Abstract

A simple phase estimation employing cubic and average interpolations to solve the oversampling problem in smooth modulated phase images is described. In the context of a general phase-shifting process, without phase-unwrapping, the modulated phase images are employed to recover wavefront shapes with high fringe density. The problem of the phase reconstruction by line integration of its gradient requires a form appropriate to the calculation of partial derivatives, especially when the phase to recover has higher-order aberration values. This is achieved by oversampling the modulated phase images, and many interpolations can be implemented. Here an oversampling procedure based on the analysis of a quadratic cost functional for phase recovery, in a particular case, is proposed.

© 2012 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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2010 (1)

2002 (1)

2000 (1)

1999 (1)

1997 (1)

1985 (1)

1982 (1)

Bertero, M.

M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (Institute of Physics, 1998).

Boccacci, P.

M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (Institute of Physics, 1998).

Ghatak, A. K.

Gonzalez, R. C.

R. C. Gonzalez and R. E. Woods, Digital Image Processing (Prentice-Hall, 2008).

Higham, D. J.

D. J. Higham and N. J. Higham, Matlab Guide (SIAM, 2005).

Higham, N. J.

D. J. Higham and N. J. Higham, Matlab Guide (SIAM, 2005).

Kumar, D. V.

Legarda-Sáenz, R.

Malacara, D.

D. Malacara, Optical Shop Testing (Wiley-Interscience, 2007).

D. Malacara, M. Servín, and Z. Malacara, Interferogram Analysis for Optical Testing (CRC, 2005).

Malacara, Z.

D. Malacara, M. Servín, and Z. Malacara, Interferogram Analysis for Optical Testing (CRC, 2005).

Malacara-Doblado, D.

Nakamura, S.

S. Nakamura, Numerical Analysis and Graphical Visualization with Matlab (Prentice-Hall, 2001).

Páez, G.

Rivera, M.

Rodríguez-Vera, R.

Servín, M.

D. Malacara, M. Servín, and Z. Malacara, Interferogram Analysis for Optical Testing (CRC, 2005).

Sharma, A.

Strojnik, M.

Téllez-Quiñones, A.

Trujillo-Schiaffino, G.

Vogel, C. R.

C. R. Vogel, Computational Methods for Inverse Problems (SIAM, 2002).

Woods, R. E.

R. C. Gonzalez and R. E. Woods, Digital Image Processing (Prentice-Hall, 2008).

Appl. Opt. (4)

J. Opt. Soc. Am. A (1)

Opt. Lett. (2)

Other (7)

C. R. Vogel, Computational Methods for Inverse Problems (SIAM, 2002).

M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (Institute of Physics, 1998).

D. Malacara, Optical Shop Testing (Wiley-Interscience, 2007).

D. Malacara, M. Servín, and Z. Malacara, Interferogram Analysis for Optical Testing (CRC, 2005).

S. Nakamura, Numerical Analysis and Graphical Visualization with Matlab (Prentice-Hall, 2001).

D. J. Higham and N. J. Higham, Matlab Guide (SIAM, 2005).

R. C. Gonzalez and R. E. Woods, Digital Image Processing (Prentice-Hall, 2008).

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Figures (8)

Fig. 1
Fig. 1

Flow diagram description of the principal steps for the phase reconstruction process by increasing the resolution of the modulated images De and Nu given by Eq. (2).

Fig. 2.
Fig. 2.

Synthetic phases and intensity patterns s1 for the cases: κ=1, items (a) and (d); κ=2, items (b) and (e); κ=3, items (c) and (f).

Fig. 3.
Fig. 3.

Estimated phases for the case κ=1: (a) without interpolation; (b) bi-cubic interpolation; (c) quadratic cost functional interpolation; (d) cubic and average interpolation.

Fig. 4.
Fig. 4.

Absolute phase errors in radians for the case κ=1: (a) without interpolation; (b) bi-cubic interpolation; (c) quadratic cost functional interpolation; (d) cubic and average interpolation.

Fig. 5.
Fig. 5.

Absolute phase errors in radians for the case κ=2: (a) without interpolation; (b) bi-cubic interpolation; (c) quadratic cost functional interpolation; (d) cubic and average interpolation.

Fig. 6.
Fig. 6.

Absolute phase errors in radians for the case κ=3: (a) without interpolation; (b) bi-cubic interpolation; (c) quadratic cost functional interpolation; (d) cubic and average interpolation.

Fig. 7.
Fig. 7.

Frequency dependent root mean square error functions for the three main interpolation methods discussed. Local perspective around κ=1.45.

Fig. 8.
Fig. 8.

Intensity pattern s1 for κ=1.6 and 256×256 pixels. (a) Without noise. (b) With 10% of multiplicative and additive random noise.

Tables (2)

Tables Icon

Table 1. Root Mean Square Errors in Radians for the Four Approximations in Six Relative Frequencies Around κ=1.45

Tables Icon

Table 2. Root Mean Square Errors in Radians for the Four Approximations in Three Relative Frequencies Considering 10% of Random Noise in the Interferogramsa

Equations (26)

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sk(x,y)=a(x,y)+b(x,y)cos(ϕ(x,y)αk),k=1,,K3,
Nu=k=1K1BkΔskk+1,De=k=1K1AkΔskk+1,
bsinϕ=Nuk=1K1BkΔsinαkk+1,bcosϕ=Dek=1K1AkΔcosαkk+1,
k=1K1Ak=k=1K1Bk=0.
Nux=τNu[bxsinϕ+bcosϕϕx],
Dex=τDe[bxcosϕbsinϕϕx],
ϕx=(bcosϕτNu)(Nux)(bsinϕτDe)(Dex)(bsinϕ)2+(bcosϕ)2,
ϕy=(bcosϕτNu)(Nuy)(bsinϕτDe)(Dey)(bsinϕ)2+(bcosϕ)2.
f0=3f432f3+168f2672f1+672f1168f2+32f33f4840Δ,
h(x,y)=x0xϕxdx=ϕ(x,y)ϕ(x0,y),
g(x,y)=y0yϕydy=ϕ(x,y)ϕ(x,y0),
ϕ^1(x,y)=h(x,y)+σh(y)=ϕ(x,y)ϕ(x0,y0),
ϕ^2(x,y)=g(x,y)+σg(x)=ϕ(x,y)ϕ(x0,y0).
R(x,y)=[x,y,f(x,y)].
T(x,y)=Rx×Ry=[fx,fy,1],
U(f^)=Sk,l|f^(k,l)f(k,l)|2+Sk,l|f^(k+1,l)f^(k1,l)2Δyfy(k,l)|2+Sk,l|f^(k,l+1)f^(k,l1)2Δxfx(k,l)|2+ρ(1Sk,l)|f^(k+1,l)2f^(k,l)+f^(k1,l)|2+ρ(1Sk,l)|f^(k,l+1)2f^(k,l)+f^(k,l1)|2,
f(x)=f0+f1x+f2x2+f3x3,
f^(xj1)=f(xj)+{f1Δx+[2(j1)+1]Δx2f2+[3(j1)2+3(j1)1]Δx3f3},
f^(xj+1)=f(xj)+{f1Δx+[2(j1)+1]Δx2f2+[3(j1)2+3(j1)+1]Δx3f3}
εj=|f^(xj+1)f^(xj1)2Δxf(xj)|=Δx2|f3|,
f^(k,l)=f^(k1,l)+f^(k+1,l)+f^(k,l1)+f^(k,l+1)4.
fork=1,M,l=1,,Nf^(k1,l1)=f(k1,l1),f^(k1,l+1)=f(k1,l+1),f^(k+1,l1)=f(k+1,l1),f^(k+1,l+1)=f(k+1,l+1),f^(k1,l)=cubic interpolation[f(k1,l1),f(k1,l+1),(f/x)(k1,l1),(f/x)(k1,l+1)],f^(k+1,l)=cubic interpolation[f(k+1,l1),f(k+1,l+1),(f/x)(k+1,l1),(f/x)(k+1,l+1)],f^(k,l1)=cubic interpolation[f(k1,l1),f(k+1,l1),(f/y)(k1,l1),(f/y)(k+1,l1)],f^(k,l+1)=cubic interpolation[f(k1,l+1),f(k+1,l+1),(f/y)(k1,l+1),(f/y)(k+1,l+1)],f^(k,l)=(1/4)[f^(k1,l)+f^(k+1,l)+f^(k,l1)+f^(k,l+1)],end
b(x,y)=10exp[(x2+y2)],
v(x,y)=exp[ln(0.63)(x2+y2)],a(x,y)=b(x,y)/v(x,y).
ϕ(x,y)=κ{2.63.9x(2.6)[16y26x2+6y4+12x2y2+6x4]+(6.93)[5xy410x3y2+x5]+(0.86)[3x12xy212x3+10xy4+20x3y2+10x5]+(5.2)[4y3+12x2y+5y510x2y315x4y]},
snoisy=ε0+ε1s,ε0(x,y)=(ξ/2)+ξrand(x,y),ε1(x,y)=1(η/2)+ηrand(x,y),

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