Abstract

Least-squares (LS)-based integration computes the function values by solving a set of integration equations (IEs) in LS sense, and is widely used in wavefront reconstruction and other fields where the measured data forms a slope. It is considered that the applications of IEs with smaller truncation errors (TEs) will improve the reconstruction accuracy. This paper proposes a general method based on the Taylor theorem to derive all kinds of IEs, and finds that an IE with a smaller TE has a higher-order TE. Three specific IEs with higher-order TEs in the Southwell geometry are deduced using this method, and three LS-based integration algorithms corresponding to these three IEs are formulated. A series of simulations demonstrate the validity of applying IEs with higher-order TEs in improving reconstruction accuracy. In addition, the IEs with higher-order TEs in the Hudgin and Fried geometries are also deduced using the proposed method, and the performances of these IEs in wavefront reconstruction are presented.

© 2013 Optical Society of America

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References

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2012

2011

2008

2007

2006

2005

2004

M. C. Knauer, J. Kaminski, and G. Häusler, “Phase measuring deflectometry: a new approach to measure specular free-form surfaces,” Proc. SPIE 5457, 366–376 (2004).
[CrossRef]

T. Bothe, W. Li, C. von Kopylow, and W. P. O. Jüptner, “High resolution 3D shape measurement on specular surfaces by fringe reflection,” Proc. SPIE 5457, 411–422 (2004).
[CrossRef]

2002

2000

1995

1993

1991

1986

1983

1980

1979

1977

1976

Artal, P.

Asundi, A.

Bahk, S.-W.

Boreman, G. D.

Bothe, T.

T. Bothe, W. Li, C. von Kopylow, and W. P. O. Jüptner, “High resolution 3D shape measurement on specular surfaces by fringe reflection,” Proc. SPIE 5457, 411–422 (2004).
[CrossRef]

Brase, J. M.

Campos, J.

Dainty, J. C.

Ellerbroek, B. L.

Freischlad, K.

Fried, D. L.

Gavel, D. T.

Goelz, S.

Gotwols, B. L.

Hahn, J.-W.

Häusler, G.

M. C. Knauer, J. Kaminski, and G. Häusler, “Phase measuring deflectometry: a new approach to measure specular free-form surfaces,” Proc. SPIE 5457, 366–376 (2004).
[CrossRef]

Hermann, J.

Huang, L.

Hudgin, R. H.

Hunt, B. R.

Jüptner, W. P. O.

T. Bothe, W. Li, C. von Kopylow, and W. P. O. Jüptner, “High resolution 3D shape measurement on specular surfaces by fringe reflection,” Proc. SPIE 5457, 411–422 (2004).
[CrossRef]

Kaminski, J.

M. C. Knauer, J. Kaminski, and G. Häusler, “Phase measuring deflectometry: a new approach to measure specular free-form surfaces,” Proc. SPIE 5457, 366–376 (2004).
[CrossRef]

Keller, W. C.

Knauer, M. C.

M. C. Knauer, J. Kaminski, and G. Häusler, “Phase measuring deflectometry: a new approach to measure specular free-form surfaces,” Proc. SPIE 5457, 366–376 (2004).
[CrossRef]

Koliopoulos, C. L.

Lee, J.-S.

Li, Q.

Li, W.

T. Bothe, W. Li, C. von Kopylow, and W. P. O. Jüptner, “High resolution 3D shape measurement on specular surfaces by fringe reflection,” Proc. SPIE 5457, 411–422 (2004).
[CrossRef]

Mahajan, V. N.

V. N. Mahajan, “Zernike polynomial and wavefront fitting,” in Optical Shop Testing, D. Malacara, ed. (Wiley, 2007), pp. 498–546.

Moreno, A.

Nicholls, T. W.

Noll, R. J.

Poyneer, L. A.

Prieto, P.

Ribak, E. N.

Roddier, C.

Roddier, F.

Rolland, J. P.

Southwell, W. H.

Sun, S.

Talmi, A.

Tang, S.

Vargas-Martín, F.

von Kopylow, C.

T. Bothe, W. Li, C. von Kopylow, and W. P. O. Jüptner, “High resolution 3D shape measurement on specular surfaces by fringe reflection,” Proc. SPIE 5457, 411–422 (2004).
[CrossRef]

Wu, J.

Yang, H.-S.

Yaroslavsky, L. P.

Yzuel, M. J.

Zhang, Z.

Zhao, M.

Zorich, V. A.

V. A. Zorich, Mathematical Analysis I (Springer, 2004).

Zou, W.

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Express

Opt. Lett.

Proc. SPIE

M. C. Knauer, J. Kaminski, and G. Häusler, “Phase measuring deflectometry: a new approach to measure specular free-form surfaces,” Proc. SPIE 5457, 366–376 (2004).
[CrossRef]

T. Bothe, W. Li, C. von Kopylow, and W. P. O. Jüptner, “High resolution 3D shape measurement on specular surfaces by fringe reflection,” Proc. SPIE 5457, 411–422 (2004).
[CrossRef]

Other

V. A. Zorich, Mathematical Analysis I (Springer, 2004).

V. N. Mahajan, “Zernike polynomial and wavefront fitting,” in Optical Shop Testing, D. Malacara, ed. (Wiley, 2007), pp. 498–546.

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Figures (8)

Fig. 1.
Fig. 1.

Row of four estimated points in the Southwell geometry. The unknown function values (marked by closed circles) are coincident with the slopes (marked by crosses). Intermediate “imaginary” positions are indicated with open circles; h is the interval between two adjacent slope measurements.

Fig. 2.
Fig. 2.

Square domain is discretized into a 50 by 50 grid of subapertures whose centers are marked by dots. The circle is the incircle of the square. Only the subapertures whose centers are within the circle are considered in the simulation.

Fig. 3.
Fig. 3.

Relative RMS reconstruction errors of Southwell’s algorithm (SA) and A1–A3 for the reconstructions of the second to the 105th Zernike polynomials are displayed in (a) a linear scale and (b) a log scale. The corresponding powers of the Zernike polynomials are from 1 to 13, which are also labeled.

Fig. 4.
Fig. 4.

Error between the reconstructed and the ideal shape for (a) SA, (b) A1, (c) A2, and (d) A3.

Fig. 5.
Fig. 5.

EPCs of the four algorithms for the (a) circular reconstruction domain and (b) square reconstruction domain. N is the discretized subaperture number on one dimension, which is from 10 to 50. The circular domain is the incircle of the square as shown in Fig. 2.

Fig. 6.
Fig. 6.

Relative RMS reconstruction error for the reconstructions of the second to the 105th Zernike polynomials in the presence of noise. (a) SNR of the simulated slopes is 10 and (b) SNR of the simulated slopes is 30. The powers (1–13) of the Zernike polynomials are also labeled.

Fig. 7.
Fig. 7.

Total reconstruction errors of (a) SA, (b) A1, (c) A2, and (d) A3.

Fig. 8.
Fig. 8.

(a) Row of four estimated points for Hudgin geometry; (b) two rows of four estimated points for the Fried geometry. The closed circles indicate the positions of the function values to be computed; the horizontal dashes indicate the positions of the known x slopes; the vertical dashes indicate the positions of the known y slopes.

Tables (2)

Tables Icon

Table 1. Std. and PV of the Reconstruction Error for Each Algorithm

Tables Icon

Table 2. Std. of Algorithm Error (AE), Noise-Induced Error (NE), and Total Reconstruction Rrror (RE) of SA and A1–A3

Equations (62)

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f(x)=Pn(x0;x)+rn(x0;x),
Pn(x0;x)=k=0nf(k)(x0)k!(xx0)k,
rn(x0;x)=f(n+1)(ξ)(n+1)!(xx0)n+1,
f2f1=h2(f1+f2),
(f2f1)h2(f1+f2)=(fξ1(4)24Δξfξ2(3)12)h3,
Af2+Bf3=h(Cf1+Df2+Ef3+Ff4).
A{f2.5f2.5h/2+O[(h/2)5]}+B{f2.5+f2.5h/2++O[(h/2)5]}=h(C{f2.5f2.53h/2++O[(3h/2)4]}+D{f2.5f2.5h/2++O[(h/2)4]}+E{f2.5+f2.5h/2++O[(h/2)4]}+F{f2.5+f2.53h/2++O[(3h/2)4]}).
A=k,B=k,C=124k,D=1324k,E=1324k,F=124k,
f3f2=13h24(113f1+f2+f3113f4).
19f4+f3f219f1=2h3(f2+f3).
1127f4+f3f21127f1=h(19f1+f2+f3+19f4).
Wi,j+1Wi,j=13h24(113Si,j1x+Si,jx+Si,j+1x113Si,j+2x),
Wi,jWi+1,j=13h24(113Si1,jy+Si,jy+Si+1,jy113Si+2,jy),
[P1xP1y]W=[S1xS1y].
W|m,1=Wrem(m,N),ceil(m/N),
S1x|m,1=13h24(113Scm,rmx+Scm,rm+1x+Scm,rm+2x113Scm,rm+3x),
S1y|m,1=13h24(113Srm,cmy+Srm+1,cmy+Srm+2,cmy113Srm+3,cmy),
P1x|m,n={1,cm=rn&rm+1=cn1,cm=rn&rm+2=cn0,otherwise,
P1y|m,n={1,cm=cn&rm+1=rn1,cm=cn&rm+2=rn0,otherwise,
19Wi,j+2+Wi,j+1Wi,j19Wi,j1=2h3(Si,jx+Si,j+1x),
19Wi1,j+Wi,jWi+1,j19Wi+2,j=2h3(Si,jy+Si+1,jy),
[P2xP2y]W=[S2xS2y],
S2x|m,1=2h3(Scm,rm+1x+Scm,rm+2x),
S2y|m,1=2h3(Srm+1,cmy+Srm+2,cmy),
P2x|m,n={1/9,cm=rn&rm=cn1,cm=rn&rm+1=cn1,cm=rn&rm+2=cn1/9,cm=rn&rm+3=cn0,otherwise,
P2y|m,n={1/9,cm=cn&rm=rn1,cm=cn&rm+1=rn1,cm=cn&rm+2=rn1/9,cm=cn&rm+3=rn0,otherwise,
1127Wi,j+2+Wi,j+1Wi,j1127Wi,j1=h(19Si,j1x+Si,jx+Si,j+1x+19Si,j+2x),
1127Wi1,j+Wi,jWi+1,j1127Wi+2,j=h(19Si1,jy+Si,jy+Si+1,jy+19Si+2,jy),
[P3xP3y]W=[S3xS3y],
S3x|m,1=h(19Scm,rmx+Scm,rm+1x+Scm,rm+2x+19Scm,rm+3x),
S3y|m,1=h(19Srm,cmy+Srm+1,cmy+Srm+2,cmy+19Srm+3,cmy),
P3x|m,n={11/27,cm=rn&rm=cn1,cm=rn&rm+1=cn1,cm=rn&rm+2=cn11/27,cm=rn&rm+3=cn0otherwise,
Pxy|m,n={11/27,cm=cn&rm=rn1,cm=cn&rm+1=rn1,cm=cn&rm+2=rn11/27,cm=cn&rm+3=rn0otherwise,
PW=S,
P=[PixPiy],andS=[SixSiy],
PTPW=PTS,
Wi,j+2Wi,j=h3(Si,jx+4Si,j+1x+Si,j+2x),
Wi,jWi+2,j=h3(Si,jy+4Si+1,jy+Si+2,jy),
[PsxPsy]W=[SsxSsy],
Ssx|m,1={h3(Scm2,1x+4Scm2,2x+Scm2,3x),mis oddh3(Scm2,N2x+4Scm2,N1x+Scm2,Nx),mis even,
Ssy|m,1={h3(S1,cm2y+4S2,cm2y+S3,cm2y),mis oddh3(SN2,cm2y+4SN1,cm2y+SN,cm2y),mis even,
Psx|m,n={1,cm2=rn&cn=1,N21,cm2=rn&cn=3,N0,otherwise,
Psy|m,n={1,cm2=cn&rn=1,N21,cm2=cn&rn=3,N0,otherwise,
P0W=0,
P0|1,m={1,ceil(m,N)=rem(m,N)=ceil(N/2)0,otherwise,
PeW=Se,
Pe=[PPsxPsyP0],andSe=[SSsxSsy0].
W=(PeTPe)1PeTSe,
ΔW=WW0,
Se=Se0+ne,
ΔW=ΔW0+ΔWn,
ΔW0=(PeTPe)1PeTSe0W0,
ΔWn=(PeTPe)1PeTne,
R=[(ΔW0·ΔW0)/(W0·W0)]1/2,
z(x,y)=0.3cos(0.4x2+2x)cos(0.4y2+2y)+0.7cos[(x3+y2)/4π].
SNR=[(S0·S0)/(n·n)]1/2,
z(x,y)=3(1x)2·ex2(y+1)210(x5x3y5)·ex2y213e(x+1)2y2.
f2f1=hf1.5;
f12f2+f3=h(f2.5f1.5);
1727f4+7f37f21727f1=h(f1.5+629f2.5+f3.5).
f1,2+f2,2(f1,1+f2,1)=2hf1.5,1.5x,
f1,1+f2,12(f1,2+f2,2)+f1,3+f2,3=2h(f2.5,2.5xf1.5,1.5x).

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