Abstract

Creating a large-scale synthetic aperture makes it possible to break the resolution boundaries dictated by the wave nature of light of common optical systems. However, their implementation is challenging, since the generation of a large size continuous mosaic synthetic aperture composed of many patterns is complicated in terms of both phase matching and time-multiplexing duration. In this study we present an advanced configuration for an incoherent holographic imaging system with super resolution qualities that creates a partial synthetic aperture. The new system, termed sparse synthetic aperture with Fresnel elements (S-SAFE), enables significantly decreasing the number of the recorded elements, and it is free from positional constrains on their location. Additionally, in order to obtain the best image quality we propose an optimal mosaicking structure derived on the basis of physical and numerical considerations, and introduce three reconstruction approaches which are compared and discussed. The super-resolution capabilities of the proposed scheme and its limitations are analyzed, numerically simulated and experimentally demonstrated.

© 2015 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Could SAFE concept be applied for designing a new synthetic aperture telescope?

Barak Katz and Joseph Rosen
Opt. Express 19(6) 4924-4936 (2011)

Coded aperture correlation holography–a new type of incoherent digital holograms

A. Vijayakumar, Yuval Kashter, Roy Kelner, and Joseph Rosen
Opt. Express 24(11) 12430-12441 (2016)

References

  • View by:
  • |
  • |
  • |

  1. E. Abbe, “Beiträge zur theorie des mikroskops und der mikroskopischen wahrnehmung,” Archiv. Microskopische Anat. 9(1), 413–418 (1873).
    [Crossref]
  2. A. A. Michelson, “On the application of interference methods to astronomical measurements,” Astrophys. J. 51, 257–262 (1920).
    [Crossref] [PubMed]
  3. P. R. Lawson, Selected Paper on Long Baseline Stellar Interferometry (SPIE Press Book, 1997).
  4. S. M. Beck, J. R. Buck, W. F. Buell, R. P. Dickinson, D. A. Kozlowski, N. J. Marechal, and T. J. Wright, “Synthetic-aperture imaging laser radar: laboratory demonstration and signal processing,” Appl. Opt. 44(35), 7621–7629 (2005).
    [Crossref] [PubMed]
  5. V. Micó, Z. Zalevsky, P. García-Martínez, and J. García, “Synthetic aperture superresolution with multiple off-axis holograms,” J. Opt. Soc. Am. A 23(12), 3162–3170 (2006).
    [Crossref] [PubMed]
  6. G. Indebetouw, Y. Tada, J. Rosen, and G. Brooker, “Scanning holographic microscopy with resolution exceeding the Rayleigh limit of the objective by superposition of off-axis holograms,” Appl. Opt. 46(6), 993–1000 (2007).
    [Crossref] [PubMed]
  7. L. Martínez-León and B. Javidi, “Synthetic aperture single-exposure on-axis digital holography,” Opt. Express 16(1), 161–169 (2008).
    [Crossref] [PubMed]
  8. L. Granero, V. Micó, Z. Zalevsky, and J. García, “Synthetic aperture superresolved microscopy in digital lensless Fourier holography by time and angular multiplexing of the object information,” Appl. Opt. 49(5), 845–857 (2010).
    [Crossref] [PubMed]
  9. K. Ji, P. Gao, J. Min, R. Guo, and N. Menke, “A synthetic aperture telescope based on a pair of gratings,” J. Mod. Opt. 60(15), 1229–1233 (2013).
    [Crossref]
  10. B. Katz and J. Rosen, “Super-resolution in incoherent optical imaging using synthetic aperture with Fresnel elements,” Opt. Express 18(2), 962–972 (2010).
    [Crossref] [PubMed]
  11. J. Rosen and G. Brooker, “Digital spatially incoherent Fresnel holography,” Opt. Lett. 32(8), 912–914 (2007).
    [Crossref] [PubMed]
  12. J. Rosen and G. Brooker, “Non-scanning motionless fluorescence three-dimensional holographic microscopy,” Nat. Photonics 2(3), 190–195 (2008).
    [Crossref]
  13. B. Katz and J. Rosen, “Could SAFE concept be applied for designing a new synthetic aperture telescope?” Opt. Express 19(6), 4924–4936 (2011).
    [Crossref] [PubMed]
  14. Y. Kashter and J. Rosen, “Enhanced-resolution using modified configuration of Fresnel incoherent holographic recorder with synthetic aperture,” Opt. Express 22(17), 20551–20565 (2014).
    [Crossref] [PubMed]
  15. J. Rosen, N. Siegel, and G. Brooker, “Theoretical and experimental demonstration of resolution beyond the Rayleigh limit by FINCH fluorescence microscopic imaging,” Opt. Express 19(27), 26249–26268 (2011).
    [Crossref] [PubMed]
  16. B. Katz, J. Rosen, R. Kelner, and G. Brooker, “Enhanced resolution and throughput of Fresnel incoherent correlation holography (FINCH) using dual diffractive lenses on a spatial light modulator (SLM),” Opt. Express 20(8), 9109–9121 (2012).
    [Crossref] [PubMed]
  17. A. E. Tippie, A. Kumar, and J. R. Fienup, “High-resolution synthetic-aperture digital holography with digital phase and pupil correction,” Opt. Express 19(13), 12027–12038 (2011).
    [Crossref] [PubMed]
  18. N. J. Miller, M. P. Dierking and B. D. Duncan, “Optical sparse aperture imaging,” Appl. Opt. 46(23), 0003–6935(2007).
    [Crossref]
  19. Y. Rivenson, A. Stern, and B. Javidi, “Compressive Fresnel holography,” J. Disp. Technol. 6(10), 506–509 (2010).
    [Crossref]
  20. E. Candès and J. Romberg, “Sparsity and incoherence in compressive sampling,” Inverse Probl. 23(3), 969–985 (2007).
    [Crossref]
  21. E. Candès and M. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag. 25(2), 21–30 (2008).
    [Crossref]
  22. G. Brooker, N. Siegel, V. Wang, and J. Rosen, “Optimal resolution in Fresnel incoherent correlation holographic fluorescence microscopy,” Opt. Express 19(6), 5047–5062 (2011).
    [Crossref] [PubMed]
  23. J. W. Goodman, Introduction to Fourier optics, 3rd Ed. (Roberts and Company Publishers, 2005).
  24. Y. Rivenson, A. Stern, and B. Javidi, “Overview of compressive sensing techniques applied in holography [Invited],” Appl. Opt. 52(1), A423–A432 (2013).
    [Crossref] [PubMed]
  25. J. M. Bioucas-Dias and M. A. T. Figueiredo, “A new TwIST: two-step iterative shrinkage/thresholding algorithms for image restoration,” IEEE Trans. Image Process. 16(12), 2992–3004 (2007).
    [Crossref] [PubMed]
  26. P. Chatterjee, S. Mukherjee, S. Chaudhuri, and G. Seetharaman, “Application of Papoulis-Gerchberg method in image super-resolution and inpainting,” Comput. J. 52(1), 80–89 (2009).
    [Crossref]
  27. Y. Rivenson, A. Rot, S. Balber, A. Stern, and J. Rosen, “Recovery of partially occluded objects by applying compressive Fresnel holography,” Opt. Lett. 37(10), 1757–1759 (2012).
    [Crossref] [PubMed]
  28. B. K. Jennison, J. P. Allebach, and D. W. Sweeney, “Iterative approaches to computer-generated holography,” Opt. Eng. 28(6), 286629 (1989).
    [Crossref]

2014 (1)

2013 (2)

K. Ji, P. Gao, J. Min, R. Guo, and N. Menke, “A synthetic aperture telescope based on a pair of gratings,” J. Mod. Opt. 60(15), 1229–1233 (2013).
[Crossref]

Y. Rivenson, A. Stern, and B. Javidi, “Overview of compressive sensing techniques applied in holography [Invited],” Appl. Opt. 52(1), A423–A432 (2013).
[Crossref] [PubMed]

2012 (2)

2011 (4)

2010 (3)

2009 (1)

P. Chatterjee, S. Mukherjee, S. Chaudhuri, and G. Seetharaman, “Application of Papoulis-Gerchberg method in image super-resolution and inpainting,” Comput. J. 52(1), 80–89 (2009).
[Crossref]

2008 (3)

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

L. Martínez-León and B. Javidi, “Synthetic aperture single-exposure on-axis digital holography,” Opt. Express 16(1), 161–169 (2008).
[Crossref] [PubMed]

J. Rosen and G. Brooker, “Non-scanning motionless fluorescence three-dimensional holographic microscopy,” Nat. Photonics 2(3), 190–195 (2008).
[Crossref]

2007 (4)

E. Candès and J. Romberg, “Sparsity and incoherence in compressive sampling,” Inverse Probl. 23(3), 969–985 (2007).
[Crossref]

G. Indebetouw, Y. Tada, J. Rosen, and G. Brooker, “Scanning holographic microscopy with resolution exceeding the Rayleigh limit of the objective by superposition of off-axis holograms,” Appl. Opt. 46(6), 993–1000 (2007).
[Crossref] [PubMed]

J. Rosen and G. Brooker, “Digital spatially incoherent Fresnel holography,” Opt. Lett. 32(8), 912–914 (2007).
[Crossref] [PubMed]

J. M. Bioucas-Dias and M. A. T. Figueiredo, “A new TwIST: two-step iterative shrinkage/thresholding algorithms for image restoration,” IEEE Trans. Image Process. 16(12), 2992–3004 (2007).
[Crossref] [PubMed]

2006 (1)

2005 (1)

1989 (1)

B. K. Jennison, J. P. Allebach, and D. W. Sweeney, “Iterative approaches to computer-generated holography,” Opt. Eng. 28(6), 286629 (1989).
[Crossref]

1920 (1)

A. A. Michelson, “On the application of interference methods to astronomical measurements,” Astrophys. J. 51, 257–262 (1920).
[Crossref] [PubMed]

1873 (1)

E. Abbe, “Beiträge zur theorie des mikroskops und der mikroskopischen wahrnehmung,” Archiv. Microskopische Anat. 9(1), 413–418 (1873).
[Crossref]

Abbe, E.

E. Abbe, “Beiträge zur theorie des mikroskops und der mikroskopischen wahrnehmung,” Archiv. Microskopische Anat. 9(1), 413–418 (1873).
[Crossref]

Allebach, J. P.

B. K. Jennison, J. P. Allebach, and D. W. Sweeney, “Iterative approaches to computer-generated holography,” Opt. Eng. 28(6), 286629 (1989).
[Crossref]

Balber, S.

Beck, S. M.

Bioucas-Dias, J. M.

J. M. Bioucas-Dias and M. A. T. Figueiredo, “A new TwIST: two-step iterative shrinkage/thresholding algorithms for image restoration,” IEEE Trans. Image Process. 16(12), 2992–3004 (2007).
[Crossref] [PubMed]

Brooker, G.

Buck, J. R.

Buell, W. F.

Candès, E.

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

E. Candès and J. Romberg, “Sparsity and incoherence in compressive sampling,” Inverse Probl. 23(3), 969–985 (2007).
[Crossref]

Chatterjee, P.

P. Chatterjee, S. Mukherjee, S. Chaudhuri, and G. Seetharaman, “Application of Papoulis-Gerchberg method in image super-resolution and inpainting,” Comput. J. 52(1), 80–89 (2009).
[Crossref]

Chaudhuri, S.

P. Chatterjee, S. Mukherjee, S. Chaudhuri, and G. Seetharaman, “Application of Papoulis-Gerchberg method in image super-resolution and inpainting,” Comput. J. 52(1), 80–89 (2009).
[Crossref]

Dickinson, R. P.

Fienup, J. R.

Figueiredo, M. A. T.

J. M. Bioucas-Dias and M. A. T. Figueiredo, “A new TwIST: two-step iterative shrinkage/thresholding algorithms for image restoration,” IEEE Trans. Image Process. 16(12), 2992–3004 (2007).
[Crossref] [PubMed]

Gao, P.

K. Ji, P. Gao, J. Min, R. Guo, and N. Menke, “A synthetic aperture telescope based on a pair of gratings,” J. Mod. Opt. 60(15), 1229–1233 (2013).
[Crossref]

García, J.

García-Martínez, P.

Granero, L.

Guo, R.

K. Ji, P. Gao, J. Min, R. Guo, and N. Menke, “A synthetic aperture telescope based on a pair of gratings,” J. Mod. Opt. 60(15), 1229–1233 (2013).
[Crossref]

Indebetouw, G.

Javidi, B.

Jennison, B. K.

B. K. Jennison, J. P. Allebach, and D. W. Sweeney, “Iterative approaches to computer-generated holography,” Opt. Eng. 28(6), 286629 (1989).
[Crossref]

Ji, K.

K. Ji, P. Gao, J. Min, R. Guo, and N. Menke, “A synthetic aperture telescope based on a pair of gratings,” J. Mod. Opt. 60(15), 1229–1233 (2013).
[Crossref]

Kashter, Y.

Katz, B.

Kelner, R.

Kozlowski, D. A.

Kumar, A.

Marechal, N. J.

Martínez-León, L.

Menke, N.

K. Ji, P. Gao, J. Min, R. Guo, and N. Menke, “A synthetic aperture telescope based on a pair of gratings,” J. Mod. Opt. 60(15), 1229–1233 (2013).
[Crossref]

Michelson, A. A.

A. A. Michelson, “On the application of interference methods to astronomical measurements,” Astrophys. J. 51, 257–262 (1920).
[Crossref] [PubMed]

Micó, V.

Min, J.

K. Ji, P. Gao, J. Min, R. Guo, and N. Menke, “A synthetic aperture telescope based on a pair of gratings,” J. Mod. Opt. 60(15), 1229–1233 (2013).
[Crossref]

Mukherjee, S.

P. Chatterjee, S. Mukherjee, S. Chaudhuri, and G. Seetharaman, “Application of Papoulis-Gerchberg method in image super-resolution and inpainting,” Comput. J. 52(1), 80–89 (2009).
[Crossref]

Rivenson, Y.

Romberg, J.

E. Candès and J. Romberg, “Sparsity and incoherence in compressive sampling,” Inverse Probl. 23(3), 969–985 (2007).
[Crossref]

Rosen, J.

Y. Kashter and J. Rosen, “Enhanced-resolution using modified configuration of Fresnel incoherent holographic recorder with synthetic aperture,” Opt. Express 22(17), 20551–20565 (2014).
[Crossref] [PubMed]

Y. Rivenson, A. Rot, S. Balber, A. Stern, and J. Rosen, “Recovery of partially occluded objects by applying compressive Fresnel holography,” Opt. Lett. 37(10), 1757–1759 (2012).
[Crossref] [PubMed]

B. Katz, J. Rosen, R. Kelner, and G. Brooker, “Enhanced resolution and throughput of Fresnel incoherent correlation holography (FINCH) using dual diffractive lenses on a spatial light modulator (SLM),” Opt. Express 20(8), 9109–9121 (2012).
[Crossref] [PubMed]

J. Rosen, N. Siegel, and G. Brooker, “Theoretical and experimental demonstration of resolution beyond the Rayleigh limit by FINCH fluorescence microscopic imaging,” Opt. Express 19(27), 26249–26268 (2011).
[Crossref] [PubMed]

G. Brooker, N. Siegel, V. Wang, and J. Rosen, “Optimal resolution in Fresnel incoherent correlation holographic fluorescence microscopy,” Opt. Express 19(6), 5047–5062 (2011).
[Crossref] [PubMed]

B. Katz and J. Rosen, “Could SAFE concept be applied for designing a new synthetic aperture telescope?” Opt. Express 19(6), 4924–4936 (2011).
[Crossref] [PubMed]

B. Katz and J. Rosen, “Super-resolution in incoherent optical imaging using synthetic aperture with Fresnel elements,” Opt. Express 18(2), 962–972 (2010).
[Crossref] [PubMed]

J. Rosen and G. Brooker, “Non-scanning motionless fluorescence three-dimensional holographic microscopy,” Nat. Photonics 2(3), 190–195 (2008).
[Crossref]

J. Rosen and G. Brooker, “Digital spatially incoherent Fresnel holography,” Opt. Lett. 32(8), 912–914 (2007).
[Crossref] [PubMed]

G. Indebetouw, Y. Tada, J. Rosen, and G. Brooker, “Scanning holographic microscopy with resolution exceeding the Rayleigh limit of the objective by superposition of off-axis holograms,” Appl. Opt. 46(6), 993–1000 (2007).
[Crossref] [PubMed]

Rot, A.

Seetharaman, G.

P. Chatterjee, S. Mukherjee, S. Chaudhuri, and G. Seetharaman, “Application of Papoulis-Gerchberg method in image super-resolution and inpainting,” Comput. J. 52(1), 80–89 (2009).
[Crossref]

Siegel, N.

Stern, A.

Sweeney, D. W.

B. K. Jennison, J. P. Allebach, and D. W. Sweeney, “Iterative approaches to computer-generated holography,” Opt. Eng. 28(6), 286629 (1989).
[Crossref]

Tada, Y.

Tippie, A. E.

Wakin, M.

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

Wang, V.

Wright, T. J.

Zalevsky, Z.

Appl. Opt. (4)

Archiv. Microskopische Anat. (1)

E. Abbe, “Beiträge zur theorie des mikroskops und der mikroskopischen wahrnehmung,” Archiv. Microskopische Anat. 9(1), 413–418 (1873).
[Crossref]

Astrophys. J. (1)

A. A. Michelson, “On the application of interference methods to astronomical measurements,” Astrophys. J. 51, 257–262 (1920).
[Crossref] [PubMed]

Comput. J. (1)

P. Chatterjee, S. Mukherjee, S. Chaudhuri, and G. Seetharaman, “Application of Papoulis-Gerchberg method in image super-resolution and inpainting,” Comput. J. 52(1), 80–89 (2009).
[Crossref]

IEEE Signal Process. Mag. (1)

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

IEEE Trans. Image Process. (1)

J. M. Bioucas-Dias and M. A. T. Figueiredo, “A new TwIST: two-step iterative shrinkage/thresholding algorithms for image restoration,” IEEE Trans. Image Process. 16(12), 2992–3004 (2007).
[Crossref] [PubMed]

Inverse Probl. (1)

E. Candès and J. Romberg, “Sparsity and incoherence in compressive sampling,” Inverse Probl. 23(3), 969–985 (2007).
[Crossref]

J. Disp. Technol. (1)

Y. Rivenson, A. Stern, and B. Javidi, “Compressive Fresnel holography,” J. Disp. Technol. 6(10), 506–509 (2010).
[Crossref]

J. Mod. Opt. (1)

K. Ji, P. Gao, J. Min, R. Guo, and N. Menke, “A synthetic aperture telescope based on a pair of gratings,” J. Mod. Opt. 60(15), 1229–1233 (2013).
[Crossref]

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

Nat. Photonics (1)

J. Rosen and G. Brooker, “Non-scanning motionless fluorescence three-dimensional holographic microscopy,” Nat. Photonics 2(3), 190–195 (2008).
[Crossref]

Opt. Eng. (1)

B. K. Jennison, J. P. Allebach, and D. W. Sweeney, “Iterative approaches to computer-generated holography,” Opt. Eng. 28(6), 286629 (1989).
[Crossref]

Opt. Express (8)

G. Brooker, N. Siegel, V. Wang, and J. Rosen, “Optimal resolution in Fresnel incoherent correlation holographic fluorescence microscopy,” Opt. Express 19(6), 5047–5062 (2011).
[Crossref] [PubMed]

B. Katz and J. Rosen, “Could SAFE concept be applied for designing a new synthetic aperture telescope?” Opt. Express 19(6), 4924–4936 (2011).
[Crossref] [PubMed]

Y. Kashter and J. Rosen, “Enhanced-resolution using modified configuration of Fresnel incoherent holographic recorder with synthetic aperture,” Opt. Express 22(17), 20551–20565 (2014).
[Crossref] [PubMed]

J. Rosen, N. Siegel, and G. Brooker, “Theoretical and experimental demonstration of resolution beyond the Rayleigh limit by FINCH fluorescence microscopic imaging,” Opt. Express 19(27), 26249–26268 (2011).
[Crossref] [PubMed]

B. Katz, J. Rosen, R. Kelner, and G. Brooker, “Enhanced resolution and throughput of Fresnel incoherent correlation holography (FINCH) using dual diffractive lenses on a spatial light modulator (SLM),” Opt. Express 20(8), 9109–9121 (2012).
[Crossref] [PubMed]

A. E. Tippie, A. Kumar, and J. R. Fienup, “High-resolution synthetic-aperture digital holography with digital phase and pupil correction,” Opt. Express 19(13), 12027–12038 (2011).
[Crossref] [PubMed]

L. Martínez-León and B. Javidi, “Synthetic aperture single-exposure on-axis digital holography,” Opt. Express 16(1), 161–169 (2008).
[Crossref] [PubMed]

B. Katz and J. Rosen, “Super-resolution in incoherent optical imaging using synthetic aperture with Fresnel elements,” Opt. Express 18(2), 962–972 (2010).
[Crossref] [PubMed]

Opt. Lett. (2)

Other (3)

J. W. Goodman, Introduction to Fourier optics, 3rd Ed. (Roberts and Company Publishers, 2005).

P. R. Lawson, Selected Paper on Long Baseline Stellar Interferometry (SPIE Press Book, 1997).

N. J. Miller, M. P. Dierking and B. D. Duncan, “Optical sparse aperture imaging,” Appl. Opt. 46(23), 0003–6935(2007).
[Crossref]

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (12)

Fig. 1
Fig. 1 Schematic configurations of S-SAFE system: L0 collimation lens; P1 and P2, polarizers; SLM, SLM1 and SLM2, spatial light modulators; CCD, charged couple device; BPF, bandpass filter. These setups create (a) a continuous central holographic element by implementing the FINCH method, f1 is the focal length of the diffractive lens displayed on the SLM (b) a marginal holographic element, where SLM1 and SLM2 are shifted in two symmetrical viewpoints in front of the collimation lens L0; a diffractive lens with focal length f1 is displayed on SLM1.The symbols ⦿, ↑ and represent polarization directions, perpendicular, parallel and 45° with respect to the plane of the page, respectively.
Fig. 2
Fig. 2 (a) For the index n = 0, the central mask (Ax × Ay in size) consists of a diffractive lens with a focal length of f1. (b) For n>0, two masks ( A ˜ x × A ˜ y in size) are displayed on SLM1 and SLM2; one is a diffractive lens with a focal length of f1 and the other has a constant phase. SLM1 and SLM2 are located at (Xn,Yn) and (-Xn,-Yn), respectively. (c) An example for a sparse aperture of the recorded hologram. The diameter DH represents the aperture of the total digital hologram. The red arrows in (b) represent radial and angular symmetrical movements of SLM1 and SLM2. In this example Ax,y = 250, A ˜ x,y =17, DH = 1024 pixels.
Fig. 3
Fig. 3 Example of sparse holographic imaging of an object point: (a) a complete hologram of an object point; (b) an example of the mask Λ; (c) a sparse hologram; (d) a reconstructed image using FBP; (e) a reconstructed image using the TwIST algorithm; (f) The horizontal cross-sections intensity of the images in (d) and (e). The blue and the red colors represent the FBP and CS reconstructions, respectively. The scale bar is 60 pixels.
Fig. 4
Fig. 4 Block diagram of the iterative PG algorithm.
Fig. 5
Fig. 5 Simulation results: (a1) the original object (satellite). The scale bar is 50 pixels. (a2) measurement mask that consists of 5.22% of the active area, (a3) the corresponding Fresnel hologram magnitude, (a4-a6) the reconstructed images corresponding to the hologram (a3) using (a4) FBP, (a5) PG and (a6) CS reconstructions. (b1-b6) The same as (a1-a6) for the object (land texture) of (b1) and a measurement mask (b2) of 4.32% of the active areas. (c1-c6) The same as (a1-a6) for the object of (c1) and a measurement mask (c2) of a rectangular aperture with 4.6% of the active area. The scale bar is 64 pixels. (d1-d6) The same as (a1-a6) for the out of the center object of (d1) and a measurement mask (d2) of 5.22% of the active areas. (e1-e6) The same as (a1-a6) for the object of (e1) and a measurement mask (e2) of random active areas distributed uniformly on 5.7% of the total area. (f1-f6) The same as (a1-a6) for the object of (f1) and a measurement mask (f2) of equally spaced peripheral elements mask on 7.4% of the total area.
Fig. 6
Fig. 6 Experimental setup of classical FINCH: L0, lens; P1 and P2, polarizers; SLM, spatial light modulator; CCD, charged couple device; BS, beam splitter; BPF, bandpass filter.
Fig. 7
Fig. 7 Experimental results obtained by recording holograms of an RC 25 cycles/mm with different percentages of the hologram active area: (a1-f1) the images obtained with FBP with 31%-15% measurement area; (a2-f2) same as (a1-f1) for PG; (a3-f3) same as (a1-f1) for CS ; (a4-f4) the average intensity cross-sections of the three techniques along the horizontal gratings. The blue, the green and the red colors represent the FBP, PG and CS reconstructions, respectively; (a5-f5) same as (a4-f4) for the vertical gratings. The scale bar is 0.18mm.
Fig. 8
Fig. 8 Average visibility versus the percentage of the measurement area along the (a) horizontal and (b) vertical gratings. The blue rings, the green dots and the red diamonds represent the FBP, PG and CS algorithms, respectively.
Fig. 9
Fig. 9 Experimental setups: L0 lens; P1 and P2, polarizers; SLM, spatial light modulator; CCD, charged couple device; BS, beam splitter; BPF, bandpass filter, PH, pinhole. (a) Configuration for the central holographic element; (b) configuration for the peripheral holographic elements. The piecewise object and the pinhole are illuminated separately in two different experiments.
Fig. 10
Fig. 10 (a) Mask Λ1 of the first cluster; (b) mask Λ2 of the second cluster; (c) the quadratic phase pattern Q1; (d) the quadratic phase pattern Q2.
Fig. 11
Fig. 11 (a) The active areas revealed by reconstructing the SLM plane illuminated by a laser; (b) the calibrated digital mask Λ ˜ matched to the actual measurements.
Fig. 12
Fig. 12 Experimental results obtained by recording an S-SAFE hologram with 30% of the SA area: (a),(b),(c) the images obtained by FBP, PG and CS, respectively; (d) The image obtained by FBP of the central element; (e), (f) the average cross-sections of the three techniques along the horizontal and vertical gratings, respectively. The blue, the green and the red colors represent the FBP, PG and CS reconstructions, respectively; (g), (h) average visibility corresponding to the three techniques along the horizontal and vertical gratings, respectively.

Equations (21)

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

H( r ¯ 0 ; θ j )= I s ( r ¯ s )I( r ¯ 0 ; r ¯ s ; θ j ),
u H ( r ¯ 0 )=H( r ¯ 0 ; θ 1 )[ exp( i θ 3 )exp( i θ 2 ) ] +H( r ¯ 0 ; θ 2 )[ exp( i θ 1 )exp( i θ 3 ) ] +H( r ¯ 0 ; θ 3 )[ exp( i θ 2 )exp( i θ 1 ) ],
u REC = u H Q[ 1 z r ],
u H =Λ{ fQ[ 1 z r ] },
min f ^ TV( f ^ ) s.t. f ^ = u H Q[ 1 z r ] ,
u H i = u REC i Q[ 1 z r ],
u ^ H i = u H i ( 1Λ )+ u H .
u initial =( { u H +( 1Λ )A e i2π φ rand }Q[ 1 z r ] )S,
P k,j (x,y)= Λ k Q k [ 1/ f 1 ]exp( i θ j ),
T n ( x,y; θ j )={ C 1 Q( 1 f 1 )exp( i θ j )rect( x A x , y A y   )+ C 2 rect( x A x , y A y   ) n=0 C 1 Q( 1 f 1 )exp( i θ j )rect( x X n A ˜ x , y Y n A ˜ y   )+ C 2 rect( x+ X n A ˜ x , y+ Y n A ˜ y   ) Otherwise,
I( r ¯ 0 ; r ¯ s ; θ j )= n=0 N | I s C( r ¯ s )L( - r ¯ s f 0 )Q( 1 f 0 )Q( - 1 f 0 )*Q( 1 d ) × T n ( x,y; θ j )*Q( 1 z h ) | 2 ,
I( r ¯ 0 ; r ¯ s ; θ j )=( C 3 + C 4 ( r ¯ s )Q( - 1 z r )L( - r ¯ r z r )exp( -i θ j )+ C 4 * ( r ¯ s )Q( 1 z r )L( r ¯ r z r )exp( i θ j ) )×{ rect( x A x , y A y   )+ n=1 N rect( x X n A ˜ x , y Y n A ˜ y   ) },
z r =± z h 2 .
r ¯ r =( x r , y r )=( x s z h f 0 , y s z h f 0 ).
D H =2 X N 2 + Y N 2 .
Δ θ MIN = 1 2 2λ z r M T D H f 0 ,
Δ θ MIN = λ 4 X N 2 + Y N 2 .
e ¯ rms = e 1/2 MAX mn | f ˜ mn | ,
e= 1 MN m M n N | f ˜ mn γ h ¯ mn | 2 ,
γ= m M n N f ˜ mn h ¯ mn * m M n N | h ¯ mn | 2 .
Δλ< λ 2 4 f 1 2 + D H 2 2 f 1 .

Metrics