Abstract

In contrast to the standard digital image processing, which operates over the detected image intensity, we propose to perform amplitude image processing. Amplitude processing, like low pass or high pass filtering, is carried out using diffractive optics elements (DOE) since it allows to operate over the field complex amplitude before it has been detected. We show the procedure for designing the DOE that corresponds to each operation. Furthermore, we accomplish an analysis of amplitude image processing performances. In particular, a DOE Laplacian filter is applied to simulated astronomical images for detecting two stars one Airy ring apart. We also check by numerical simulations that the use of a Laplacian amplitude filter produces less noisy images than the standard digital image processing.

© 2016 Optical Society of America

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References

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  1. R. C. Gonzalez and R. E. Woods, Digital Image Processing (Prentice Hall, 2007).
  2. P. J. Valle, J. E. Oti, V. F. Canales, and M. P. Cagigal, “Multiple coaxial foci generation by phase-only pupil filters,” Opt. Commun. 272(2), 325–329 (2007).
    [Crossref]
  3. J. Ares García, S. Bará, M. Gomez García, Z. Jaroszewicz, A. Kolodziejczyk, and K. Petelczyc, “Imaging with extended focal depth by means of the refractive light sword optical element,” Opt. Express 16(22), 18371–18378 (2008).
    [Crossref] [PubMed]
  4. P. J. Valle, V. F. Canales, and M. P. Cagigal, “Focal modulation using rotating phase filters,” Opt. Express 18(8), 7820–7826 (2010).
    [Crossref] [PubMed]
  5. V. Arrizón, U. Ruiz, D. Sánchez-de-la-Llave, G. Mellado-Villaseñor, and A. S. Ostrovsky, “Optimum generation of annular vortices using phase diffractive optical elements,” Opt. Lett. 40(7), 1173–1176 (2015).
    [Crossref] [PubMed]
  6. S. Thibault, A. Arfaoui, and P. Desaulniers, “Cross-diffractive optical elements for wide angle geometric camera calibration,” Opt. Lett. 36(24), 4770–4772 (2011).
    [Crossref] [PubMed]
  7. J. Pérez-Vizcaíno, O. Mendoza-Yero, G. Mínguez-Vega, R. Martínez-Cuenca, P. Andrés, and J. Lancis, “Dispersion management in two-photon microscopy by using diffractive optical elements,” Opt. Lett. 38(4), 440–442 (2013).
    [Crossref] [PubMed]
  8. M. A. Golub, “Laser Beam Splitting by Diffractive Optics,” Opt. Photonics News 15(2), 36–41 (2004).
  9. P. J. Valle and M. P. Cagigal, “Analytic design of multiple-axis, multifocal diffractive lenses,” Opt. Lett. 37(6), 1121–1123 (2012).
    [Crossref] [PubMed]
  10. M. P. Cagigal and P. J. Valle, “Wavefront sensing using diffractive elements,” Opt. Lett. 37(18), 3813–3815 (2012).
    [Crossref] [PubMed]
  11. K. Iizuka, Engineering Optics (Springer, 2008).
  12. P. J. Valle, M. A. Cagigas, and M. P. Cagigal, “Diffractive optical elements to improve the quality of aberrated images,” J. Opt. 16(40), 1655–1658 (2014).
  13. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).
  14. J. L. Starck, A. Bijaoui, I. Valtchanov, and F. Murtagh, “A combined approach for object detection and deconvolution,” Astron. Astrophys. Suppl. Ser. 147(1), 139–149 (2000).
    [Crossref]
  15. J. L. Starck, E. Pantin, and F. Murtagh, “Deconvolution in Astronomy: A Review,” Publ. Astron. Soc. Pac. 114(800), 1051–1069 (2002).
    [Crossref]
  16. R. Stevenson, J. Kleyna, and D. Jewitt, “Transient fragments in outbursting comet,” Astrophys. J. 139, 2230–2240 (2010).
  17. L. M. Lara, R. Rodrigo, G. P. Tozzi, H. Boehnhardt, and P. Leisy, “The gas and dust coma of Comet C/1999 H1 (Lee),” Astron. Astrophys. 420(1), 371–382 (2004).
    [Crossref]
  18. V. F. Canales, D. M. de Juana, and M. P. Cagigal, “Superresolution in compensated telescopes,” Opt. Lett. 29(9), 935–937 (2004).
    [Crossref] [PubMed]
  19. V. F. Canales, J. E. Oti, P. J. Valle, M. P. Cagigal, and N. Devaney, “Reduction of the diffraction pattern in segmented apertures,” Opt. Eng. 45(9), 098001 (2006).
    [Crossref]
  20. N. A. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng. 29(10), 1174–1180 (1990).
    [Crossref]
  21. J. W. Hardy, Adaptive Optics for Astronomical telescopes (Oxford University, 1998).

2015 (1)

2014 (1)

P. J. Valle, M. A. Cagigas, and M. P. Cagigal, “Diffractive optical elements to improve the quality of aberrated images,” J. Opt. 16(40), 1655–1658 (2014).

2013 (1)

2012 (2)

2011 (1)

2010 (2)

P. J. Valle, V. F. Canales, and M. P. Cagigal, “Focal modulation using rotating phase filters,” Opt. Express 18(8), 7820–7826 (2010).
[Crossref] [PubMed]

R. Stevenson, J. Kleyna, and D. Jewitt, “Transient fragments in outbursting comet,” Astrophys. J. 139, 2230–2240 (2010).

2008 (1)

2007 (1)

P. J. Valle, J. E. Oti, V. F. Canales, and M. P. Cagigal, “Multiple coaxial foci generation by phase-only pupil filters,” Opt. Commun. 272(2), 325–329 (2007).
[Crossref]

2006 (1)

V. F. Canales, J. E. Oti, P. J. Valle, M. P. Cagigal, and N. Devaney, “Reduction of the diffraction pattern in segmented apertures,” Opt. Eng. 45(9), 098001 (2006).
[Crossref]

2004 (3)

L. M. Lara, R. Rodrigo, G. P. Tozzi, H. Boehnhardt, and P. Leisy, “The gas and dust coma of Comet C/1999 H1 (Lee),” Astron. Astrophys. 420(1), 371–382 (2004).
[Crossref]

V. F. Canales, D. M. de Juana, and M. P. Cagigal, “Superresolution in compensated telescopes,” Opt. Lett. 29(9), 935–937 (2004).
[Crossref] [PubMed]

M. A. Golub, “Laser Beam Splitting by Diffractive Optics,” Opt. Photonics News 15(2), 36–41 (2004).

2002 (1)

J. L. Starck, E. Pantin, and F. Murtagh, “Deconvolution in Astronomy: A Review,” Publ. Astron. Soc. Pac. 114(800), 1051–1069 (2002).
[Crossref]

2000 (1)

J. L. Starck, A. Bijaoui, I. Valtchanov, and F. Murtagh, “A combined approach for object detection and deconvolution,” Astron. Astrophys. Suppl. Ser. 147(1), 139–149 (2000).
[Crossref]

1990 (1)

N. A. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng. 29(10), 1174–1180 (1990).
[Crossref]

Andrés, P.

Ares García, J.

Arfaoui, A.

Arrizón, V.

Bará, S.

Bijaoui, A.

J. L. Starck, A. Bijaoui, I. Valtchanov, and F. Murtagh, “A combined approach for object detection and deconvolution,” Astron. Astrophys. Suppl. Ser. 147(1), 139–149 (2000).
[Crossref]

Boehnhardt, H.

L. M. Lara, R. Rodrigo, G. P. Tozzi, H. Boehnhardt, and P. Leisy, “The gas and dust coma of Comet C/1999 H1 (Lee),” Astron. Astrophys. 420(1), 371–382 (2004).
[Crossref]

Cagigal, M. P.

P. J. Valle, M. A. Cagigas, and M. P. Cagigal, “Diffractive optical elements to improve the quality of aberrated images,” J. Opt. 16(40), 1655–1658 (2014).

P. J. Valle and M. P. Cagigal, “Analytic design of multiple-axis, multifocal diffractive lenses,” Opt. Lett. 37(6), 1121–1123 (2012).
[Crossref] [PubMed]

M. P. Cagigal and P. J. Valle, “Wavefront sensing using diffractive elements,” Opt. Lett. 37(18), 3813–3815 (2012).
[Crossref] [PubMed]

P. J. Valle, V. F. Canales, and M. P. Cagigal, “Focal modulation using rotating phase filters,” Opt. Express 18(8), 7820–7826 (2010).
[Crossref] [PubMed]

P. J. Valle, J. E. Oti, V. F. Canales, and M. P. Cagigal, “Multiple coaxial foci generation by phase-only pupil filters,” Opt. Commun. 272(2), 325–329 (2007).
[Crossref]

V. F. Canales, J. E. Oti, P. J. Valle, M. P. Cagigal, and N. Devaney, “Reduction of the diffraction pattern in segmented apertures,” Opt. Eng. 45(9), 098001 (2006).
[Crossref]

V. F. Canales, D. M. de Juana, and M. P. Cagigal, “Superresolution in compensated telescopes,” Opt. Lett. 29(9), 935–937 (2004).
[Crossref] [PubMed]

Cagigas, M. A.

P. J. Valle, M. A. Cagigas, and M. P. Cagigal, “Diffractive optical elements to improve the quality of aberrated images,” J. Opt. 16(40), 1655–1658 (2014).

Canales, V. F.

P. J. Valle, V. F. Canales, and M. P. Cagigal, “Focal modulation using rotating phase filters,” Opt. Express 18(8), 7820–7826 (2010).
[Crossref] [PubMed]

P. J. Valle, J. E. Oti, V. F. Canales, and M. P. Cagigal, “Multiple coaxial foci generation by phase-only pupil filters,” Opt. Commun. 272(2), 325–329 (2007).
[Crossref]

V. F. Canales, J. E. Oti, P. J. Valle, M. P. Cagigal, and N. Devaney, “Reduction of the diffraction pattern in segmented apertures,” Opt. Eng. 45(9), 098001 (2006).
[Crossref]

V. F. Canales, D. M. de Juana, and M. P. Cagigal, “Superresolution in compensated telescopes,” Opt. Lett. 29(9), 935–937 (2004).
[Crossref] [PubMed]

de Juana, D. M.

Desaulniers, P.

Devaney, N.

V. F. Canales, J. E. Oti, P. J. Valle, M. P. Cagigal, and N. Devaney, “Reduction of the diffraction pattern in segmented apertures,” Opt. Eng. 45(9), 098001 (2006).
[Crossref]

Golub, M. A.

M. A. Golub, “Laser Beam Splitting by Diffractive Optics,” Opt. Photonics News 15(2), 36–41 (2004).

Gomez García, M.

Jaroszewicz, Z.

Jewitt, D.

R. Stevenson, J. Kleyna, and D. Jewitt, “Transient fragments in outbursting comet,” Astrophys. J. 139, 2230–2240 (2010).

Kleyna, J.

R. Stevenson, J. Kleyna, and D. Jewitt, “Transient fragments in outbursting comet,” Astrophys. J. 139, 2230–2240 (2010).

Kolodziejczyk, A.

Lancis, J.

Lara, L. M.

L. M. Lara, R. Rodrigo, G. P. Tozzi, H. Boehnhardt, and P. Leisy, “The gas and dust coma of Comet C/1999 H1 (Lee),” Astron. Astrophys. 420(1), 371–382 (2004).
[Crossref]

Leisy, P.

L. M. Lara, R. Rodrigo, G. P. Tozzi, H. Boehnhardt, and P. Leisy, “The gas and dust coma of Comet C/1999 H1 (Lee),” Astron. Astrophys. 420(1), 371–382 (2004).
[Crossref]

Martínez-Cuenca, R.

Mellado-Villaseñor, G.

Mendoza-Yero, O.

Mínguez-Vega, G.

Murtagh, F.

J. L. Starck, E. Pantin, and F. Murtagh, “Deconvolution in Astronomy: A Review,” Publ. Astron. Soc. Pac. 114(800), 1051–1069 (2002).
[Crossref]

J. L. Starck, A. Bijaoui, I. Valtchanov, and F. Murtagh, “A combined approach for object detection and deconvolution,” Astron. Astrophys. Suppl. Ser. 147(1), 139–149 (2000).
[Crossref]

Ostrovsky, A. S.

Oti, J. E.

P. J. Valle, J. E. Oti, V. F. Canales, and M. P. Cagigal, “Multiple coaxial foci generation by phase-only pupil filters,” Opt. Commun. 272(2), 325–329 (2007).
[Crossref]

V. F. Canales, J. E. Oti, P. J. Valle, M. P. Cagigal, and N. Devaney, “Reduction of the diffraction pattern in segmented apertures,” Opt. Eng. 45(9), 098001 (2006).
[Crossref]

Pantin, E.

J. L. Starck, E. Pantin, and F. Murtagh, “Deconvolution in Astronomy: A Review,” Publ. Astron. Soc. Pac. 114(800), 1051–1069 (2002).
[Crossref]

Pérez-Vizcaíno, J.

Petelczyc, K.

Roddier, N. A.

N. A. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng. 29(10), 1174–1180 (1990).
[Crossref]

Rodrigo, R.

L. M. Lara, R. Rodrigo, G. P. Tozzi, H. Boehnhardt, and P. Leisy, “The gas and dust coma of Comet C/1999 H1 (Lee),” Astron. Astrophys. 420(1), 371–382 (2004).
[Crossref]

Ruiz, U.

Sánchez-de-la-Llave, D.

Starck, J. L.

J. L. Starck, E. Pantin, and F. Murtagh, “Deconvolution in Astronomy: A Review,” Publ. Astron. Soc. Pac. 114(800), 1051–1069 (2002).
[Crossref]

J. L. Starck, A. Bijaoui, I. Valtchanov, and F. Murtagh, “A combined approach for object detection and deconvolution,” Astron. Astrophys. Suppl. Ser. 147(1), 139–149 (2000).
[Crossref]

Stevenson, R.

R. Stevenson, J. Kleyna, and D. Jewitt, “Transient fragments in outbursting comet,” Astrophys. J. 139, 2230–2240 (2010).

Thibault, S.

Tozzi, G. P.

L. M. Lara, R. Rodrigo, G. P. Tozzi, H. Boehnhardt, and P. Leisy, “The gas and dust coma of Comet C/1999 H1 (Lee),” Astron. Astrophys. 420(1), 371–382 (2004).
[Crossref]

Valle, P. J.

P. J. Valle, M. A. Cagigas, and M. P. Cagigal, “Diffractive optical elements to improve the quality of aberrated images,” J. Opt. 16(40), 1655–1658 (2014).

P. J. Valle and M. P. Cagigal, “Analytic design of multiple-axis, multifocal diffractive lenses,” Opt. Lett. 37(6), 1121–1123 (2012).
[Crossref] [PubMed]

M. P. Cagigal and P. J. Valle, “Wavefront sensing using diffractive elements,” Opt. Lett. 37(18), 3813–3815 (2012).
[Crossref] [PubMed]

P. J. Valle, V. F. Canales, and M. P. Cagigal, “Focal modulation using rotating phase filters,” Opt. Express 18(8), 7820–7826 (2010).
[Crossref] [PubMed]

P. J. Valle, J. E. Oti, V. F. Canales, and M. P. Cagigal, “Multiple coaxial foci generation by phase-only pupil filters,” Opt. Commun. 272(2), 325–329 (2007).
[Crossref]

V. F. Canales, J. E. Oti, P. J. Valle, M. P. Cagigal, and N. Devaney, “Reduction of the diffraction pattern in segmented apertures,” Opt. Eng. 45(9), 098001 (2006).
[Crossref]

Valtchanov, I.

J. L. Starck, A. Bijaoui, I. Valtchanov, and F. Murtagh, “A combined approach for object detection and deconvolution,” Astron. Astrophys. Suppl. Ser. 147(1), 139–149 (2000).
[Crossref]

Astron. Astrophys. (1)

L. M. Lara, R. Rodrigo, G. P. Tozzi, H. Boehnhardt, and P. Leisy, “The gas and dust coma of Comet C/1999 H1 (Lee),” Astron. Astrophys. 420(1), 371–382 (2004).
[Crossref]

Astron. Astrophys. Suppl. Ser. (1)

J. L. Starck, A. Bijaoui, I. Valtchanov, and F. Murtagh, “A combined approach for object detection and deconvolution,” Astron. Astrophys. Suppl. Ser. 147(1), 139–149 (2000).
[Crossref]

Astrophys. J. (1)

R. Stevenson, J. Kleyna, and D. Jewitt, “Transient fragments in outbursting comet,” Astrophys. J. 139, 2230–2240 (2010).

J. Opt. (1)

P. J. Valle, M. A. Cagigas, and M. P. Cagigal, “Diffractive optical elements to improve the quality of aberrated images,” J. Opt. 16(40), 1655–1658 (2014).

Opt. Commun. (1)

P. J. Valle, J. E. Oti, V. F. Canales, and M. P. Cagigal, “Multiple coaxial foci generation by phase-only pupil filters,” Opt. Commun. 272(2), 325–329 (2007).
[Crossref]

Opt. Eng. (2)

V. F. Canales, J. E. Oti, P. J. Valle, M. P. Cagigal, and N. Devaney, “Reduction of the diffraction pattern in segmented apertures,” Opt. Eng. 45(9), 098001 (2006).
[Crossref]

N. A. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng. 29(10), 1174–1180 (1990).
[Crossref]

Opt. Express (2)

Opt. Lett. (6)

Opt. Photonics News (1)

M. A. Golub, “Laser Beam Splitting by Diffractive Optics,” Opt. Photonics News 15(2), 36–41 (2004).

Publ. Astron. Soc. Pac. (1)

J. L. Starck, E. Pantin, and F. Murtagh, “Deconvolution in Astronomy: A Review,” Publ. Astron. Soc. Pac. 114(800), 1051–1069 (2002).
[Crossref]

Other (4)

R. C. Gonzalez and R. E. Woods, Digital Image Processing (Prentice Hall, 2007).

J. W. Hardy, Adaptive Optics for Astronomical telescopes (Oxford University, 1998).

K. Iizuka, Engineering Optics (Springer, 2008).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

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Figures (8)

Fig. 1
Fig. 1 AMIGO setup consisting on a diffractive optical element (DOE) placed at the entrance pupil of lens L1 which forms an image onto the CCD (in some applications after crossing an aberrating medium, here represented by a phase screen, PS).
Fig. 2
Fig. 2 Image replica positions and the corresponding weighting coefficients according to Eq. (3).
Fig. 3
Fig. 3 (a) One-pixel radius averaging mask. (b) Mask used for obtaining image replicas, Eq. (3), and (c) displaced replicas of the image for the case of a plain incoming wavefront (the displacement, α = β = 4π, is exaggerated to be easily appreciated), 2C0 = a = b = c = d = 0.22.
Fig. 4
Fig. 4 (a) One-pixel radius Gaussian mask. (b) Mask described by Eq. (7) used for obtaining image replicas. (c) Displaced and weighted replicas of the image for the case of a Gaussian mask (α = β = 4π).
Fig. 5
Fig. 5 (a) One-pixel radius Laplacian filter. (b) Mask described by Eq. (8) with α = β = 4π used for obtaining image replicas. (c) Displaced and weighted replicas of the amplitude image for the case of a Laplacian mask. Grey scale range at (c) reproduces the range values at the filter in (a).
Fig. 6
Fig. 6 (a) PSF of the clear pupil. (b) PSF of the pupil with the mask of Eq. (10) (c) Transversal cut of the PSF for the clear pupil (solid curve) and with the mask (dashed curve).
Fig. 7
Fig. 7 (a) Amplitude-only object. (b) Result of convolving this object with a 3x3 Laplacian filter in digital image processing. (c) The result of using the mask given by Eq. (10) in the AMIGO setup.
Fig. 8
Fig. 8 (a) Shift-and-add of 100 aberrated and digitally processed PSF with a clear pupil. (b) Shift-and-add of 100 aberrated PSF with a Laplacian AMIGO pupil.

Equations (17)

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M( x,y )=cos( αx )+cos( α y ).
M( x,y )= C 0 + n,m { a n cos( α n x )+ b n cos( α n y ) + c m cos[ β m (x+y) ]+ d m cos[ β m (xy) ] } ,
M( x,y )= C 0 +acos( αx )+bcos( αy )+ccos[ β(x+y) ]+dcos[ β(xy) ],
PM(x,y)=exp[ iM(x,y) ].
I 0 (u,v)= | A(u,v)FTM(u,v) | 2 ,
I D (u,v)= | A(u,v) | 2 FTM(u,v).
M( x,y )=1+ cos( αx )+cos( αy ) 2 + cos[ β(x+y) ]+cos[ β(xy) ] 2 ,
M( x,y )= 1 6 { 4 cos ( αx ) cos ( αy ) cos [ β(x+y) ] cos [ β(xy) ] }.
M( x,y )=α cos 2 [ β( x 2 + y 2 ) ].
M( x,y )= sin 2 [ β( x 2 + y 2 ) ],
A 0 = 1 N mn w mn A mn = 1 N mn w mn R mn +i 1 N mn w mn I mn wR ¯ +i wI ¯ ,
I 0 =| 1 N mn w mn A mn | 2 = wR ¯ 2 + wI ¯ 2 ,
I D = 1 N mn w mn I mn = 1 N mn w mn R mn 2 + 1 N mn w mn I mn 2 = w R 2 ¯ + w I 2 ¯ ,
I D I 0 = σ 2 (R)+ σ 2 (I),
I 0 =| 1 N mn w mn A mn | 2 + r 0 = wR ¯ 2 + wI ¯ 2 + r 0 ,
I D = 1 N mn w mn ( I mn + r mn ) = w R 2 ¯ + w I 2 ¯ + wr ¯ ,
wr ¯ = 1 N mn w mn r mn .

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