Abstract

Phase retrieval is a wavefront sensing method that uses a series of intensity images to reconstruct the wavefront. The resolution of phase retrieval testing is limited mainly by the resolution of intensity images captured by CCD cameras. A subpixel phase retrieval method is presented to retrieve the wave field at subpixel resolution by using the information of a sequence of low-resolution images captured along the propagation direction. In this method, the sampling interval for the wave field under test is smaller than the CCD pixel size in phase reconstruction. The wave field is recovered at subpixel resolution by utilizing the energy conservation relationship between CCD pixels and their subpixels by the subpixel phase retrieval (SPR) algorithm. Numerical experiments have shown that more than a fourfold resolution enhancement can be achieved. The method has also been studied in some experiments under noisy and off-axis conditions. A mirror surface testing experiment was conducted to demonstrate the performance of SPR in the real world. The results of these experiments have shown the effectiveness and robustness of this method. SPR allows low-resolution images to be used to retrieve high-resolution wave fields and will be useful in testing wave fields from large objects.

© 2008 Optical Society of America

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References

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2006 (3)

2005 (1)

2004 (2)

H. I. Campbell and A. H. S. Zhang, “Greenaway, generalized phase diversity for wave-front sensing,” Opt. Lett. 29, 2707-2709 (2004).
[Crossref] [PubMed]

M. O. Catherine, A. F. Jessica, “Phase retrieval camera optical testing of the Advanced Mirror System Demonstrator (AMSD),” Proc. SPIE 5487, 1744-1753 (2004).

2003 (1)

S. C. Park, M. K. Park, and M. G. Kang, “Superresolution image reconstruction: a technical overview,” IEEE Signal Process. Mag. 20(3), 21-36 (2003).
[Crossref]

2001 (2)

D. Rajan and S. Chaudhuri, “Generalized interpolation and its applications in super-resolution imaging,” Image Vision Comput. 19, 957-969 (2001).
[Crossref]

M. Elad and Y. Hel-Or, “A fast super-resolution reconstruction algorithm for pure translational motion and common space-invariant blur,” IEEE Trans. Image Process. 10, 1187-1193(2001).
[Crossref]

2000 (1)

M. S. Alam, J. G. Bognar, R. C. Hardie, and B. J. Yasuda, “Infrared image registration and high-resolution reconstruction using multiple translationally shifted aliased video frames,” IEEE Trans. Instrum. Meas. 49, 915-923 (2000).
[Crossref]

1999 (1)

1997 (1)

M. Elad and A. Feuer, “Restoration of a single superresolution image from several blurred, noisy, and undersampled measured images,” IEEE Trans. Image Process. 6, 1646-1658 (1997).
[Crossref]

1994 (1)

1993 (1)

1990 (1)

D. J. Shpak and A. Antoniou, “A generalized Remez method for the design of FIR digital filters,” IEEE Trans. Circuits Syst. 37, 161-174 (1990).
[Crossref]

1989 (1)

1988 (1)

M. Bierling, “Displacement estimation by hierarchical block matching,” Proc. SPIE 1001, 942-951 (1988).

1982 (1)

1972 (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237-246 (1972).

Alam, M. S.

M. S. Alam, J. G. Bognar, R. C. Hardie, and B. J. Yasuda, “Infrared image registration and high-resolution reconstruction using multiple translationally shifted aliased video frames,” IEEE Trans. Instrum. Meas. 49, 915-923 (2000).
[Crossref]

Almoro, P.

P. Almoro, G. Pedrini, and W. Osten, “Complete wavefront reconstruction using sequential intensity measurements of a volume speckle field,” Appl. Opt 45, 8596-8605(2006).
[Crossref] [PubMed]

Antoniou, A.

D. J. Shpak and A. Antoniou, “A generalized Remez method for the design of FIR digital filters,” IEEE Trans. Circuits Syst. 37, 161-174 (1990).
[Crossref]

Bertero, M.

M. Bertero and C. Demol, “Superresolution by data inversion,” in Progress in Optics, E. Wolf, ed. (Elsevier North-Holland, 1996), Vol. 36, pp. 129-178.
[Crossref]

Bierling, M.

M. Bierling, “Displacement estimation by hierarchical block matching,” Proc. SPIE 1001, 942-951 (1988).

Bognar, J. G.

M. S. Alam, J. G. Bognar, R. C. Hardie, and B. J. Yasuda, “Infrared image registration and high-resolution reconstruction using multiple translationally shifted aliased video frames,” IEEE Trans. Instrum. Meas. 49, 915-923 (2000).
[Crossref]

Borman, S.

S. Borman and R. L. Stevenson, “Superresolution from image sequences--a review,” Proceedings of the 1998 Midwest Symposium on Circuits and Systems (IEEE, 1998), pp. 374-378.

Brady, G. R.

Campbell, H. I.

Catherine, M. O.

M. O. Catherine, A. F. Jessica, “Phase retrieval camera optical testing of the Advanced Mirror System Demonstrator (AMSD),” Proc. SPIE 5487, 1744-1753 (2004).

Chaudhuri, S.

D. Rajan and S. Chaudhuri, “Generalized interpolation and its applications in super-resolution imaging,” Image Vision Comput. 19, 957-969 (2001).
[Crossref]

Demol, C.

M. Bertero and C. Demol, “Superresolution by data inversion,” in Progress in Optics, E. Wolf, ed. (Elsevier North-Holland, 1996), Vol. 36, pp. 129-178.
[Crossref]

Dong, B.

Driggers, R. G.

Elad, M.

M. Elad and Y. Hel-Or, “A fast super-resolution reconstruction algorithm for pure translational motion and common space-invariant blur,” IEEE Trans. Image Process. 10, 1187-1193(2001).
[Crossref]

M. Elad and A. Feuer, “Restoration of a single superresolution image from several blurred, noisy, and undersampled measured images,” IEEE Trans. Image Process. 6, 1646-1658 (1997).
[Crossref]

Erosy, O. K.

Feuer, A.

M. Elad and A. Feuer, “Restoration of a single superresolution image from several blurred, noisy, and undersampled measured images,” IEEE Trans. Image Process. 6, 1646-1658 (1997).
[Crossref]

Fienup, J. R.

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237-246 (1972).

Gu, B.

Hardie, R. C.

M. S. Alam, J. G. Bognar, R. C. Hardie, and B. J. Yasuda, “Infrared image registration and high-resolution reconstruction using multiple translationally shifted aliased video frames,” IEEE Trans. Instrum. Meas. 49, 915-923 (2000).
[Crossref]

Hel-Or, Y.

M. Elad and Y. Hel-Or, “A fast super-resolution reconstruction algorithm for pure translational motion and common space-invariant blur,” IEEE Trans. Image Process. 10, 1187-1193(2001).
[Crossref]

Jessica, A. F.

M. O. Catherine, A. F. Jessica, “Phase retrieval camera optical testing of the Advanced Mirror System Demonstrator (AMSD),” Proc. SPIE 5487, 1744-1753 (2004).

Kang, M. G.

S. C. Park, M. K. Park, and M. G. Kang, “Superresolution image reconstruction: a technical overview,” IEEE Signal Process. Mag. 20(3), 21-36 (2003).
[Crossref]

Marron, J. C.

Oskoui, P.

Osten, W.

P. Almoro, G. Pedrini, and W. Osten, “Complete wavefront reconstruction using sequential intensity measurements of a volume speckle field,” Appl. Opt 45, 8596-8605(2006).
[Crossref] [PubMed]

G. Pedrini, W. Osten and Y. Zhang. “Wave-front reconstruction from a sequence of interferograms recorded at different planes,” Opt. Lett. 30, 833-835 (2005).
[Crossref] [PubMed]

Park, M. K.

S. C. Park, M. K. Park, and M. G. Kang, “Superresolution image reconstruction: a technical overview,” IEEE Signal Process. Mag. 20(3), 21-36 (2003).
[Crossref]

Park, S. C.

S. C. Park, M. K. Park, and M. G. Kang, “Superresolution image reconstruction: a technical overview,” IEEE Signal Process. Mag. 20(3), 21-36 (2003).
[Crossref]

Pedrini, G.

P. Almoro, G. Pedrini, and W. Osten, “Complete wavefront reconstruction using sequential intensity measurements of a volume speckle field,” Appl. Opt 45, 8596-8605(2006).
[Crossref] [PubMed]

G. Pedrini, W. Osten and Y. Zhang. “Wave-front reconstruction from a sequence of interferograms recorded at different planes,” Opt. Lett. 30, 833-835 (2005).
[Crossref] [PubMed]

Rajan, D.

D. Rajan and S. Chaudhuri, “Generalized interpolation and its applications in super-resolution imaging,” Image Vision Comput. 19, 957-969 (2001).
[Crossref]

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237-246 (1972).

Schulz, T. J.

Seldin, J. H.

Shpak, D. J.

D. J. Shpak and A. Antoniou, “A generalized Remez method for the design of FIR digital filters,” IEEE Trans. Circuits Syst. 37, 161-174 (1990).
[Crossref]

Stark, H.

Stevenson, R. L.

S. Borman and R. L. Stevenson, “Superresolution from image sequences--a review,” Proceedings of the 1998 Midwest Symposium on Circuits and Systems (IEEE, 1998), pp. 374-378.

Yang, G.

Yasuda, B. J.

M. S. Alam, J. G. Bognar, R. C. Hardie, and B. J. Yasuda, “Infrared image registration and high-resolution reconstruction using multiple translationally shifted aliased video frames,” IEEE Trans. Instrum. Meas. 49, 915-923 (2000).
[Crossref]

Young, S. S.

Zhang, A. H. S.

Zhang., Y.

Zhuang, J.

Appl. Opt (1)

P. Almoro, G. Pedrini, and W. Osten, “Complete wavefront reconstruction using sequential intensity measurements of a volume speckle field,” Appl. Opt 45, 8596-8605(2006).
[Crossref] [PubMed]

Appl. Opt. (4)

IEEE Signal Process. Mag. (1)

S. C. Park, M. K. Park, and M. G. Kang, “Superresolution image reconstruction: a technical overview,” IEEE Signal Process. Mag. 20(3), 21-36 (2003).
[Crossref]

IEEE Trans. Circuits Syst. (1)

D. J. Shpak and A. Antoniou, “A generalized Remez method for the design of FIR digital filters,” IEEE Trans. Circuits Syst. 37, 161-174 (1990).
[Crossref]

IEEE Trans. Image Process. (2)

M. Elad and Y. Hel-Or, “A fast super-resolution reconstruction algorithm for pure translational motion and common space-invariant blur,” IEEE Trans. Image Process. 10, 1187-1193(2001).
[Crossref]

M. Elad and A. Feuer, “Restoration of a single superresolution image from several blurred, noisy, and undersampled measured images,” IEEE Trans. Image Process. 6, 1646-1658 (1997).
[Crossref]

IEEE Trans. Instrum. Meas. (1)

M. S. Alam, J. G. Bognar, R. C. Hardie, and B. J. Yasuda, “Infrared image registration and high-resolution reconstruction using multiple translationally shifted aliased video frames,” IEEE Trans. Instrum. Meas. 49, 915-923 (2000).
[Crossref]

Image Vision Comput. (1)

D. Rajan and S. Chaudhuri, “Generalized interpolation and its applications in super-resolution imaging,” Image Vision Comput. 19, 957-969 (2001).
[Crossref]

J. Opt. Soc. Am. A (2)

Opt. Express (1)

Opt. Lett. (2)

Optik (Stuttgart) (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237-246 (1972).

Proc. SPIE (2)

M. O. Catherine, A. F. Jessica, “Phase retrieval camera optical testing of the Advanced Mirror System Demonstrator (AMSD),” Proc. SPIE 5487, 1744-1753 (2004).

M. Bierling, “Displacement estimation by hierarchical block matching,” Proc. SPIE 1001, 942-951 (1988).

Other (2)

S. Borman and R. L. Stevenson, “Superresolution from image sequences--a review,” Proceedings of the 1998 Midwest Symposium on Circuits and Systems (IEEE, 1998), pp. 374-378.

M. Bertero and C. Demol, “Superresolution by data inversion,” in Progress in Optics, E. Wolf, ed. (Elsevier North-Holland, 1996), Vol. 36, pp. 129-178.
[Crossref]

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Figures (10)

Fig. 1
Fig. 1

Experimental setup for SPR.

Fig. 2
Fig. 2

Subpixel intensity constraint function in the one-dimensional cases of (a) step SICF p and (b) filtered SICF L p .

Fig. 3
Fig. 3

(a) Phase map to be tested and CCD images at different resolutions at z = 5 mm and resolutions (b)  128 × 128 , (c)  64 × 64 , (d)  32 × 32 .

Fig. 4
Fig. 4

Phase map retrieved results: (a), (c), (e) traditional PR results at 128 × 128 , 64 × 64 , and 32 × 32 resolution; (b), (d), (f) SPR results based on different images at 128 × 128 , 64 × 64 , and 32 × 32 resolution. (g) Image error E k versus iteration k at different resolutions.

Fig. 5
Fig. 5

Phase map retrieved results with different noise levels: (a) 1% and RMS error 0.012 λ ; (b) 2% and RMS error 0.019 λ ; (c) 5% and RMS error 0.035 λ ; (d) 10% and RMS error 0.061 λ . (e) Image error E k versus iteration k at different noise levels.

Fig. 6
Fig. 6

Phase map retrieved results with 5% noise and different quantization accuracies: (a) no quantization error; (b)  10   bits and RMS error 0.012 λ ; (c)  8 bits and RMS error 0.015 λ ; (d)  6   bits and RMS error 0.023 λ .

Fig. 7
Fig. 7

Phase map retrieved results with different off-axis deviations: (a)  Δ = 1 / 8   pixel and RMS error 0.016 λ ; (b)  Δ = 1 / 4   pixel and RMS error 0.034 λ ; (c)  Δ = 1 / 2   pixel and RMS error 0.084 ; (d)  Δ = 1   pixel and RMS error 0.14 λ ; (e) Image error E k versus iteration k at different off-axis deviation levels.

Fig. 8
Fig. 8

Ball mirror testing setup.

Fig. 9
Fig. 9

ϕ 250 mm ball mirror diffraction images at (a)  4 2.5 mm in front of the focus; (b)  4 2.5 mm behind the focus.

Fig. 10
Fig. 10

ϕ 250 mm ball mirror testing results: (a) interferometer testing; (b) PR testing; (c) SPR testing.

Tables (1)

Tables Icon

Table 1 Correlations of SPR Reconstruction under Different Numbers of Images and Measurement Intervals at 64 × 64 Resolution

Equations (8)

Equations on this page are rendered with MathJax. Learn more.

U p + 1 ( x , y ) = I 1 ( I { U p ( x , y ) } exp [ j 2 π λ ( d p + 1 d p ) ( 1 λ 2 f x 2 λ 2 f y 2 ) 1 / 2 ] ) ,
U p + 1 ( k Δ x , l Δ y ) = A p + 1 ( k Δ x , l Δ y ) exp [ j φ p + 1 ( k Δ x , l Δ y ) ] = IFF t ( FF t { [ I p ( m Δ x , n Δ y ) ] 1 / 2 exp [ j φ ( k Δ x , l Δ y , d p ) ] } exp [ j 2 π λ ( d p + 1 d p ) ( 1 λ 2 m 2 Δ x 2 λ 2 n 2 Δ y 2 ) 1 / 2 ] ) ,
I ( m , n ) = a = 1 N b = 1 N I s ( m , n ; a , b ) ,
I s p ^ ( m , n ; a , b ) = I s p ( m , n ; a , b ) + c [ I p ( m , n ) a = 1 N b = 1 N I s p ( m , n ; a , b ) ] / N 2 ,
SICF L p ( m , n ; a , b ) = SICF p ( m , n ) * F L ( m , n ; a , b ) ,
SICF L W p ( m , n ; a , b ) = SICF L p ( m , n ; a , b ) × W p ( m , n ; a , b ) ,
W p ( m , n ; a , b ) = { I s p ( m , n ; a , b ) / a = 1 N b = 1 N I s p ( m , n ; a , b ) if     SICF L p ( m , n ) < 0 1 I s p ( m , n ; a , b ) / a = 1 N b = 1 N I s p ( m , n ; a , b if     SICF L p ( m , n ) > 0 .
E k = ln { p = 1 P m , n = 1 M | I p ( m , n ) a = 1 N b = 1 N I s p ( m , n ; a , b ) ] | } ,

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