant, D. I.

Gao, P.

García-Márquez, J.

Geist, E.

González-Jorge, H.

Harder, I.

Hayes, J.

Hibino, K.

Kemao, Q.

Q. Kemao, H. S. Seah, and A. K. Asundi, “Algorithm for directly retrieving the phase difference: a generalization,” Opt. Eng. 42, 1721–1724 (2003).
[CrossRef]

Lai, G.

Larkin, K. G.

Lindlein, N.

Liu, Q.

Malacara, D.

M. Servín, D. Malacara, J. L. Marroquín, and F. J. Cuevas, “Complex linear filters for phase shifting with very low detuning sensitivity,” J. Mod. Opt. 44, 1269–1278 (1997).
[CrossRef]

Malacara-Doblado, D.

Mantel, K.

Marroquín, J. L.

M. Servín, D. Malacara, J. L. Marroquín, and F. J. Cuevas, “Complex linear filters for phase shifting with very low detuning sensitivity,” J. Mod. Opt. 44, 1269–1278 (1997).
[CrossRef]

Meneses-Fabian, C.

Millerd, J.

Miranda, M.

M. Miranda, B. V. Dorrío, J. Blanci, and J. Diz-Bugarin, “Linear error analysis of differential phase shifting algorithms,” Optik 124, 710–717 (2013).
[CrossRef]

M. Miranda, B. V. Dorrío, J. Blanco, J. Diz-Bugarin, and F. Ribas, “Characteristic polynomial theory of two-stage phase shifting algorithms,” Opt. Lasers Eng. 50, 522–528 (2012).
[CrossRef]

M. Miranda and B. V. Dorrío, “Monte Carlo based techniques of two-stage phase shifting algorithms,” Opt. Lasers Eng. 49, 439–444 (2011).
[CrossRef]

M. Miranda and B. V. Dorrío, “Fourier analysis of two-stage phase-shifting algorithms,” J. Opt. Soc. Am. A 27, 276–285 (2010).
[CrossRef]

M. Miranda, V. Álvarez-Valado, B. V. Dorrío, and H. González-Jorge, “Error propagation in differential phase evaluation,” Opt. Express 18, 3199–3209 (2010).
[CrossRef]

North-Morris, M.

Novak, M.

Oreb, B. F.

Páez, G.

Ribas, F.

M. Miranda, B. V. Dorrío, J. Blanco, J. Diz-Bugarin, and F. Ribas, “Characteristic polynomial theory of two-stage phase shifting algorithms,” Opt. Lasers Eng. 50, 522–528 (2012).
[CrossRef]

Rodriguez-Zurita, G.

Schmit, J.

Seah, H. S.

Q. Kemao, H. S. Seah, and A. K. Asundi, “Algorithm for directly retrieving the phase difference: a generalization,” Opt. Eng. 42, 1721–1724 (2003).
[CrossRef]

Servín, M.

M. Servín, D. Malacara, J. L. Marroquín, and F. J. Cuevas, “Complex linear filters for phase shifting with very low detuning sensitivity,” J. Mod. Opt. 44, 1269–1278 (1997).
[CrossRef]

Strojnik, M.

Surrel, Y.

Téllez-Quiñones, A.

Toto-Arellano, N. I.

Vazquez-Castillo, J. F.

Wyant, J.

Yang, X. L.

Yao, B.

Yatagai, T.

Appl. Opt. (5)

J. Mod. Opt. (1)

M. Servín, D. Malacara, J. L. Marroquín, and F. J. Cuevas, “Complex linear filters for phase shifting with very low detuning sensitivity,” J. Mod. Opt. 44, 1269–1278 (1997).
[CrossRef]

J. Opt. Soc. Am. A (7)

Opt. Eng. (1)

Q. Kemao, H. S. Seah, and A. K. Asundi, “Algorithm for directly retrieving the phase difference: a generalization,” Opt. Eng. 42, 1721–1724 (2003).
[CrossRef]

Opt. Express (2)

Opt. Lasers Eng. (2)

M. Miranda and B. V. Dorrío, “Monte Carlo based techniques of two-stage phase shifting algorithms,” Opt. Lasers Eng. 49, 439–444 (2011).
[CrossRef]

M. Miranda, B. V. Dorrío, J. Blanco, J. Diz-Bugarin, and F. Ribas, “Characteristic polynomial theory of two-stage phase shifting algorithms,” Opt. Lasers Eng. 50, 522–528 (2012).
[CrossRef]

Opt. Lett. (2)

Optik (1)

M. Miranda, B. V. Dorrío, J. Blanci, and J. Diz-Bugarin, “Linear error analysis of differential phase shifting algorithms,” Optik 124, 710–717 (2013).
[CrossRef]

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Figures (7)

Fig. 1.
Fig. 1.

Graphic description of the use of two phase-shifting arrays (Twymann–Green configurations) to recover phase sums or phase differences with TSPS algorithms.

Fig. 2.
Fig. 2.

Case 1: (a) Intensity pattern μ 1 , 1 , (b) intensity pattern μ 2 , 1 , and (c) synthetic phase difference w .

Fig. 3.
Fig. 3.

Case 1: (a) Estimated phase difference w ^ n c with the noncompensated algorithm, (b) estimated phase difference w ^ c with the compensated algorithm, (c) absolute errors | w ^ n c w | , and (d) absolute errors | w ^ c w | .

Fig. 4.
Fig. 4.

Case 2: (a) Intensity pattern μ 1 , 1 and (b) synthetic phase difference w .

Fig. 5.
Fig. 5.

Case 2: (a) Estimated phase difference w ^ n c , (b) estimated phase difference w ^ c , (c) absolute errors | w ^ n c w | , and (d) absolute errors | w ^ c w | .

Fig. 6.
Fig. 6.

Case 3: (a) Intensity pattern μ 1 , 1 , (b) intensity pattern μ 2 , 1 , and (c) synthetic phase difference w .

Fig. 7.
Fig. 7.

Case 3: (a) Estimated phase difference w ^ n c with the noncompensated algorithm, (b) estimated phase difference w ^ c with the compensated algorithm, (c) absolute errors | w ^ n c w | , and (d) absolute errors | w ^ c w | .

Equations (31)

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μ 1 , k = m = 0 M a m cos [ m ( ϕ α 1 , k ) ] , μ 2 , p = n = 0 N b n cos [ n ( ϕ + Δ ϕ α 2 , p ) ] ,
tan ( ϕ ) = k = 1 K 1 B 1 , k Δ μ 1 , k k + 1 k = 1 K 1 A 1 , k Δ μ 1 , k k + 1 , tan ( ϕ + Δ ϕ ) = p = 1 P 1 B 2 , p Δ μ 2 , p p + 1 p = 1 P 1 A 2 , p Δ μ 2 , p p + 1 ,
tan ( Δ ϕ ) = ( B 2 , p Δ μ 2 , p p + 1 ) ( A 1 , k Δ μ 1 , k k + 1 ) ( B 1 , k Δ μ 1 , k k + 1 ) ( A 2 , p Δ μ 2 , p p + 1 ) ( A 2 , p Δ μ 2 , p p + 1 ) ( A 1 , k Δ μ 1 , k k + 1 ) + ( B 2 , p Δ μ 2 , p p + 1 ) ( B 1 , k Δ μ 1 , k k + 1 ) ,
g N 1 ( t 1 ) = k = 1 K 1 B 1 , k [ δ ( t 1 t 1 , k + 1 ) δ ( t 1 t 1 , k ) ] , g D 1 ( t 1 ) = k = 1 K 1 A 1 , k [ δ ( t 1 t 1 , k + 1 ) δ ( t 1 t 1 , k ) ] , g N 2 ( t 2 ) = p = 1 P 1 B 2 , p [ δ ( t 2 t 2 , p + 1 ) δ ( t 2 t 2 , p ) ] , g D 2 ( t 2 ) = p = 1 P 1 A 2 , p [ δ ( t 2 t 2 , p + 1 ) δ ( t 2 t 2 , p ) ] ,
tan ( w ) = μ ^ 1 ( τ 1 ) μ ^ 2 ( τ 2 ) [ g ^ D 1 * ( τ 1 ) g ^ N 2 * ( τ 2 ) g ^ N 1 * ( τ 1 ) g ^ D 2 * ( τ 2 ) ] d τ 1 d τ 2 μ ^ 1 ( τ 1 ) μ ^ 2 ( τ 2 ) [ g ^ D 1 * ( τ 1 ) g ^ D 2 * ( τ 2 ) + g ^ N 1 * ( τ 1 ) g ^ N 2 * ( τ 2 ) ] d τ 1 d τ 2 = μ ^ ( τ 1 , τ 2 ) g ^ N * ( τ 1 , τ 2 ) d τ 1 d τ 2 μ ^ ( τ 1 , τ 2 ) g ^ D * ( τ 1 , τ 2 ) d τ 1 d τ 2 ,
μ ^ ( τ ) = μ ( t ) e i 2 π τ t d t , g ^ ( τ ) = g ( t ) e i 2 π τ t d t
μ ( t ) g ( t ) d t = μ ^ ( τ ) g ^ * ( τ ) d τ ,
B 2 , p Δ μ 2 , p p + 1 = μ 2 ( t 2 ) g N 2 ( t 2 ) d t 2 = μ ^ 2 ( τ 2 ) g ^ N 2 * ( τ 2 ) d τ 2 .
g ^ N 1 * ( τ 1 ) = B 1 , k Δ cos ( τ 1 α 1 / τ 1 , r ) k k + 1 + i B 1 , k Δ sin ( τ 1 α 1 / τ 1 , r ) k k + 1 , g ^ D 1 * ( τ 1 ) = A 1 , k Δ cos ( τ 1 α 1 / τ 1 , r ) k k + 1 + i A 1 , k Δ sin ( τ 1 α 1 / τ 1 , r ) k k + 1 , g ^ N 2 * ( τ 2 ) = B 2 , p Δ cos ( τ 2 α 2 / τ 2 , r ) p p + 1 + i B 2 , p Δ sin ( τ 2 α 2 / τ 2 , r ) p p + 1 , g ^ D 2 * ( τ 2 ) = A 2 , p Δ cos ( τ 2 α 2 / τ 2 , r ) p p + 1 + i A 2 , p Δ sin ( τ 2 α 2 / τ 2 , r ) p p + 1 .
g ^ N 1 * ( τ 1 ) = i B 1 , k Δ sin ( τ 1 α 1 / τ 1 , r ) k k + 1 , g ^ D 1 * ( τ 1 ) = A 1 , k Δ cos ( τ 1 α 1 / τ 1 , r ) k k + 1 , g ^ N 2 * ( τ 2 ) = i B 2 , p Δ sin ( τ 2 α 2 / τ 2 , r ) p p + 1 , g ^ D 2 * ( τ 2 ) = A 2 , p Δ cos ( τ 2 α 2 / τ 2 , r ) p p + 1 ,
g ^ N * ( τ 1 , τ 2 ) = i [ ( B 2 , p Δ sin ( τ 2 α 2 / τ 2 , r ) p p + 1 ) ( A 1 , k Δ cos ( τ 1 α 1 / τ 1 , r ) k k + 1 ) ( B 1 , k Δ sin ( τ 1 α 1 / τ 1 , r ) k k + 1 ) ( A 2 , p Δ cos ( τ 2 α 2 / τ 2 , r ) p p + 1 ) ] = A m N ( τ 1 , τ 2 ) e i ( π / 2 ) , g ^ D * ( τ 1 , τ 2 ) = ( A 2 , p Δ cos ( τ 2 α 2 / τ 2 , r ) p p + 1 ) ( A 1 , k Δ cos ( τ 1 α 1 / τ 1 , r ) k k + 1 ) ( B 2 , p Δ sin ( τ 2 α 2 / τ 2 , r ) p p + 1 ) ( B 1 , k Δ sin ( τ 1 α 1 / τ 1 , r ) k k + 1 ) = A m D ( τ 1 , τ 2 ) ,
g ^ N * ( 0 , 0 ) = 0 , g ^ D * ( 0 , 0 ) = 0 .
g ^ N * ( τ 1 , r , τ 2 , r ) = i [ ( B 2 , p Δ sin ( α 2 ) p p + 1 ) ( A 1 , k Δ cos ( α 1 ) k k + 1 ) ( B 1 , k Δ sin ( α 1 ) k k + 1 ) ( A 2 , p Δ cos ( α 2 ) p p + 1 ) ] = i [ ( 1 / 2 ) ( 1 / 2 ) ( 1 / 2 ) ( 1 / 2 ) ] = 0 , g ^ D * ( τ 1 , r , τ 2 , r ) = ( A 2 , p Δ cos ( α 2 ) p p + 1 ) ( A 1 , k Δ cos ( α 1 ) k k + 1 ) ( B 2 , p Δ sin ( α 2 ) p p + 1 ) ( B 1 , k Δ sin ( α 1 ) k k + 1 ) = ( 1 / 2 ) ( 1 / 2 ) ( 1 / 2 ) ( 1 / 2 ) = 0 .
g ^ N * ( τ 1 , r , τ 2 , r ) = i [ ( B 2 , p Δ sin ( α 2 ) p p + 1 ) ( A 1 , k Δ cos ( α 1 ) k k + 1 ) ( B 1 , k Δ sin ( α 1 ) k k + 1 ) ( A 2 , p Δ cos ( α 2 ) p p + 1 ) ] = i [ ( 1 / 2 ) ( 1 / 2 ) ( 1 / 2 ) ( 1 / 2 ) ] = ( 1 / 2 ) e i π / 2 , g ^ D * ( τ 1 , r , τ 2 , r ) = ( A 2 , p Δ cos ( α 2 ) p p + 1 ) ( A 1 , k Δ cos ( α 1 ) k k + 1 ) ( B 2 , p Δ sin ( α 2 ) p p + 1 ) ( B 1 , k Δ sin ( α 1 ) k k + 1 ) = ( 1 / 2 ) ( 1 / 2 ) ( 1 / 2 ) ( 1 / 2 ) = ( 1 / 2 ) .
| g ^ N * ( τ 1 , r , τ 2 , r ) | = | g ^ D * ( τ 1 , r , τ 2 , r ) | = ( 1 / 2 ) .
μ ^ 1 ( τ 1 ) = a 0 δ ( τ 1 ) + ( a 1 / 2 ) e i w 1 δ ( τ 1 + τ 1 , r ) + ( a 1 / 2 ) e i w 1 δ ( τ 1 τ 1 , r ) , μ ^ 2 ( τ 2 ) = b 0 δ ( τ 2 ) + ( b 1 / 2 ) e i w 2 δ ( τ 2 + τ 2 , r ) + ( b 1 / 2 ) e i w 2 δ ( τ 2 τ 2 , r ) ,
μ ^ 1 ( τ 1 ) = a 0 δ ( τ 1 ) + ( a 1 / 2 ) e i w 1 δ ( τ 1 + τ 1 , r ) + ( a 1 / 2 ) e i w 1 δ ( τ 1 τ 1 , r ) = a 0 · 0 + ( a 1 / 2 ) e i w 1 δ ( Δ τ 1 ) + ( a 1 / 2 ) e i w 1 δ ( 2 τ 1 , r Δ τ 1 ) = ( a 1 / 2 ) e i w 1 δ ( Δ τ 1 ) + ( a 1 / 2 ) e i w 1 · 0 = ( a 1 / 2 ) e i w 1 δ ( τ 1 + τ 1 , r ) .
μ ^ 2 ( τ 2 ) = ( b 1 / 2 ) e i w 2 δ ( τ 2 τ 2 , r ) .
tan ( w Δ w ) = μ ^ ( τ 1 , τ 2 ) g ^ N * ( τ 1 , τ 2 ) d ( τ 1 ) d τ 2 μ ^ ( τ 1 , τ 2 ) g ^ D * ( τ 1 , τ 2 ) d ( τ 1 ) d τ 2 = g ^ N * ( τ 1 , τ 2 ) δ ( ( τ 1 ) ( τ 1 , r ) ) δ ( τ 2 τ 2 , r ) d ( τ 1 ) d τ 2 g ^ D * ( τ 1 , τ 2 ) δ ( ( τ 1 ) ( τ 1 , r ) ) δ ( τ 2 τ 2 , r ) d ( τ 1 ) d τ 2 = g ^ N * ( τ 1 , r , τ 2 , r ) g ^ D * ( τ 1 , r , τ 2 , r ) = A m N e i γ N A m D e i γ D = ρ e i γ N e i γ D ,
tan ( Δ w ) = cos ( γ D ) sin ( w ) ρ cos ( γ N ) cos ( w ) ρ cos ( γ N ) sin ( w ) + cos ( γ D ) cos ( w ) .
Δ w = cos ( w ) sin ( w ) ρ sin ( w ) cos ( w ) = sin ( 2 w ) [ 1 ρ 2 ] ,
α 1 = ( π / 2 ) [ 3 , 2 , 1 , 0 , 1 , 2 , 3 ] + ( π / 2 ) [ 0 , d 1 , 2 d 1 , 3 d 1 , 4 d 1 , 5 d 1 , 6 d 1 ]
α 2 = ( π / 4 ) [ 3 , 2 , 1 , 0 , 1 , 2 , 3 ] + ( π / 4 ) [ 0 , d 2 , 2 d 2 , 3 d 2 , 4 d 2 , 5 d 2 , 6 d 2 ] ,
B 1 = [ 0.1666667 0.1666667 0.1666667 0.1666667 0.1666667 0.1666667 ] , A 1 = [ 0.1666667 0.1666667 0.1666667 0.1666667 0.1666667 0.1666667 ] , B 2 = [ 0.4459029 0.4459029 1.0765048 1.0765048 0.4459029 0.4459029 ] , A 2 = [ 0.3867295 0.3867295 0.1601886 0.1601886 0.3867295 0.3867295 ] .
B 1 = [ 0.1517857 0.2053571 0.1428571 0.1428571 0.2053571 0.1517857 ] , A 1 = [ 0.0267857 0.2232143 0.25 0.25 0.2232143 0.0267857 ] , B 2 = [ 0.3535534 0.3535534 0.7071068 0.7071068 0.3535534 0.3535534 ] , A 2 = [ 2.0606602 3.767767 5.8284271 5.8284271 3.767767 2.0606602 ] .
a 0 = 0.01 e ( x 2 + y 2 ) , b 0 = 0.07 sin ( x ) cos ( y ) .
a 1 = sin ( x 2 + y 2 ) , b 1 = cos ( x 2 ) + e y 2 .
a 2 = 0.01 cos ( x 2 y 2 ) , b 2 = 0.02 sin ( y ) cos ( x ) .
w ( x , y ) = 3 x + 2 y , ϕ ( x , y ) = 3 ( x 2 + y 2 ) 2 x y + 5 x .
w ( x , y ) = 3 ( x 2 + y 2 ) 2 x y + 5 x , ϕ ( x , y ) = 3 x + 2 y .
w ( x , y ) = sin ( 3 x + 2 y ) , ϕ ( x , y ) = 3 ( x 2 + y 2 ) 2 x y + 5 x .

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