Abstract

In fringe analysis, or in projected grids for shape-from-shade, deviations from periodicity are used for finding phase changes. In another example, a Hartmann–Shack sensor produces a deformed grid of spots on a camera. The gradients of the original wavefront are calculated from that image by centroiding the spots or by demodulating them. The computation time rises linearly with the number of pixels in the image. We introduce a method to reduce the size of the image without loss of accuracy prior to the calculation to reduce the total processing time. The compressed result is superior to an image measured with reduced resolution. Hence, higher accuracy and speed are obtained by oversampling the image and reducing it correctly prior to calculations. Compression or expansion coefficients are calculated through the requirement to maintain the integrity of the original phase data.

© 2010 Optical Society of America

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References

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2009

2008

2007

2006

A. Talmi and E. N. Ribak, “Wavefront reconstruction from its gradients,” J. Opt. Soc. Am. A 23, 288–97 (2006).
[CrossRef]

S. Thomas, T. Fusco, A. Tokovinin, M. Nicolle, V. Michau, and G. Rousset, “Comparison of centroid computation algorithms in Shack–Hartmann sensor,” Mon. Not. R. Astron. Soc. 371, 323–336 (2006).
[CrossRef]

2005

2004

2003

L. Poyneer, “Scene-based Shack–Hartmann wave-front sensing: analysis and simulation,” Appl. Opt. 42, 5807–15(2003).
[CrossRef] [PubMed]

Y. Carmon and E. N. Ribak, “Phase retrieval by demodulation of a Hartmann–Shack sensor,” Opt. Commun. 215, 285–288(2003).
[CrossRef]

2002

T. Berkefeld, D. Soltau, and O. von der Luhe, “Multi-conjugate adaptive optics at the Vacuum Tower Telescope, Tenerife,” Proc. SPIE 4839, 66–76 (2002).
[CrossRef]

2001

2000

G. S. Spagnolo, G. Guattaria, C. Sapia, D. Ambrosini, D. Paoletti, and G. Accardo, “Contouring of artwork surface by fringe projection and FFT analysis,” Opt. Lasers Eng. 33, 141–156 (2000).
[CrossRef]

1970

Accardo, G.

G. S. Spagnolo, G. Guattaria, C. Sapia, D. Ambrosini, D. Paoletti, and G. Accardo, “Contouring of artwork surface by fringe projection and FFT analysis,” Opt. Lasers Eng. 33, 141–156 (2000).
[CrossRef]

Albanis, V.

Alfalou, A.

Allen, J. B.

Ambrosini, D.

G. S. Spagnolo, G. Guattaria, C. Sapia, D. Ambrosini, D. Paoletti, and G. Accardo, “Contouring of artwork surface by fringe projection and FFT analysis,” Opt. Lasers Eng. 33, 141–156 (2000).
[CrossRef]

Ang, K.

Berkefeld, T.

T. Berkefeld, D. Soltau, and O. von der Luhe, “Multi-conjugate adaptive optics at the Vacuum Tower Telescope, Tenerife,” Proc. SPIE 4839, 66–76 (2002).
[CrossRef]

Brosseau, C.

Canovas, C.

Carmon, Y.

Y. Carmon and E. N. Ribak, “Centroid distortion of a wavefront with varying amplitude due to asymmetry in lens diffraction,” J. Opt. Soc. Am. A 26, 85–90 (2009).
[CrossRef]

Y. Carmon and E. N. Ribak, “Phase retrieval by demodulation of a Hartmann–Shack sensor,” Opt. Commun. 215, 285–288(2003).
[CrossRef]

Cimet, E. M.

Cornia, A.

Coskun, E.

Dandy, S.

Fusco, T.

L. M. Mugnier, J.-F. Sauvage, T. Fusco, A. Cornia, and S. Dandy, “On-line long-exposure phase diversity: a powerful tool for sensing quasi-static aberrations of extreme adaptive optics imaging systems,” Opt. Express 16, 18406–18416 (2008).
[CrossRef] [PubMed]

S. Thomas, T. Fusco, A. Tokovinin, M. Nicolle, V. Michau, and G. Rousset, “Comparison of centroid computation algorithms in Shack–Hartmann sensor,” Mon. Not. R. Astron. Soc. 371, 323–336 (2006).
[CrossRef]

Göktas, H.

Guattaria, G.

G. S. Spagnolo, G. Guattaria, C. Sapia, D. Ambrosini, D. Paoletti, and G. Accardo, “Contouring of artwork surface by fringe projection and FFT analysis,” Opt. Lasers Eng. 33, 141–156 (2000).
[CrossRef]

Johnson, W. O.

Kocahan, Ö.

Meadows, D. M.

Michau, V.

S. Thomas, T. Fusco, A. Tokovinin, M. Nicolle, V. Michau, and G. Rousset, “Comparison of centroid computation algorithms in Shack–Hartmann sensor,” Mon. Not. R. Astron. Soc. 371, 323–336 (2006).
[CrossRef]

Mugnier, L. M.

Ng, T.

Nicolle, M.

S. Thomas, T. Fusco, A. Tokovinin, M. Nicolle, V. Michau, and G. Rousset, “Comparison of centroid computation algorithms in Shack–Hartmann sensor,” Mon. Not. R. Astron. Soc. 371, 323–336 (2006).
[CrossRef]

Özder, S.

Paoletti, D.

G. S. Spagnolo, G. Guattaria, C. Sapia, D. Ambrosini, D. Paoletti, and G. Accardo, “Contouring of artwork surface by fringe projection and FFT analysis,” Opt. Lasers Eng. 33, 141–156 (2000).
[CrossRef]

Poyneer, L.

Ribak, E. N.

Rousset, G.

S. Thomas, T. Fusco, A. Tokovinin, M. Nicolle, V. Michau, and G. Rousset, “Comparison of centroid computation algorithms in Shack–Hartmann sensor,” Mon. Not. R. Astron. Soc. 371, 323–336 (2006).
[CrossRef]

Sapia, C.

G. S. Spagnolo, G. Guattaria, C. Sapia, D. Ambrosini, D. Paoletti, and G. Accardo, “Contouring of artwork surface by fringe projection and FFT analysis,” Opt. Lasers Eng. 33, 141–156 (2000).
[CrossRef]

Sauvage, J.-F.

Soltau, D.

T. Berkefeld, D. Soltau, and O. von der Luhe, “Multi-conjugate adaptive optics at the Vacuum Tower Telescope, Tenerife,” Proc. SPIE 4839, 66–76 (2002).
[CrossRef]

Spagnolo, G. S.

G. S. Spagnolo, G. Guattaria, C. Sapia, D. Ambrosini, D. Paoletti, and G. Accardo, “Contouring of artwork surface by fringe projection and FFT analysis,” Opt. Lasers Eng. 33, 141–156 (2000).
[CrossRef]

Stup, A.

Talmi, A.

Thomas, S.

S. Thomas, T. Fusco, A. Tokovinin, M. Nicolle, V. Michau, and G. Rousset, “Comparison of centroid computation algorithms in Shack–Hartmann sensor,” Mon. Not. R. Astron. Soc. 371, 323–336 (2006).
[CrossRef]

Tokovinin, A.

S. Thomas, T. Fusco, A. Tokovinin, M. Nicolle, V. Michau, and G. Rousset, “Comparison of centroid computation algorithms in Shack–Hartmann sensor,” Mon. Not. R. Astron. Soc. 371, 323–336 (2006).
[CrossRef]

Tyson, R. K.

R. K. Tyson, Principles of Adaptive Optics, 2nd ed.(Academic, 1998).

von der Luhe, O.

T. Berkefeld, D. Soltau, and O. von der Luhe, “Multi-conjugate adaptive optics at the Vacuum Tower Telescope, Tenerife,” Proc. SPIE 4839, 66–76 (2002).
[CrossRef]

Adv. Opt. Photon.

Appl. Opt.

J. Opt. Soc. Am. A

Mon. Not. R. Astron. Soc.

S. Thomas, T. Fusco, A. Tokovinin, M. Nicolle, V. Michau, and G. Rousset, “Comparison of centroid computation algorithms in Shack–Hartmann sensor,” Mon. Not. R. Astron. Soc. 371, 323–336 (2006).
[CrossRef]

Opt. Commun.

Y. Carmon and E. N. Ribak, “Phase retrieval by demodulation of a Hartmann–Shack sensor,” Opt. Commun. 215, 285–288(2003).
[CrossRef]

Opt. Express

Opt. Lasers Eng.

G. S. Spagnolo, G. Guattaria, C. Sapia, D. Ambrosini, D. Paoletti, and G. Accardo, “Contouring of artwork surface by fringe projection and FFT analysis,” Opt. Lasers Eng. 33, 141–156 (2000).
[CrossRef]

Opt. Lett.

Proc. SPIE

T. Berkefeld, D. Soltau, and O. von der Luhe, “Multi-conjugate adaptive optics at the Vacuum Tower Telescope, Tenerife,” Proc. SPIE 4839, 66–76 (2002).
[CrossRef]

Other

D.Malacara, ed., Optical Shop Testing (Wiley, 1978).

R. K. Tyson, Principles of Adaptive Optics, 2nd ed.(Academic, 1998).

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Figures (7)

Fig. 1
Fig. 1

(a) Wavefront slopes shift the focal spots of the lenslets. (b) An additional time-variable aberration widens the shifted focal spots during long integration—the fixed error can still be recovered. (c) Wider, even overlapping, spots can be formed from an extended object.

Fig. 2
Fig. 2

Each source pixel j contributes to three destination pixels, l = [ j M ] 1 , [ j M ] , and [ j M ] + 1 . The weights of the contributions are set by conservation of location and intensity of the original pattern.

Fig. 3
Fig. 3

Phase errors: differences of phase between resized and original vectors, along the array of Hartmann spots. The parameters are as follows: pitch, D = 13 pixels; spot width, σ = 1.5 pixels; and magnification M = 0.7 , with the phase gradient growing over the array from 0 to 0.1 rad . The number of smoothing passes is two (blue, broken curve), three (red, dotted curve), and four (green, full curve). Three-term fits (lower curves) are slightly less accurate than five-term fits (curves shifted up by 0.01). The difference is that slope is expressed in radians (the phase of the phasor) proportional to the gradient of the wavefront.

Fig. 4
Fig. 4

Hartmann pattern, including Poisson noise, with (a) 0–6 photons per pixel (negative image), the source being an asterism, the reference a long exposure of the same asterism. The data were extended beyond the edge [6], and the (b) x phase slope was calculated. On the left and bottom are pixel numbers; on the right are (a) photons and (b) phase (radians).

Fig. 5
Fig. 5

Errors introduced by shrinking. The Hartmann array of Fig. 4 was shrunk by 0.71 in x and 0.61 in y (notice coordinates). The results are too similar to Fig. 4b for changes to be visible. Instead, the differences are drawn for (a), (b) a two-pass smoothing (varying gray scale), (a) three-term interpolation, and (b) five-term interpolation. The corresponding RMS of the differences in phase are 0.0024 and 0.0015 rad . Scale bar is expressed in radians.

Fig. 6
Fig. 6

Errors introduced in reduction from Fig. 4 by 0.71 in x and 0.61 in y, as in Fig. 5, but with further smoothing and thus lower errors. The difference between the original and the reduced slope is shown for (a), (b) a four-pass smoothing (varying gray scale), (a) three-term interpolation, and (b) five-term interpolation. The corresponding RMS differences are 0.0014 and 0.0012 radians.

Fig. 7
Fig. 7

Root mean square shift in the location of the peaks before and after reduction, in differently reduced images and different noise levels. An input image (as in Figs. 4, 5, 6) with a pitch of 16 pixels was reduced to various pitches as marked, both with N = 3 and N = 5 terms formulas. The indicated peak intensities I are realizations of 3 ± 3 , 12 ± 12 and 36 ± 36 photons. Results were calculated using a four-pass smoothing, and are expressed in pixels of the reduced images.

Equations (23)

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U x ( r ) = I ( r ) exp ( i k x · r ) .
W x ( r ) = F S ( r ) U x ( r ) = ( D x D y ) 1 u = 1 D x ν = 1 D y U x ( u + x 1 2 D x , ν + y 1 2 D y ) .
r ¯ μ , ν = r ¯ ( μ D x , ν D y ) = x , y r F ( μ D x x , ν D y y ) I ( x , y ) x , y F ( μ D x x , ν D y y ) I ( x , y ) , x μ , ν = lim w 0 1 w arg { F ( r ) I ( r ) e i w x } , y μ , ν = lim w 0 1 w arg { F ( r ) I ( r ) e i w y } .
t = I ( t ) e ( t c ) 2 / 2 σ 2 g = I ( g ) M e ( g c M ) 2 / 2 σ 2 M 2 .
δ φ ( c ) = φ ( c ) φ ( c M ) arg { W ( c ) } arg { W ( c M ) } .
I ( g ) = p = 1 1 δ g , [ t M ] + p t = 1 n I ( t ) K ( p , t ) = p = 1 1 t = 1 n I ( t ) K ( p , s ) δ g p , [ t M ] ,
I ( g ) = d = 0 R 1 p = 1 1 t = 0 N / R I ( t R + d ) K ( p , d ) δ g , t + [ d M ] + p ,
W ( c ) = t = I ( t ) e i k t F ( t c D ) , W ( c M ) = g = I ( g ) M e i k g / M F ( g c M D M ) ,
W ( c ) = t = I ( t ) e i k t F ( t c D ) , W ( c M ) = t , g = p = 1 1 I ( t ) K ( p , s ) δ g p , [ t M ] e i k g / M F ( g c M D M ) .
W ( c M ) = t = p = 1 1 I ( t ) F ( t M + p s c M D M ) K ( p , s ) e i k ( t M + p s ) / M = t I ( t ) e i k t p F ( t c D + p s D M ) K ( p , s ) e + i k ( p s ) / M .
F ( t c D + p s D M ) = F ( t c D ) + p s D M F ( t c D ) + 1 2 ( p s D M ) 2 F ( t c D + α ) ,
I ( t ) e i k t F ( t c D ) = I ( t ) e i k t F ( t c D ) p K ( p , s ) e i k ( p s ) / M .
0 = I ( t ) e i k t F ( t c D ) p p s D M K ( p , s ) e i k ( p s ) / M .
1 = p = 1 1 K ( p , s ) e i k ( p s ) ,
0 = p = 1 1 ( p s ) K ( p , s ) e i k ( p s ) ,
1 = p = 1 1 K ( p , s ) cos [ k ( p s ) ] , 0 = p = 1 1 K ( p , s ) sin [ k ( p s ) ] , 0 = p = 1 1 ( p + s ) K ( p , s ) sin [ k ( p s ) ] .
K ( p , s ) = 3 p 2 2 sin [ k ( p s ) ] · 1 2 cotan ( k s ) + cotan [ k ( 1 s ) ] + cotan [ k ( 1 s ) ] .
K ( 1 , s ) = ( | s | s ) / 2 , K ( 0 , s ) = 1 | s | , K ( 1 , s ) = ( | s | + s ) / 2.
a 1 = sin ( k s ) / sin k , B 2 = 1 2 [ s cos ( k s ) a 1 cos k ] , B 1 = 1 2 a 1 2 B 2 cos k , A 2 = 2 s B 2 , A 1 = 1 2 a 1 4 A 2 cos k , A 0 = cos ( k s ) 2 A 1 cos k 2 A 2 cos ( 2 k ) , K ( 0 , s ) = A 0 , K ( ± 1 , s ) = A 1 ± B 1 , K ( ± 2 , s ) = A 2 ± B 2 .
ε = 1 2 ( p + s D M ) 2 F ( t c D + α ) / F ( t c D ) .
δ φ ( c ) = arg { W ( c M ) } arg { W ( c ) } = arg p , g , t I ( t ) I ( g ) F ( t c D ) F ( g c + ( p s ) / M D ) K ( p , s ) e i k ( g t ) + i k ( p s ) / M = arg p , g , t I ( t ) I ( g ) [ D | t c | ] [ D | g c ( p s ) / M | ] K ( p , s ) e i k ( g t ) + i k ( p s ) / M .
1 = K ( p , s ) e i q 1 · ( p s ) , 1 = K ( p , s ) e i q 2 · ( p s ) , 0 = ( p s ) K ( p , s ) e i q 1 · ( p s ) , 0 = ( p s ) K ( p , s ) e i q 2 · ( p s ) ,
0 = K ( p , s ) sin [ q 1 · ( p s ) ] , 0 = K ( p , s ) sin [ q 2 · ( p s ) ] , 0 = K ( p , s ) ( p x s x ) sin [ q 1 · ( p s ) ] , 0 = K ( p , s ) ( p x s x ) sin [ q 2 · ( p s ) ] , 0 = K ( p , s ) ( p y s y ) sin [ q 1 · ( p s ) ] , 0 = K ( p , s ) ( p y s y ) sin [ q 2 · ( p s ) ] , 1 = K ( p , s ) { cos [ q 1 · ( p s ) ] + cos [ q 2 · ( p s ) ] } / 2.

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