Abstract

In this paper we introduce an imaging system based on a reflective phase-only spatial light modulator (SLM) in order to perform imaging with improved geometric resolution. By using the SLM, we combine the realization of two main abilities: a lens with a tunable focus and a phase function that, after proper free-space propagation, is projected as an amplitude distribution on top of the inspected object. The first ability is related to the realization of a lens function combined with a tunable prism that yields a microscanning of the inspected object. This by itself improves the spatial sampling density. The second ability is related to a projection of a phase function that is computed using an iterative beam-shaping Gerchberg–Saxton algorithm. After the free-space propagation from the SLM toward the inspected object, an amplitude pattern is generated on top of the object. This projected pattern and a set of low- resolution images with relative shift are interlaced and, after applying the proper regularization method, a geometrically superresolved image is reconstructed.

© 2011 Optical Society of America

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References

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  1. Z. Zalevsky and D. Mendelovic, Optical Superresolution (Springer-Verlag, 2004).
  2. Z. Zalevsky, D. Mendlovic, and E. Marom, “Special sensor masking for exceeding system geometrical resolving power,” Opt. Eng. 39, 1936–1942 (2000).
    [CrossRef]
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    [CrossRef] [PubMed]
  4. A. Borkowski, Z. Zalevsky, and B. Javidi, “Geometrical superresolved imaging using nonperiodic spatial masking,” J. Opt. Soc. Am. A 26, 589–601 (2009).
    [CrossRef]
  5. J. Fortin, P. Chevrette, and R. Plante, “Evaluation of the microscanning process,” Proc. SPIE 2269, 271–279(1994).
    [CrossRef]
  6. S. Park, M. Park, and M. G. Kang, “Super-resolution image reconstruction, a technical overview,” IEEE Signal Process. Mag. 20(3), 21–36 (2003).
    [CrossRef]
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  8. D. C. O’Shea, T. J. Suleski, A. D. Kathman, and D. W. Prather, Diffractive Optics: Design, Fabrication, and Test (SPIE, 2003).
  9. C. Kohler, X. Schwab, and W. Osten, “Optimally tuned spatial light modulators for digital holography,” Appl. Opt. 45, 960–967 (2006).
    [CrossRef] [PubMed]
  10. E. Carcolé, J. Campos, and S. Bosch, “Diffraction theory of Fresnel lenses encoded in low-resolution devices,” Appl. Opt. 33, 162–174 (1994).
    [CrossRef] [PubMed]
  11. D. M. Cottrell, J. A. Davis, T. R. Hedman, and R. A. Lilly, “Multiple imaging phase-encoded optical elements written as programmable spatial light modulators,” Appl. Opt. 29, 2505–2509 (1990).
    [CrossRef] [PubMed]
  12. V. Laude, “Twisted-nematic liquid-crystal pixelated active lens,” Opt. Commun. 153, 134–152 (1998).
    [CrossRef]
  13. M. S. Millán, J. Otón, and E. Pérez-Cabré, “Chromatic compensation of programmable Fresnel lenses,” Opt. Express 14, 6226–6242 (2006).
    [CrossRef] [PubMed]
  14. A. Márquez, C. Iemmi, J. Campos, and M. J. Yzuel, “Achromatic diffractive lens written onto a liquid crystal display,” Opt. Lett. 31, 392–394 (2006).
    [CrossRef] [PubMed]
  15. M. S. Millán, J. Otón, and E. Pérez-Cabré, “Dynamic compensation of chromatic aberration in a programmable diffractive lens,” Opt. Express 14, 9103–9012 (2006).
    [CrossRef] [PubMed]
  16. C. Iemmi, J. Campos, J. C. Escalera, O. Lopez-Coronado, R. Gimeno, and M. J. Yzuel, “Depth of focus increase by multiplexing programmable diffractive lenses,” Opt. Express 14, 10207–10217 (2006).
    [CrossRef] [PubMed]
  17. E. B. Eliezer, Z. Zalevsky, E. Marom, and N. Konforti, “All optical extended depth of field imaging system,” J. Opt. A 5, S164–S169 (2003).
    [CrossRef]
  18. R. W. Gerchberg and W. O. Saxton, “Phase determination for image and diffraction plane pictures in the electron microscope,” Optik 34, 275–284 (1971).
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  20. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).
  21. Z. Zalevsky, D. Mendlovic, and R. G. Dorsch, “Gerchberg–Saxton algorithm applied in the fractional Fourier or the Fresnel domain,” Opt. Lett. 21, 842–844 (1996).
    [CrossRef] [PubMed]
  22. D. Mendlovic, Z. Zalevsky, and N. Konforti, “Computing considerations and fast algorithms for calculating the diffraction integral,” J. Mod. Opt. 44, 407–414 (1997).
    [CrossRef]
  23. S. Chaudhur, Super Resolution Imaging (Kluwer Academic, 2001).
  24. S. Borman and R. Stevenson, “Super-resolution from image sequences—a review,” in Proceedings of the 1998 Midwest Symposium on Circuits and Systems (IEEE Circuits and Systems Society, 1998), pp. 374–378.
  25. S. Borman, “Topics in multiframe superresolution restoration,” Ph.D. dissertation (University of Notre Dame, 2004).
  26. Y. Lu, M. Inamura, and M. Valdes, “Super-resolution of the undersampled and subpixel shifted image sequence by a neural network,” Int. J. Imaging Syst. Technol. 14, 8–15 (2004).
    [CrossRef]
  27. A. C. Yau and M. K. Ng, “Super-resolution image restoration from blurred low-resolution images,” J. Math. Imaging Vision 23, 367–378 (2005).
    [CrossRef]
  28. A. N. Rajagopalan and S. Chaudhuri, “A block shift-variant blur model for recovering depth from defocused images,” in International Conference on Image Processing (ICIP’95) (1995), pp. 636–639.
  29. J. Hadamard, Lectures on the Cauchy Problem in Linear Partial Differential Equations (Yale Univ. Press, 1923).
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    [CrossRef]
  31. A. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems (Winston, 1977).
  32. G. de Villiers, B. McNally, and E. Pike, “Positive solutions to linear inverse problems,” Inverse Probl. 15, 615–635 (1999).
    [CrossRef]
  33. A. Borkowski, E. Marom, Z. Zalevsky, and B. Javidi, “Enhanced geometrical super resolved imaging with moving binary random mask,” J. Opt. Soc. Am. A 28, 566–575 (2011).
    [CrossRef]

2011 (1)

2009 (1)

2006 (5)

2005 (2)

J. Solomon, Z. Zalevsky, and D. Mendlovic, “Geometric superresolution by code division multiplexing,” Appl. Opt. 44, 32–40 (2005).
[CrossRef] [PubMed]

A. C. Yau and M. K. Ng, “Super-resolution image restoration from blurred low-resolution images,” J. Math. Imaging Vision 23, 367–378 (2005).
[CrossRef]

2004 (1)

Y. Lu, M. Inamura, and M. Valdes, “Super-resolution of the undersampled and subpixel shifted image sequence by a neural network,” Int. J. Imaging Syst. Technol. 14, 8–15 (2004).
[CrossRef]

2003 (2)

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

E. B. Eliezer, Z. Zalevsky, E. Marom, and N. Konforti, “All optical extended depth of field imaging system,” J. Opt. A 5, S164–S169 (2003).
[CrossRef]

2000 (1)

Z. Zalevsky, D. Mendlovic, and E. Marom, “Special sensor masking for exceeding system geometrical resolving power,” Opt. Eng. 39, 1936–1942 (2000).
[CrossRef]

1999 (1)

G. de Villiers, B. McNally, and E. Pike, “Positive solutions to linear inverse problems,” Inverse Probl. 15, 615–635 (1999).
[CrossRef]

1998 (1)

V. Laude, “Twisted-nematic liquid-crystal pixelated active lens,” Opt. Commun. 153, 134–152 (1998).
[CrossRef]

1997 (1)

D. Mendlovic, Z. Zalevsky, and N. Konforti, “Computing considerations and fast algorithms for calculating the diffraction integral,” J. Mod. Opt. 44, 407–414 (1997).
[CrossRef]

1996 (1)

1994 (2)

J. Fortin, P. Chevrette, and R. Plante, “Evaluation of the microscanning process,” Proc. SPIE 2269, 271–279(1994).
[CrossRef]

E. Carcolé, J. Campos, and S. Bosch, “Diffraction theory of Fresnel lenses encoded in low-resolution devices,” Appl. Opt. 33, 162–174 (1994).
[CrossRef] [PubMed]

1990 (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 35, 227–246 (1972).

1971 (1)

R. W. Gerchberg and W. O. Saxton, “Phase determination for image and diffraction plane pictures in the electron microscope,” Optik 34, 275–284 (1971).

1970 (1)

Arsenin, V. Y.

A. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems (Winston, 1977).

Borkowski, A.

Borman, S.

S. Borman and R. Stevenson, “Super-resolution from image sequences—a review,” in Proceedings of the 1998 Midwest Symposium on Circuits and Systems (IEEE Circuits and Systems Society, 1998), pp. 374–378.

S. Borman, “Topics in multiframe superresolution restoration,” Ph.D. dissertation (University of Notre Dame, 2004).

Bosch, S.

Campos, J.

Carcolé, E.

Chaudhur, S.

S. Chaudhur, Super Resolution Imaging (Kluwer Academic, 2001).

Chaudhuri, S.

A. N. Rajagopalan and S. Chaudhuri, “A block shift-variant blur model for recovering depth from defocused images,” in International Conference on Image Processing (ICIP’95) (1995), pp. 636–639.

Chevrette, P.

J. Fortin, P. Chevrette, and R. Plante, “Evaluation of the microscanning process,” Proc. SPIE 2269, 271–279(1994).
[CrossRef]

Cottrell, D. M.

Davis, J. A.

de Villiers, G.

G. de Villiers, B. McNally, and E. Pike, “Positive solutions to linear inverse problems,” Inverse Probl. 15, 615–635 (1999).
[CrossRef]

Dorsch, R. G.

Eliezer, E. B.

E. B. Eliezer, Z. Zalevsky, E. Marom, and N. Konforti, “All optical extended depth of field imaging system,” J. Opt. A 5, S164–S169 (2003).
[CrossRef]

Escalera, J. C.

Fortin, J.

J. Fortin, P. Chevrette, and R. Plante, “Evaluation of the microscanning process,” Proc. SPIE 2269, 271–279(1994).
[CrossRef]

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 35, 227–246 (1972).

R. W. Gerchberg and W. O. Saxton, “Phase determination for image and diffraction plane pictures in the electron microscope,” Optik 34, 275–284 (1971).

Gimeno, R.

Goodman, W.

W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

Hadamard, J.

J. Hadamard, Lectures on the Cauchy Problem in Linear Partial Differential Equations (Yale Univ. Press, 1923).

Hedman, T. R.

Hirsch, P. M.

Iemmi, C.

Inamura, M.

Y. Lu, M. Inamura, and M. Valdes, “Super-resolution of the undersampled and subpixel shifted image sequence by a neural network,” Int. J. Imaging Syst. Technol. 14, 8–15 (2004).
[CrossRef]

Javidi, B.

Jordan, J. A.

Kang, M. G.

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

Kathman, A. D.

D. C. O’Shea, T. J. Suleski, A. D. Kathman, and D. W. Prather, Diffractive Optics: Design, Fabrication, and Test (SPIE, 2003).

Kirsch, A.

A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, Vol.  120 of Applied Mathematical Sciences (Springer, 1996).
[CrossRef]

Kohler, C.

Konforti, N.

E. B. Eliezer, Z. Zalevsky, E. Marom, and N. Konforti, “All optical extended depth of field imaging system,” J. Opt. A 5, S164–S169 (2003).
[CrossRef]

D. Mendlovic, Z. Zalevsky, and N. Konforti, “Computing considerations and fast algorithms for calculating the diffraction integral,” J. Mod. Opt. 44, 407–414 (1997).
[CrossRef]

Laude, V.

V. Laude, “Twisted-nematic liquid-crystal pixelated active lens,” Opt. Commun. 153, 134–152 (1998).
[CrossRef]

Lesem, L. B.

Lilly, R. A.

Lopez-Coronado, O.

Lu, Y.

Y. Lu, M. Inamura, and M. Valdes, “Super-resolution of the undersampled and subpixel shifted image sequence by a neural network,” Int. J. Imaging Syst. Technol. 14, 8–15 (2004).
[CrossRef]

Marom, E.

A. Borkowski, E. Marom, Z. Zalevsky, and B. Javidi, “Enhanced geometrical super resolved imaging with moving binary random mask,” J. Opt. Soc. Am. A 28, 566–575 (2011).
[CrossRef]

E. B. Eliezer, Z. Zalevsky, E. Marom, and N. Konforti, “All optical extended depth of field imaging system,” J. Opt. A 5, S164–S169 (2003).
[CrossRef]

Z. Zalevsky, D. Mendlovic, and E. Marom, “Special sensor masking for exceeding system geometrical resolving power,” Opt. Eng. 39, 1936–1942 (2000).
[CrossRef]

Márquez, A.

McNally, B.

G. de Villiers, B. McNally, and E. Pike, “Positive solutions to linear inverse problems,” Inverse Probl. 15, 615–635 (1999).
[CrossRef]

Mendelovic, D.

Z. Zalevsky and D. Mendelovic, Optical Superresolution (Springer-Verlag, 2004).

Mendlovic, D.

J. Solomon, Z. Zalevsky, and D. Mendlovic, “Geometric superresolution by code division multiplexing,” Appl. Opt. 44, 32–40 (2005).
[CrossRef] [PubMed]

Z. Zalevsky, D. Mendlovic, and E. Marom, “Special sensor masking for exceeding system geometrical resolving power,” Opt. Eng. 39, 1936–1942 (2000).
[CrossRef]

D. Mendlovic, Z. Zalevsky, and N. Konforti, “Computing considerations and fast algorithms for calculating the diffraction integral,” J. Mod. Opt. 44, 407–414 (1997).
[CrossRef]

Z. Zalevsky, D. Mendlovic, and R. G. Dorsch, “Gerchberg–Saxton algorithm applied in the fractional Fourier or the Fresnel domain,” Opt. Lett. 21, 842–844 (1996).
[CrossRef] [PubMed]

Millán, M. S.

Ng, M. K.

A. C. Yau and M. K. Ng, “Super-resolution image restoration from blurred low-resolution images,” J. Math. Imaging Vision 23, 367–378 (2005).
[CrossRef]

O’Shea, D. C.

D. C. O’Shea, T. J. Suleski, A. D. Kathman, and D. W. Prather, Diffractive Optics: Design, Fabrication, and Test (SPIE, 2003).

Osten, W.

Otón, J.

Park, M.

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

Pérez-Cabré, E.

Pike, E.

G. de Villiers, B. McNally, and E. Pike, “Positive solutions to linear inverse problems,” Inverse Probl. 15, 615–635 (1999).
[CrossRef]

Plante, R.

J. Fortin, P. Chevrette, and R. Plante, “Evaluation of the microscanning process,” Proc. SPIE 2269, 271–279(1994).
[CrossRef]

Prather, D. W.

D. C. O’Shea, T. J. Suleski, A. D. Kathman, and D. W. Prather, Diffractive Optics: Design, Fabrication, and Test (SPIE, 2003).

Rajagopalan, A. N.

A. N. Rajagopalan and S. Chaudhuri, “A block shift-variant blur model for recovering depth from defocused images,” in International Conference on Image Processing (ICIP’95) (1995), pp. 636–639.

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 35, 227–246 (1972).

R. W. Gerchberg and W. O. Saxton, “Phase determination for image and diffraction plane pictures in the electron microscope,” Optik 34, 275–284 (1971).

Schwab, X.

Solomon, J.

Stevenson, R.

S. Borman and R. Stevenson, “Super-resolution from image sequences—a review,” in Proceedings of the 1998 Midwest Symposium on Circuits and Systems (IEEE Circuits and Systems Society, 1998), pp. 374–378.

Suleski, T. J.

D. C. O’Shea, T. J. Suleski, A. D. Kathman, and D. W. Prather, Diffractive Optics: Design, Fabrication, and Test (SPIE, 2003).

Tikhonov, A.

A. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems (Winston, 1977).

Valdes, M.

Y. Lu, M. Inamura, and M. Valdes, “Super-resolution of the undersampled and subpixel shifted image sequence by a neural network,” Int. J. Imaging Syst. Technol. 14, 8–15 (2004).
[CrossRef]

Van Rooy, D. L.

Yau, A. C.

A. C. Yau and M. K. Ng, “Super-resolution image restoration from blurred low-resolution images,” J. Math. Imaging Vision 23, 367–378 (2005).
[CrossRef]

Yzuel, M. J.

Zalevsky, Z.

A. Borkowski, E. Marom, Z. Zalevsky, and B. Javidi, “Enhanced geometrical super resolved imaging with moving binary random mask,” J. Opt. Soc. Am. A 28, 566–575 (2011).
[CrossRef]

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

J. Solomon, Z. Zalevsky, and D. Mendlovic, “Geometric superresolution by code division multiplexing,” Appl. Opt. 44, 32–40 (2005).
[CrossRef] [PubMed]

E. B. Eliezer, Z. Zalevsky, E. Marom, and N. Konforti, “All optical extended depth of field imaging system,” J. Opt. A 5, S164–S169 (2003).
[CrossRef]

Z. Zalevsky, D. Mendlovic, and E. Marom, “Special sensor masking for exceeding system geometrical resolving power,” Opt. Eng. 39, 1936–1942 (2000).
[CrossRef]

D. Mendlovic, Z. Zalevsky, and N. Konforti, “Computing considerations and fast algorithms for calculating the diffraction integral,” J. Mod. Opt. 44, 407–414 (1997).
[CrossRef]

Z. Zalevsky, D. Mendlovic, and R. G. Dorsch, “Gerchberg–Saxton algorithm applied in the fractional Fourier or the Fresnel domain,” Opt. Lett. 21, 842–844 (1996).
[CrossRef] [PubMed]

Z. Zalevsky and D. Mendelovic, Optical Superresolution (Springer-Verlag, 2004).

Appl. Opt. (5)

IEEE Signal Process. Mag. (1)

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

Int. J. Imaging Syst. Technol. (1)

Y. Lu, M. Inamura, and M. Valdes, “Super-resolution of the undersampled and subpixel shifted image sequence by a neural network,” Int. J. Imaging Syst. Technol. 14, 8–15 (2004).
[CrossRef]

Inverse Probl. (1)

G. de Villiers, B. McNally, and E. Pike, “Positive solutions to linear inverse problems,” Inverse Probl. 15, 615–635 (1999).
[CrossRef]

J. Math. Imaging Vision (1)

A. C. Yau and M. K. Ng, “Super-resolution image restoration from blurred low-resolution images,” J. Math. Imaging Vision 23, 367–378 (2005).
[CrossRef]

J. Mod. Opt. (1)

D. Mendlovic, Z. Zalevsky, and N. Konforti, “Computing considerations and fast algorithms for calculating the diffraction integral,” J. Mod. Opt. 44, 407–414 (1997).
[CrossRef]

J. Opt. A (1)

E. B. Eliezer, Z. Zalevsky, E. Marom, and N. Konforti, “All optical extended depth of field imaging system,” J. Opt. A 5, S164–S169 (2003).
[CrossRef]

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

Opt. Commun. (1)

V. Laude, “Twisted-nematic liquid-crystal pixelated active lens,” Opt. Commun. 153, 134–152 (1998).
[CrossRef]

Opt. Eng. (1)

Z. Zalevsky, D. Mendlovic, and E. Marom, “Special sensor masking for exceeding system geometrical resolving power,” Opt. Eng. 39, 1936–1942 (2000).
[CrossRef]

Opt. Express (3)

Opt. Lett. (2)

Optik (2)

R. W. Gerchberg and W. O. Saxton, “Phase determination for image and diffraction plane pictures in the electron microscope,” Optik 34, 275–284 (1971).

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 227–246 (1972).

Proc. SPIE (1)

J. Fortin, P. Chevrette, and R. Plante, “Evaluation of the microscanning process,” Proc. SPIE 2269, 271–279(1994).
[CrossRef]

Other (10)

Z. Zalevsky and D. Mendelovic, Optical Superresolution (Springer-Verlag, 2004).

W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

D. C. O’Shea, T. J. Suleski, A. D. Kathman, and D. W. Prather, Diffractive Optics: Design, Fabrication, and Test (SPIE, 2003).

S. Chaudhur, Super Resolution Imaging (Kluwer Academic, 2001).

S. Borman and R. Stevenson, “Super-resolution from image sequences—a review,” in Proceedings of the 1998 Midwest Symposium on Circuits and Systems (IEEE Circuits and Systems Society, 1998), pp. 374–378.

S. Borman, “Topics in multiframe superresolution restoration,” Ph.D. dissertation (University of Notre Dame, 2004).

A. N. Rajagopalan and S. Chaudhuri, “A block shift-variant blur model for recovering depth from defocused images,” in International Conference on Image Processing (ICIP’95) (1995), pp. 636–639.

J. Hadamard, Lectures on the Cauchy Problem in Linear Partial Differential Equations (Yale Univ. Press, 1923).

A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, Vol.  120 of Applied Mathematical Sciences (Springer, 1996).
[CrossRef]

A. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems (Winston, 1977).

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Figures (13)

Fig. 1
Fig. 1

Example of the phase function that is encoded onto the SLM display. On the right, we realize a lens function and, on the left side, a phase function that was computed using the G-S algorithm.

Fig. 2
Fig. 2

Schematic diagram of the experimental setup as it is presented in detail in Section 6. Part 1 (numbers 1 and 2): beam shaping of the binary mask on the object. Part 2 (numbers 3 and 4): imaging system with dynamic diffractive lens ( u = 37 cm , v = 43 cm , f = 20 cm ).

Fig. 3
Fig. 3

Experimental setup: the laser beam illuminates the right side of the SLM display, which encodes with the beam-shaping function. Afterward, the light that is being reflected from the object is imaged onto the camera due to the lens function (displayed in the left side of the SLM).

Fig. 4
Fig. 4

Output of the imaging system obtained using the reflective phase-only SLM. (a) Object (which was placed upside down). (b) Captured image.

Fig. 5
Fig. 5

Example for the MS technique with factor 4 using a lens that combines a tunable prism. Between each pair of images from (a) to (d), there is a 1 pixel shift along the X axis (e.g., see the left eye of the smiley that is shifted from pixel 26 to pixel 29).

Fig. 6
Fig. 6

Beam shaping results using 650 × 650 pixels at the SLM display. (a) Input of the G-S algorithm is presented in the left side. (b) Obtained experimental results.

Fig. 7
Fig. 7

(a) HR reference image. (b) LR image. (c) Image after applying the MS technique with factor of 4 and interlacing. (d) Reconstructed HR image using the Tikhonov regularization method with mask projection.

Fig. 8
Fig. 8

(a) HR reference image using a red laser ( λ = 633 nm ). (b) LR image. (c) Image after applying MS tech nique with a factor of 2 and interlacing. (d) Reconstructed HR image using the Tikhonov regularization method with the mask projection.

Fig. 9
Fig. 9

Image of the spectrum of a rosetta in decibels. (a) HR reference spectrum image. (b) Image spectrum after applying the MS technique with a factor of 4 and an interlacing. (c) and (d) Spectrum of the reconstructed HR images obtained when using the Tikhonov regularization method without and with mask projection, respectively.

Fig. 10
Fig. 10

MSE between the HR reference spectrum image and different reconstructed images (with projection over the f x axis). (a) MSE when using Tikhonov regularization with mask projection (solid curve) and without mask projection (dashed curve). (b) MSE when using Tikhonov regularization with mask projection (solid curve) and for MS without using Tikhonov regularization (dashed curve). (c) and (d) Subtraction of the MSE obtained with Tikhonov regularization and a mask (solid curve) from the MSE (dashed curve) of (a) and (b), respectively.

Fig. 11
Fig. 11

Effect of the regularization parameter on the reconstruction accuracy. (a) MSE (between reconstruction and the HR reference spectrum image) when using Tikhonov regularization parameter value α = 0.9 (solid curve) and α = 15 (dashed curve). (b) Subtraction of MSE obtained with Tikhonov regularization with α = 0.9 from MSE obtained with α = 15 .

Fig. 12
Fig. 12

(a) LR image. (b) and (c) HR reconstructed image using Tikhonov regularization (with α = 0.9 ) without and with mask projection, respectively. (d). Correlation coefficient between a reference HR image and the reconstructed image with (solid curve) and without (dashed curve) mask projection.

Fig. 13
Fig. 13

MSE between the HR reference spectrum image and different reconstructed images (with projection over the f y axis). (a) Solid curve plots the MSE obtained when using Tikhonov regularization with mask projection ( α = 0.9 ) and the dashed curve plots it without a mask ( α = 12 ). (b) Solid curve plots the MSE when using Tikhonov regularization with mask projection and the dashed curve plots it for MS without using Tikhonov regularization. (c) and (d) Subtraction between the MSE obtained with Tikhonov regularization with a mask projection and the MSE of (a) and (b), respectively.

Equations (14)

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L ( x , y ) = exp [ i π λ f ( x 2 + y 2 ) ] .
L [ m , n ] = exp [ i π d 2 λ f ( m 2 + n 2 ) ] .
Z c = N d 2 λ ,
f ( λ ) = λ 0 λ f 0 ,
b = A x + n .
[ a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 ] [ a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 ] = b .
1 4 · [ 1 1 1 1 ] ,
A = 1 4 · [ 1 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 ] .
x ^ = ( A T A + α I ) 1 A T b ,
min [ A α I ] x [ b 0 ] 2 .
[ 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 ] ,
Z c = 1080 · 8 μm 2 532 nm = 13 cm .
T ( x ) = exp ( 2 π i ν 0 x ) ,
tan ( α ) = x out z = λ d 0 = λ ν 0 x out = λ ν 0 z .

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