Abstract

Super-resolution is an important goal of many image acquisition systems. Here we demonstrate the possibility of achieving super-resolution with a single exposure by combining the well known optical scheme of double random phase encoding which has been traditionally used for encryption with results from the relatively new and emerging field of compressive sensing. It is shown that the proposed model can be applied for recovering images from a general image degrading model caused by both diffraction and geometrical limited resolution.

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  1. S. Park, M. Park, and M. Gang, “Super-resolution image reconstruction: a technical overview,” IEEE Signal Process. Mag. 20(3), 21–36 (2003).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  8. P. Réfrégier and B. Javidi, “Optical image encryption based on input plane and Fourier plane random encoding,” Opt. Lett. 20(7), 767–769 (1995).
    [CrossRef] [PubMed]
  9. B. Javidi, G. Zhang, and J. Li, “Encrypted optical memory using double-random phase encoding,” Appl. Opt. 36(5), 1054–1058 (1997).
    [CrossRef] [PubMed]
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    [CrossRef]
  11. E. Tajahuerce, O. Matoba, S. C. Verrall, and B. Javidi, “Optoelectronic information encryption with phase-shifting interferometry,” Appl. Opt. 39(14), 2313–2320 (2000).
    [CrossRef]
  12. E. Tajahuerce, J. Lancis, P. Andres, V. Climent, and B. Javidi, “Optoelectronic Information Encryption with Incoherent Light,” in Optical and Digital Techniques for Information Security, B. Javidi, ed. (Springer-Verlag, 2004).
  13. O. Matoba and B. Javidi, “Encrypted optical memory system using three-dimensional keys in the Fresnel domain,” Opt. Lett. 24(11), 762–764 (1999).
    [CrossRef]
  14. G. Unnikrishnan, J. Joseph, and K. Singh, “Optical encryption by double-random phase encoding in the fractional Fourier domain,” Opt. Lett. 25(12), 887–889 (2000).
    [CrossRef]
  15. P. C. Mogensen and J. Glückstad, “Phase-only optical encryption,” Opt. Lett. 25(8), 566–568 (2000).
    [CrossRef]
  16. B. M. Hennelly, T. J. Naughton, J. McDonald, J. T. Sheridan, G. Unnikrishnan, D. P. Kelly, and B. Javidi, “Spread-space spread-spectrum technique for secure multiplexing,” Opt. Lett. 32(9), 1060–1062 (2007).
    [CrossRef] [PubMed]
  17. O. Matoba and B. Javidi, “Encrypted optical storage with angular multiplexing,” Appl. Opt. 38(35), 7288–7293 (1999).
    [CrossRef]
  18. E. Tajahuerce and B. Javidi, “Encrypting three-dimensional information with digital holography,” Appl. Opt. 39(35), 6595–6601 (2000).
    [CrossRef]
  19. X. Tan, O. Matoba, Y. Okada-Shudo, M. Ide, T. Shimura, and K. Kuroda, “Secure optical memory system with polarization encryption,” Appl. Opt. 40(14), 2310–2315 (2001).
    [CrossRef]
  20. D. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006).
    [CrossRef]
  21. E. Candes and M. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag. 25(2), 21–30 (2008).
    [CrossRef]
  22. T. Do, T. Tran, and L. Gan, “Fast compressive sampling with structurally random matrices,” in Proc. ICASSP, 3369–3372, (2008).
  23. J. Romberg, “Compressive sensing by random convolution,” SIAM J. Imaging Sci. 2(4), 1098–1128 (2009).
    [CrossRef]
  24. Y. Rivenson, A. Stern, and B. Javidi, “Compressive Fresnel Holography,” to appear in IEEE/OSA J. on Display Technology, (2010).
  25. A. Stern and B. Javidi, “Random projections imaging with extended space-bandwidth product,” IEEE/OSA Journal on Display Technology, 3(3), 315–320 (2007).

2009

O. Matoba, T. Nomura, E. Perez-Cabre, M. S. Millan, and B. Javidi, “Optical techniques for information security,” Proc. IEEEl 97(6), 1128–1148 (2009).
[CrossRef]

J. Romberg, “Compressive sensing by random convolution,” SIAM J. Imaging Sci. 2(4), 1098–1128 (2009).
[CrossRef]

A. Borkowski, Z. Zalevsky, and B. Javidi, “Geometrical superresolved imaging using nonperiodic spatial masking,” J. Opt. Soc. Am. A 26(3), 589–601 (2009).
[CrossRef]

S. Prasad and X. Luo, “Support-assisted optical superresolution of low-resolution image sequences: the one-dimensional problem,” Opt. Express 17(25), 23213–23233 (2009).
[CrossRef]

2008

E. Candes and M. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag. 25(2), 21–30 (2008).
[CrossRef]

2007

2006

D. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006).
[CrossRef]

2005

2004

S. Farsiu, M. D. Robinson, M. Elad, and P. Milanfar, “Fast and robust multiframe super resolution,” IEEE Trans. Image Process. 13(10), 1327–1344 (2004).
[CrossRef] [PubMed]

2003

S. Park, M. Park, and M. Gang, “Super-resolution image reconstruction: a technical overview,” IEEE Signal Process. Mag. 20(3), 21–36 (2003).
[CrossRef]

2001

2000

1999

1997

1995

Ben-Dor, A.

Borkowski, A.

Candes, E.

E. Candes and M. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag. 25(2), 21–30 (2008).
[CrossRef]

Donoho, D.

D. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006).
[CrossRef]

Elad, M.

S. Farsiu, M. D. Robinson, M. Elad, and P. Milanfar, “Fast and robust multiframe super resolution,” IEEE Trans. Image Process. 13(10), 1327–1344 (2004).
[CrossRef] [PubMed]

Farsiu, S.

S. Farsiu, M. D. Robinson, M. Elad, and P. Milanfar, “Fast and robust multiframe super resolution,” IEEE Trans. Image Process. 13(10), 1327–1344 (2004).
[CrossRef] [PubMed]

Fixler, D.

Gang, M.

S. Park, M. Park, and M. Gang, “Super-resolution image reconstruction: a technical overview,” IEEE Signal Process. Mag. 20(3), 21–36 (2003).
[CrossRef]

García, J.

Glückstad, J.

Hennelly, B. M.

Ide, M.

Javidi, B.

A. Borkowski, Z. Zalevsky, and B. Javidi, “Geometrical superresolved imaging using nonperiodic spatial masking,” J. Opt. Soc. Am. A 26(3), 589–601 (2009).
[CrossRef]

O. Matoba, T. Nomura, E. Perez-Cabre, M. S. Millan, and B. Javidi, “Optical techniques for information security,” Proc. IEEEl 97(6), 1128–1148 (2009).
[CrossRef]

B. M. Hennelly, T. J. Naughton, J. McDonald, J. T. Sheridan, G. Unnikrishnan, D. P. Kelly, and B. Javidi, “Spread-space spread-spectrum technique for secure multiplexing,” Opt. Lett. 32(9), 1060–1062 (2007).
[CrossRef] [PubMed]

E. Tajahuerce and B. Javidi, “Encrypting three-dimensional information with digital holography,” Appl. Opt. 39(35), 6595–6601 (2000).
[CrossRef]

E. Tajahuerce, O. Matoba, S. C. Verrall, and B. Javidi, “Optoelectronic information encryption with phase-shifting interferometry,” Appl. Opt. 39(14), 2313–2320 (2000).
[CrossRef]

O. Matoba and B. Javidi, “Encrypted optical memory system using three-dimensional keys in the Fresnel domain,” Opt. Lett. 24(11), 762–764 (1999).
[CrossRef]

O. Matoba and B. Javidi, “Encrypted optical storage with angular multiplexing,” Appl. Opt. 38(35), 7288–7293 (1999).
[CrossRef]

B. Javidi, G. Zhang, and J. Li, “Encrypted optical memory using double-random phase encoding,” Appl. Opt. 36(5), 1054–1058 (1997).
[CrossRef] [PubMed]

P. Réfrégier and B. Javidi, “Optical image encryption based on input plane and Fourier plane random encoding,” Opt. Lett. 20(7), 767–769 (1995).
[CrossRef] [PubMed]

Joseph, J.

Kelly, D. P.

Kopeika, N. S.

Kuroda, K.

Li, J.

Luo, X.

Matoba, O.

McDonald, J.

Milanfar, P.

S. Farsiu, M. D. Robinson, M. Elad, and P. Milanfar, “Fast and robust multiframe super resolution,” IEEE Trans. Image Process. 13(10), 1327–1344 (2004).
[CrossRef] [PubMed]

Millan, M. S.

O. Matoba, T. Nomura, E. Perez-Cabre, M. S. Millan, and B. Javidi, “Optical techniques for information security,” Proc. IEEEl 97(6), 1128–1148 (2009).
[CrossRef]

Mogensen, P. C.

Naughton, T. J.

Nomura, T.

O. Matoba, T. Nomura, E. Perez-Cabre, M. S. Millan, and B. Javidi, “Optical techniques for information security,” Proc. IEEEl 97(6), 1128–1148 (2009).
[CrossRef]

Okada-Shudo, Y.

Park, M.

S. Park, M. Park, and M. Gang, “Super-resolution image reconstruction: a technical overview,” IEEE Signal Process. Mag. 20(3), 21–36 (2003).
[CrossRef]

Park, S.

S. Park, M. Park, and M. Gang, “Super-resolution image reconstruction: a technical overview,” IEEE Signal Process. Mag. 20(3), 21–36 (2003).
[CrossRef]

Perez-Cabre, E.

O. Matoba, T. Nomura, E. Perez-Cabre, M. S. Millan, and B. Javidi, “Optical techniques for information security,” Proc. IEEEl 97(6), 1128–1148 (2009).
[CrossRef]

Porat, Y.

Prasad, S.

Réfrégier, P.

Robinson, M. D.

S. Farsiu, M. D. Robinson, M. Elad, and P. Milanfar, “Fast and robust multiframe super resolution,” IEEE Trans. Image Process. 13(10), 1327–1344 (2004).
[CrossRef] [PubMed]

Romberg, J.

J. Romberg, “Compressive sensing by random convolution,” SIAM J. Imaging Sci. 2(4), 1098–1128 (2009).
[CrossRef]

Sheridan, J. T.

Shimura, T.

Singh, K.

Stern, A.

Tajahuerce, E.

Tan, X.

Unnikrishnan, G.

Verrall, S. C.

Wakin, M.

E. Candes and M. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag. 25(2), 21–30 (2008).
[CrossRef]

Zalevsky, Z.

Zhang, G.

Appl. Opt.

IEEE Signal Process. Mag.

S. Park, M. Park, and M. Gang, “Super-resolution image reconstruction: a technical overview,” IEEE Signal Process. Mag. 20(3), 21–36 (2003).
[CrossRef]

E. Candes and M. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag. 25(2), 21–30 (2008).
[CrossRef]

IEEE Trans. Image Process.

S. Farsiu, M. D. Robinson, M. Elad, and P. Milanfar, “Fast and robust multiframe super resolution,” IEEE Trans. Image Process. 13(10), 1327–1344 (2004).
[CrossRef] [PubMed]

IEEE Trans. Inf. Theory

D. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Express

Opt. Lett.

Proc. IEEEl

O. Matoba, T. Nomura, E. Perez-Cabre, M. S. Millan, and B. Javidi, “Optical techniques for information security,” Proc. IEEEl 97(6), 1128–1148 (2009).
[CrossRef]

SIAM J. Imaging Sci.

J. Romberg, “Compressive sensing by random convolution,” SIAM J. Imaging Sci. 2(4), 1098–1128 (2009).
[CrossRef]

Other

Y. Rivenson, A. Stern, and B. Javidi, “Compressive Fresnel Holography,” to appear in IEEE/OSA J. on Display Technology, (2010).

A. Stern and B. Javidi, “Random projections imaging with extended space-bandwidth product,” IEEE/OSA Journal on Display Technology, 3(3), 315–320 (2007).

T. Do, T. Tran, and L. Gan, “Fast compressive sampling with structurally random matrices,” in Proc. ICASSP, 3369–3372, (2008).

E. Tajahuerce, J. Lancis, P. Andres, V. Climent, and B. Javidi, “Optoelectronic Information Encryption with Incoherent Light,” in Optical and Digital Techniques for Information Security, B. Javidi, ed. (Springer-Verlag, 2004).

Z. Zalevsky and D. Mendlovic, Optical Super Resolution (Springer-Verlag, 2003).

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Figures (6)

Fig. 1
Fig. 1

Block diagram of the double phase encoding process. ℑ denotes Fourier transform operator.

Fig. 2
Fig. 2

DPRE Implementation of Fig. 1 using a 4f optical scheme.

Fig. 3
Fig. 3

Imaging scheme of compressed sensing [25]

Fig. 4
Fig. 4

(a) Original USAF 1024x1024 pixels resolution chart image, with pixel spacing ∆ (b) DRPE image using a 512x256 pixels, 2x4 pixels averaged captured image. (c) Image reconstructed from DRPE captured image (b). (d) zoom-in to (a). (e) Result of downsampling the image in (a); taking 512x256 regular samples from the original image averaged with a 2x4 pixels kernel and then up-sampled to size of 1024x1024 by bi-cubic interpolation. (f) Zoom in to (c) which is captured with DRPE-CS.

Fig. 5
Fig. 5

(a) The target in Fig. 4(a) blurred by an aperture with f cutoff = f nom / 6 , and additive noise yielding SNR of 37 dB. (b) Reconstruction from data captured with DRPE-CS. (c) Zoom in to the original object (Fig. 4 (a)). (d) zoom-in to (a). (e) Zoom in to (b) which is captured with DRPE-CS.

Fig. 6
Fig. 6

(a) Zoom in to the original object (Fig. 4 (a)). (b) Blurring by an aperture with f cutoff = f nom / 5 , sensor down-sampling by a 2x2 factor, and an additive measurement noise to yield 37 db SNR. (c) Reconstruction result using the DRPE-CS strategy.

Equations (11)

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u = Φ t ,
μ = N max i j | < φ i , ψ j > | ,
α ^ = min α α 1 s u b j e c t t o u = Φ Ψ α ,
M C μ 2 ( Φ , Ψ ) K log ( N ) ,
u = F * HFPt ,
F P = [ 1 1 . . 1 1 W N . . W N ( N 1 ) . . . . . . 1 W N ( N 1 ) . W N ( N 1 ) ( N 1 ) ] [ e j 2 π p 1 0 0 0 0 . e j 2 π p 2 0 0 0 . . . . . . 0 . . . e j 2 π p N ] = = [ e j 2 π p 1 e j 2 π p 2 . . e j 2 π p N e j 2 π p 1 W N e j 2 π p 2 . . W N ( N 1 ) e j 2 π p N . . . . . . e j 2 π p 1 W N ( N 1 ) e j 2 π p 2 . W N ( N 1 ) ( N 1 ) e j 2 π p N ] ,
u L ( x , y ) = D ( L ) { t ( x , y ) exp [ i 2 π p ( x , y ) ] } h ( x , y ) h s ,
u L = D ( L ) u = 1 4 S D u = 1 4 [ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 ] [ 1 1 1 1 0 0 0 0 0 1 1 1 1 0 0 0 0 0 1 1 1 1 0 0 0 0 0 1 1 1 1 0 0 0 0 0 1 1 1 1 ] u ,
min | Ψ T t | 1 + γ T V ( t ) s . t . u = D F 1 H F P t ,
T V ( x ) = i , j ( x i + 1 , j x i , j ) 2 + ( x i , j + 1 x i , j ) 2 .
min | Ψ T t | 1 + γ T V ( t ) s . t . u = F 1 H F A P t ,

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