Abstract

We present a class of binary masks that encode, in polar coordinates, the values of a Hadamard matrix of order N. For order N ≥ 2, the binary masks increase the Strehl ratio vs. focus error by the factor N, with the highest possible light throughput. Since a Strehl ratio with high tolerance to defocus does not guarantee a modulation transfer function (MTF) with low sensitivity to focus errors, then, we show that for N = 16 the binary mask reduces also the impact of focus error on the MTF. Equivalently, the discrete binary mask has Fisher information with low variations to defocus.

© 2017 Optical Society of America

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References

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    [Crossref] [PubMed]
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2017 (1)

2015 (2)

F. Soldevila, E. Salvador-Balaguer, P. Clemente, E. Tajahuerce, and J. Lancis, “High-resolution adaptive imaging with a single photodiode,” Sci. Rep. 5(1), 14300 (2015).
[Crossref] [PubMed]

J. Ojeda-Castañeda and C. M. Gómez-Sarabia, “Tuning field depth at high resolution by pupil engineering,” Adv. Opt. Photonics 7(4), 814–880 (2015).
[Crossref]

2014 (1)

2013 (2)

S. S. Gorthi, D. Schaak, and E. Schonbrun, “Fluorescence imaging of flowing cells using a temporally coded excitation,” Opt. Express 21(4), 5164–5170 (2013).
[Crossref] [PubMed]

J. Ojeda-Castañeda, S. Ledesma, and C. M. Gómez-Sarabia, “Tunable apodizers and tunable focalizers using helical pairs,” Photonics Lett. Pol. 5(1), 20–22 (2013).
[Crossref]

2010 (2)

2008 (1)

2006 (1)

2005 (1)

2003 (2)

1995 (2)

1992 (1)

1991 (1)

1990 (2)

1988 (2)

1987 (1)

1978 (1)

1972 (1)

G. Häusler, “A method to increase the depth of focus by two step image processing,” Opt. Commun. 6(1), 38–42 (1972).
[Crossref]

1969 (1)

1964 (1)

1954 (1)

Andrés, P.

Bélanger, P. A.

Berriel-Valdos, L. R.

Brady, D. J.

Cao, Q.

Cathey, W. T.

Chi, W.

Clemente, P.

F. Soldevila, E. Salvador-Balaguer, P. Clemente, E. Tajahuerce, and J. Lancis, “High-resolution adaptive imaging with a single photodiode,” Sci. Rep. 5(1), 14300 (2015).
[Crossref] [PubMed]

E. Tajahuerce, V. Durán, P. Clemente, E. Irles, F. Soldevila, P. Andrés, and J. Lancis, “Image transmission through dynamic scattering media by single-pixel photodetection,” Opt. Express 22(14), 16945–16955 (2014).
[Crossref] [PubMed]

Davidson, N.

Demenikov, M.

Dowski, E. R.

Durán, V.

Fei, Z.

Fine, T.

Friesem, A. A.

Gan, F.

Gao, X.

Gehm, M. E.

George, N.

Gómez-Sarabia, C. M.

L. Ledesma-Carrillo, R. Guzmán-Cabrera, C. M. Gómez-Sarabia, M. Torres-Cisneros, and J. Ojeda-Castañeda, “Tunable field depth: hyperbolic optical masks,” Appl. Opt. 56(1), A104–A114 (2017).
[Crossref]

J. Ojeda-Castañeda and C. M. Gómez-Sarabia, “Tuning field depth at high resolution by pupil engineering,” Adv. Opt. Photonics 7(4), 814–880 (2015).
[Crossref]

J. Ojeda-Castañeda, S. Ledesma, and C. M. Gómez-Sarabia, “Tunable apodizers and tunable focalizers using helical pairs,” Photonics Lett. Pol. 5(1), 20–22 (2013).
[Crossref]

J. Ojeda-Castañeda, J. E. A. Landgrave, and C. M. Gómez-Sarabia, “Conjugate phase plate use in analysis of the frequency response of imaging systems designed for extended depth of field,” Appl. Opt. 47(22), E99–E105 (2008).
[Crossref] [PubMed]

Gorthi, S. S.

Guzmán-Cabrera, R.

Harvey, A. R.

Harwit, M.

Hasman, E.

Häusler, G.

G. Häusler, “A method to increase the depth of focus by two step image processing,” Opt. Commun. 6(1), 38–42 (1972).
[Crossref]

Irles, E.

Jahns, J.

Lancis, J.

F. Soldevila, E. Salvador-Balaguer, P. Clemente, E. Tajahuerce, and J. Lancis, “High-resolution adaptive imaging with a single photodiode,” Sci. Rep. 5(1), 14300 (2015).
[Crossref] [PubMed]

E. Tajahuerce, V. Durán, P. Clemente, E. Irles, F. Soldevila, P. Andrés, and J. Lancis, “Image transmission through dynamic scattering media by single-pixel photodetection,” Opt. Express 22(14), 16945–16955 (2014).
[Crossref] [PubMed]

Landgrave, J. E. A.

Ledesma, S.

J. Ojeda-Castañeda, S. Ledesma, and C. M. Gómez-Sarabia, “Tunable apodizers and tunable focalizers using helical pairs,” Photonics Lett. Pol. 5(1), 20–22 (2013).
[Crossref]

Ledesma-Carrillo, L.

Lohmann, A. W.

Martínez-Corral, M.

McCain, S. T.

McCutchen, C. W.

McLeod, J. H.

Montes, E.

Muyo, G.

Noyola-Isgleas, A.

Ojeda-Castaneda, J.

Ojeda-Castañeda, J.

L. Ledesma-Carrillo, R. Guzmán-Cabrera, C. M. Gómez-Sarabia, M. Torres-Cisneros, and J. Ojeda-Castañeda, “Tunable field depth: hyperbolic optical masks,” Appl. Opt. 56(1), A104–A114 (2017).
[Crossref]

J. Ojeda-Castañeda and C. M. Gómez-Sarabia, “Tuning field depth at high resolution by pupil engineering,” Adv. Opt. Photonics 7(4), 814–880 (2015).
[Crossref]

J. Ojeda-Castañeda, S. Ledesma, and C. M. Gómez-Sarabia, “Tunable apodizers and tunable focalizers using helical pairs,” Photonics Lett. Pol. 5(1), 20–22 (2013).
[Crossref]

J. Ojeda-Castañeda, J. E. A. Landgrave, and C. M. Gómez-Sarabia, “Conjugate phase plate use in analysis of the frequency response of imaging systems designed for extended depth of field,” Appl. Opt. 47(22), E99–E105 (2008).
[Crossref] [PubMed]

A. W. Lohmann, J. Ojeda-Castañeda, and A. Serrano-Heredia, “Synthesis of analog apodizers with binary angular sectors,” Appl. Opt. 34(2), 317–322 (1995).
[Crossref] [PubMed]

J. Ojeda-Castañeda, P. Andrés, and M. Martínez-Corral, “Zero axial irradiance by annular screens with angular variation,” Appl. Opt. 31(22), 4600–4602 (1992).
[Crossref] [PubMed]

J. Ojeda-Castañeda and L. R. Berriel-Valdos, “Zone plate for arbitrarily high focal depth,” Appl. Opt. 29(7), 994–997 (1990).
[Crossref] [PubMed]

J. Ojeda-Castañeda, P. Andrés, and M. Martínez-Corral, “Zone plates with cells apodized by Legendre profiles,” Appl. Opt. 29(9), 1299–1303 (1990).
[Crossref] [PubMed]

J. Ojeda-Castañeda and L. R. Valdós, “Arbitrarily high focal depth with finite apertures,” Opt. Lett. 13(3), 183–185 (1988).
[Crossref] [PubMed]

J. Ojeda-Castañeda, L. R. Berriel-Valdos, and E. Montes, “Bessel annular apodizers: imaging characteristics,” Appl. Opt. 26(14), 2770–2772 (1987).
[Crossref] [PubMed]

Phillips, P. G.

Pitsianis, N. P.

Potuluri, P.

Ramos, R.

Rioux, M.

Salvador-Balaguer, E.

F. Soldevila, E. Salvador-Balaguer, P. Clemente, E. Tajahuerce, and J. Lancis, “High-resolution adaptive imaging with a single photodiode,” Sci. Rep. 5(1), 14300 (2015).
[Crossref] [PubMed]

Schaak, D.

Schonbrun, E.

Serrano-Heredia, A.

Sloane, N. J. A.

Soldevila, F.

F. Soldevila, E. Salvador-Balaguer, P. Clemente, E. Tajahuerce, and J. Lancis, “High-resolution adaptive imaging with a single photodiode,” Sci. Rep. 5(1), 14300 (2015).
[Crossref] [PubMed]

E. Tajahuerce, V. Durán, P. Clemente, E. Irles, F. Soldevila, P. Andrés, and J. Lancis, “Image transmission through dynamic scattering media by single-pixel photodetection,” Opt. Express 22(14), 16945–16955 (2014).
[Crossref] [PubMed]

Sullivan, M. E.

Tajahuerce, E.

F. Soldevila, E. Salvador-Balaguer, P. Clemente, E. Tajahuerce, and J. Lancis, “High-resolution adaptive imaging with a single photodiode,” Sci. Rep. 5(1), 14300 (2015).
[Crossref] [PubMed]

E. Tajahuerce, V. Durán, P. Clemente, E. Irles, F. Soldevila, P. Andrés, and J. Lancis, “Image transmission through dynamic scattering media by single-pixel photodetection,” Opt. Express 22(14), 16945–16955 (2014).
[Crossref] [PubMed]

Torres-Cisneros, M.

Tremblay, R.

Valdós, L. R.

Xu, W.

Adv. Opt. Photonics (1)

J. Ojeda-Castañeda and C. M. Gómez-Sarabia, “Tuning field depth at high resolution by pupil engineering,” Adv. Opt. Photonics 7(4), 814–880 (2015).
[Crossref]

Appl. Opt. (13)

N. J. A. Sloane, T. Fine, P. G. Phillips, and M. Harwit, “Codes for multiplex spectrometry,” Appl. Opt. 8(10), 2103–2106 (1969).
[Crossref] [PubMed]

M. Rioux, R. Tremblay, and P. A. Bélanger, “Linear, annular, and radial focusing with axicons and applications to laser machining,” Appl. Opt. 17(10), 1532–1536 (1978).
[Crossref] [PubMed]

J. Ojeda-Castaneda, R. Ramos, and A. Noyola-Isgleas, “High focal depth by apodization and digital restoration,” Appl. Opt. 27(12), 2583–2586 (1988).
[Crossref] [PubMed]

A. W. Lohmann, J. Ojeda-Castañeda, and A. Serrano-Heredia, “Synthesis of analog apodizers with binary angular sectors,” Appl. Opt. 34(2), 317–322 (1995).
[Crossref] [PubMed]

E. R. Dowski and W. T. Cathey, “Extended depth of field through wave-front coding,” Appl. Opt. 34(11), 1859–1866 (1995).
[Crossref] [PubMed]

J. Ojeda-Castañeda, P. Andrés, and M. Martínez-Corral, “Zone plates with cells apodized by Legendre profiles,” Appl. Opt. 29(9), 1299–1303 (1990).
[Crossref] [PubMed]

J. Ojeda-Castañeda and L. R. Berriel-Valdos, “Zone plate for arbitrarily high focal depth,” Appl. Opt. 29(7), 994–997 (1990).
[Crossref] [PubMed]

J. Ojeda-Castañeda, P. Andrés, and M. Martínez-Corral, “Zero axial irradiance by annular screens with angular variation,” Appl. Opt. 31(22), 4600–4602 (1992).
[Crossref] [PubMed]

J. Ojeda-Castañeda, L. R. Berriel-Valdos, and E. Montes, “Bessel annular apodizers: imaging characteristics,” Appl. Opt. 26(14), 2770–2772 (1987).
[Crossref] [PubMed]

X. Gao, Z. Fei, W. Xu, and F. Gan, “Tunable three-dimensional intensity distribution by a pure phase-shifting apodizer,” Appl. Opt. 44(23), 4870–4873 (2005).
[Crossref] [PubMed]

M. E. Gehm, S. T. McCain, N. P. Pitsianis, D. J. Brady, P. Potuluri, and M. E. Sullivan, “Static two-dimensional aperture coding for multimodal, multiplex spectroscopy,” Appl. Opt. 45(13), 2965–2974 (2006).
[Crossref] [PubMed]

J. Ojeda-Castañeda, J. E. A. Landgrave, and C. M. Gómez-Sarabia, “Conjugate phase plate use in analysis of the frequency response of imaging systems designed for extended depth of field,” Appl. Opt. 47(22), E99–E105 (2008).
[Crossref] [PubMed]

L. Ledesma-Carrillo, R. Guzmán-Cabrera, C. M. Gómez-Sarabia, M. Torres-Cisneros, and J. Ojeda-Castañeda, “Tunable field depth: hyperbolic optical masks,” Appl. Opt. 56(1), A104–A114 (2017).
[Crossref]

J. Opt. Soc. Am. (2)

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

Opt. Commun. (1)

G. Häusler, “A method to increase the depth of focus by two step image processing,” Opt. Commun. 6(1), 38–42 (1972).
[Crossref]

Opt. Express (3)

Opt. Lett. (3)

Photonics Lett. Pol. (1)

J. Ojeda-Castañeda, S. Ledesma, and C. M. Gómez-Sarabia, “Tunable apodizers and tunable focalizers using helical pairs,” Photonics Lett. Pol. 5(1), 20–22 (2013).
[Crossref]

Sci. Rep. (1)

F. Soldevila, E. Salvador-Balaguer, P. Clemente, E. Tajahuerce, and J. Lancis, “High-resolution adaptive imaging with a single photodiode,” Sci. Rep. 5(1), 14300 (2015).
[Crossref] [PubMed]

Other (4)

M. Harwit and N. J. A. Sloane, Hadamard Transform Optics (Academic, 1979)

R. Raskar, A. Agrawal, and J. Tumblin, “Coded exposure photography: motion deblurring via fluttered shutter,” in Proceedings SIGGRAPH ’06 795 (ACM, 2006), paper 804.
[Crossref]

T. M. Cover and J. A. Thomas, Elements of Information Theory (Wiley, 1991).

L. L. Scharf, Statistical Signal Processing (Addison-Wesley, 1991).

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Figures (15)

Fig. 1
Fig. 1 Geometrical strategy for implementing a circular version of a Hadamard mask of order N = 2: In (a), we show a pseudo-Cartesian display, to be discussed in section 3. In (b) we use polar coordinates for showing the circular Hadamard mask of order N = 2. We render in orange the regions where the amplitude transmittance is equal to unity. And in blue, we depict the regions where the amplitude transmittance has a phase delay of π.
Fig. 2
Fig. 2 Schematics of a classical optical processor. At the Fraunhofer plane we locate a circular Hadamard mask (of order N = 8) for generating the irradiance impulse response.
Fig. 3
Fig. 3 Schematics illustrating, in a clockwise route, the change of variables describe in Eq. (3).
Fig. 4
Fig. 4 Display of the circular Hadamard masks of order: (a) N = 2; (b) N = 4; (c) N = 8; (d) N = 16; (e) N = 32; (f) N = 64; (g) N = 128; and (h) N = 256. We render in orange the regions that have complex amplitude transmittances equal to unity; and we render in blue, the regions where there is a phase delay equal to π.
Fig. 5
Fig. 5 The circular Hadamard masks are distributed in (ζ, φ) domain as rectangular partitions, which encode the values of a Hadamard matrix of order N: In (a) N = 2; in (b) N = 4; and in (c) N = 8.
Fig. 6
Fig. 6 Variations of the Strehl ratio vs focus error for different circular Hadamard masks. The broken red line, along the horizontal axis, marks the tolerance criterion due to Rayleigh. As the order N of the circular mask increases, the tolerance to focus error is well beyond the value W 2, 0 λ / 4.
Fig. 7
Fig. 7 Irradiance point spread functions (PSF) of the circular Hadamard masks of order N = 1, 2, 4, 8, 16 and 32. Along the columns we vary the focus error coefficient. We note that the PSF do not exhibit circular symmetry, if the order N is greater than 2. We note that for the orders N = 16 and N = 32 the PSF does not change substantially with focus errors.
Fig. 8
Fig. 8 Display of the Modulation Transfer Function (MTF) of the circular Hadamard masks of order N = 1, 2, 4, 16, and 32; for three values of the focus error coefficient. As in Fig. 7, the MTF does no exhibit circular symmetry; and for the order N = 16 and 32 the MTF does not change substantially with focus errors. However, at high spatial frequencies, the MTF changes substantially with focus error. Consequently, the use of the Hadamard mask is limited to inputs with moderate spatial frequency content.
Fig. 9
Fig. 9 Graphs depicting the influence of focus error on the Modulation Transfer Function (MTF): In (a) the MTF of a clear pupil aperture. The MTF varies substantially with focus errors. In (b) the MTF (evaluated along the horizontal axis, θ = 0) of the circular Hadamard mask of order N = 16. In the range 0 ≤ ρ ≤ Ω / 5, the MTF has values above a noise level of 10%. In the same range, the MTF does not vary substantially with focus error. However, beyond the selected range of spatial frequencies, the MTF changes wildly with focus error. Then, the usefulness of the Hadamard mask is limited to inputs with moderate spatial frequency content.
Fig. 10
Fig. 10 Graphs showing the variation of the area under the MTF as the focus error coefficient changes from zero to W2, 0 = 3 λ; for the circular Hadamard masks of order N = 1, 2, 4, 8, 16, 32. Even when the values of the area have small values for the orders N = 16 and 32, the area under the MTF remains fairly constant with focus error. These numerical evaluations validate the heuristic considerations in Figs. 6, 7, 8 and 9(b).
Fig. 11
Fig. 11 Fisher information (in dB) as a function of the focus error coefficient. In blue the Fisher information of the clear pupil; in red the Fisher information of the cubic phase mask; and in black the Fisher information of the circular Hadamard mask of order N = 16.
Fig. 12
Fig. 12 Images obtained when using a clear pupil aperture, N = 1: Along line (a) we show the in-focus image as well as the defocused images; along line (b) we display the images obtained when using an inverse filter for reconstructing the images along the line (a).
Fig. 13
Fig. 13 Images obtained when using a circular Hadamard mask of order N = 16: (a) In-focus image and defocused images; (b) digital reconstruction when using an inverse filter.
Fig. 14
Fig. 14 Baboon images obtained when using a clear pupil aperture. Along line (a) we show the in-focus image as well as the defocused images; along line (b) we display the images obtained when using an inverse filter for reconstructing the images along the line (a).
Fig. 15
Fig. 15 Baboon images generated when using a circular Hadamard mask of order N = 16 and an inverse, digital post-processing filter. Along line (a) the in-focus and defocused images; along line (b) the digital reconstructions when using an inverse filter that is set at W2,0 = (3/2)λ.

Equations (30)

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p ( r , θ , z ) = 0 Ω 0 2 π P ( ρ , φ ) exp { i 2 π [ λ z 2 ρ 2 + r ρ c o s ( θ φ ) ] } ρ d ρ d φ .
p ( 0 , θ , z ) = 2 π 0 Ω { 1 2 π 0 2 π P ( ρ , φ ) d φ } exp { i 2 π [ λ z 2 ρ 2 ] } ρ d ρ .
ζ = ( ρ Ω ) 2 1 2 ; Q ( ζ , φ ) = P ( ρ , φ ) . Q ( ζ ) = 1 2 π 0 2 π Q ( ζ , φ ) d φ = 1 2 π 0 2 π P ( ρ , φ ) d φ .
p ( 0 , θ , z ) = ( π Ω 2 ) exp ( i π 2 λ Ω 2 z ) 1 2 1 2 Q ( ζ ) exp [ ( i π λ Ω 2 z ) ζ ] d ζ .
W 2 , 0 = λ 2 Ω 2 2 z ; q ( W 2 , 0 λ ) = p ( 0 , θ , z ) .
q ( W 2 , 0 λ ) = ( π Ω 2 ) exp ( i π W 2 , 0 λ ) 1 2 1 2 Q ( ζ ) exp [ i 2 π ( W 2 , 0 λ ) ζ ] d ζ .
s ( W 2 , 0 ) = | q ( W 2 , 0 λ ) | 2 | q ( 0 ) | 2 .
s ( W 2 , 0 ) = | 1 2 1 2 Q ( ζ ) exp [ i 2 π ( W 2 , 0 λ ) ζ ] d ζ | 2 | 1 2 1 2 Q ( ζ ) d ζ | 2 .
c i r c ( ρ Ω ) = { 1 i f ρ Ω 0 i f ρ > Ω .
( m 1 N ) 2 π φ ( m N ) 2 π ; m = 1 , 2 , 3 , ... N .
( n 1 N ) Ω 2 ρ 2 ( n N ) Ω 2 ; n = 1 , 2 , 3 , ... N .
1 2 + ( n 1 N ) ζ 1 2 + ( n N ) ; n = 1 , 2 , 3 , ... N .
P ( ρ , φ ) = m = 1 N n = 1 N H m , n ( N ) r e c t ( φ π N ( 2 m 1 ) 2 π N ) r e c t ( ρ 2 Ω 2 2 N ( 2 n 1 ) Ω 2 N ) c i r c ( ρ Ω ) .
Q ( ζ , φ ) = m = 1 N n = 1 N H m , n r e c t ( φ π N ( 2 m 1 ) 2 π N ) r e c t ( ζ + 1 2 ( 2 n 1 ) 2 N 1 N ) .
Q ( ζ ) = n = 1 N { 1 N m = 1 N H m , n } r e c t ( ζ + 1 2 ( 2 n 1 ) 2 N 1 N ) .
1 N m = 1 N H m , n = { 1 I f n = 1 0 I f n 1 } = δ n , 1 .
Q ( ζ ) = r e c t ( ζ + 1 2 1 2 N 1 N ) .
s ( W 2 , 0 ) = | 1 2 1 2 + 1 N exp [ i 2 π ( W 2 , 0 λ ) ζ ] d ζ | 2 1 N 2 .
s ( W 2 , 0 ) = sin c 2 ( W 2 , 0 N λ )
s ( W 2 , 0 ) 0.8 ; i f W 2 , 0 N λ 4
J ( W 2 , 0 ) = μ = 2 Ω 2 Ω | W 2 , 0 H ( μ ; W 2 , 0 ) | 2 d μ ; 0 | W 2 , 0 | 3 λ .
1 N m = 1 N H m , n = { 1 I f n = 1 0 I f n 1 } = δ n , 1
H ( 2 ) = [ H ( 1 ) H ( 1 ) H ( 1 ) H ( 1 ) ] = [ 1 1 1 1 ]
1 2 m = 1 2 H m , n ( 2 ) = { 1 I f n = 1 0 I f n = 2 }
H ( 4 ) = [ H ( 2 ) H ( 2 ) H ( 2 ) H ( 2 ) ]
1 4 m = 1 4 H m , n ( 4 ) = { m = 1 4 H m , n ( 2 ) i f n = 1 , 2 0 i f n = 3 , 4 }
1 4 m = 1 4 H m , n ( 4 ) = { 1 i f n = 1 0 i f n = 2 , 3 , 4 }
H ( 8 ) = [ H ( 4 ) H ( 4 ) H ( 4 ) H ( 4 ) ]
1 8 m = 1 8 H m , n ( 8 ) = { m = 1 8 H m , n ( 4 ) i f n = 1 , 2 , 3 , 4 0 i f n = 5 , 6 , 7 , 8 }
1 8 m = 1 8 H m , n ( 8 ) = { 1 i f n = 1 0 i f n = 2 , 3 , 4 , 5 , 6 , 7 , 8 }

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