Abstract

We present an iterative camera aperture design procedure, which determines an optimal mask pattern based on a sparse set of desired intensity distributions at different focal depths. This iterative method uses the ambiguity function as a tool to shape the camera’s response to defocus, and shares conceptual similarities with phase retrieval procedures. An analysis of algorithm convergence is presented, and experimental examples are shown to demonstrate the flexibility of the design process. This algorithm potentially ties together previous disjointed PSF design approaches under a common framework, and offers new insights for the creation of future application-specific imaging systems.

© 2010 OSA

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    [CrossRef] [PubMed]
  2. W. Chi and N. George, “Electronic imaging using a logarithmic asphere,” Opt. Lett. 26(12), 875–877 (2001).
    [CrossRef]
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    [CrossRef] [PubMed]
  4. A. Greengard, Y. Y. Schechner, and R. Piestun, “Depth from diffracted rotation,” Opt. Lett. 31(2), 181–183 (2006).
    [CrossRef] [PubMed]
  5. A. Levin, R. Fergus, F. Durand, and W. T. Freeman, “Image and depth from a conventional camera with a coded aperture,” ACM Trans. Graph. 26(3), 70 (2007).
    [CrossRef]
  6. M. D. Stenner, D. J. Townsend, and M. E. Gehm, “Static Architecture for Compressive Motion Detection in Persistent, Pervasive Surveillance Applications,” in Imaging Systems, OSA technical Digest (CD) (Optical Society of America, 2010), paper IMB2. http://www.opticsinfobase.org/abstract.cfm?URI=IS-2010-IMB2
  7. A. Ashok and M. Neifeld, “Pseudo-random phase masks for superresolution imaging from subpixel shifting,” Appl. Opt. 46(12), 2256 (2007).
    [CrossRef] [PubMed]
  8. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane figures,” Optik (Stuttg.) 35, 237–246 (1972).
  9. J. R. Fienup, “Iterative method applied to image reconstruction and to computer generated holograms,” Opt. Eng. 19, 297–305 (1980).
  10. D. Elkind, Z. Zalevsky, U. Levy, and D. Mendlovic, “Optical transfer function shaping and depth of focus by using a phase only filter,” Appl. Opt. 42(11), 1925–1931 (2003).
    [CrossRef] [PubMed]
  11. W. D. Furlan, D. Zalvidea, and G. Saavedra, “Synthesis of filters for specified axial irradiance by use of phase-space tomography,” Opt. Commun. 189(1-3), 15–19 (2001).
    [CrossRef]
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    [CrossRef]
  14. M. Testorf, B. Hennelly, and J. Ojeda-Castaneda, Phase Space Optics: Fundamentals and Applications, (McGraw-Hill, (2010).
  15. J. Ojeda-Castañeda, L. R. Berriel-Valdos, and E. Montes, “Ambiguity function as a design tool for high focal depth,” Appl. Opt. 27(4), 790–795 (1988).
    [CrossRef] [PubMed]
  16. M. G. Raymer, M. Beck, and D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72(8), 1137–1140 (1994).
    [CrossRef] [PubMed]
  17. J. Tu and S. Tamura, “Wave-field determination using tomography of the ambiguity function,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 55(2), 1946–1949 (1997).
    [CrossRef]
  18. X. Liu and K. H. Brenner, “Reconstruction of two-dimensional complex amplitudes from intensity measurements,” Opt. Commun. 225(1-3), 19–30 (2003).
    [CrossRef]
  19. M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. 73(11), 1434–1441 (1983).
    [CrossRef]
  20. L. Waller, L. Tian, and G. Barbastathis, “Transport of Intensity phase-amplitude imaging with higher order intensity derivatives,” Opt. Express 18(12), 12552–12561 (2010).
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  21. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1982), Chap. 4.
  22. E. Wolf, “New theory of partial coherence in the space-frequency domain. Part I: spectra and cross spectra of steady-state sources,” J. Opt. Soc. Am. 72(3), 343–351 (1982).
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  23. H. M. Ozaktas, S. Yüksel, and M. A. Kutay, “Linear algebraic theory of partial coherence: discrete fields and measures of partial coherence,” J. Opt. Soc. Am. A 19(8), 1563–1571 (2002).
    [CrossRef]
  24. L. Hogben, Handbook of Linear Algebra (Chapman & Hall, 2007) Chap. 5.
  25. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21(15), 2758–2769 (1982).
    [CrossRef] [PubMed]
  26. R. Piestun and J. Shamir, “Synthesis of three-dimensional light fields and applications,” Proc. IEEE 90(2), 222–244 (2002).
    [CrossRef]
  27. A. W. Lohmann, “Three-dimensional properties of wave fields,” Optik (Stuttg.) 51, 105–117 (1978).

2010 (2)

2007 (2)

A. Levin, R. Fergus, F. Durand, and W. T. Freeman, “Image and depth from a conventional camera with a coded aperture,” ACM Trans. Graph. 26(3), 70 (2007).
[CrossRef]

A. Ashok and M. Neifeld, “Pseudo-random phase masks for superresolution imaging from subpixel shifting,” Appl. Opt. 46(12), 2256 (2007).
[CrossRef] [PubMed]

2006 (1)

2003 (2)

D. Elkind, Z. Zalevsky, U. Levy, and D. Mendlovic, “Optical transfer function shaping and depth of focus by using a phase only filter,” Appl. Opt. 42(11), 1925–1931 (2003).
[CrossRef] [PubMed]

X. Liu and K. H. Brenner, “Reconstruction of two-dimensional complex amplitudes from intensity measurements,” Opt. Commun. 225(1-3), 19–30 (2003).
[CrossRef]

2002 (2)

2001 (2)

W. D. Furlan, D. Zalvidea, and G. Saavedra, “Synthesis of filters for specified axial irradiance by use of phase-space tomography,” Opt. Commun. 189(1-3), 15–19 (2001).
[CrossRef]

W. Chi and N. George, “Electronic imaging using a logarithmic asphere,” Opt. Lett. 26(12), 875–877 (2001).
[CrossRef]

1997 (1)

J. Tu and S. Tamura, “Wave-field determination using tomography of the ambiguity function,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 55(2), 1946–1949 (1997).
[CrossRef]

1995 (1)

1994 (1)

M. G. Raymer, M. Beck, and D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72(8), 1137–1140 (1994).
[CrossRef] [PubMed]

1988 (1)

1983 (2)

K. H. Brenner, A. Lohmann, and J. Ojeda-Castaneda, “The ambiguity function as a polar display of the OTF,” Opt. Commun. 44(5), 323–326 (1983).
[CrossRef]

M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. 73(11), 1434–1441 (1983).
[CrossRef]

1982 (2)

1980 (1)

J. R. Fienup, “Iterative method applied to image reconstruction and to computer generated holograms,” Opt. Eng. 19, 297–305 (1980).

1978 (1)

A. W. Lohmann, “Three-dimensional properties of wave fields,” Optik (Stuttg.) 51, 105–117 (1978).

1974 (1)

1972 (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane figures,” Optik (Stuttg.) 35, 237–246 (1972).

Ashok, A.

Barbastathis, G.

Beck, M.

M. G. Raymer, M. Beck, and D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72(8), 1137–1140 (1994).
[CrossRef] [PubMed]

Berriel-Valdos, L. R.

Brenner, K. H.

X. Liu and K. H. Brenner, “Reconstruction of two-dimensional complex amplitudes from intensity measurements,” Opt. Commun. 225(1-3), 19–30 (2003).
[CrossRef]

K. H. Brenner, A. Lohmann, and J. Ojeda-Castaneda, “The ambiguity function as a polar display of the OTF,” Opt. Commun. 44(5), 323–326 (1983).
[CrossRef]

Cathey, W. T.

Chi, W.

Dowski, E. R.

Durand, F.

A. Levin, R. Fergus, F. Durand, and W. T. Freeman, “Image and depth from a conventional camera with a coded aperture,” ACM Trans. Graph. 26(3), 70 (2007).
[CrossRef]

Elkind, D.

Fergus, R.

A. Levin, R. Fergus, F. Durand, and W. T. Freeman, “Image and depth from a conventional camera with a coded aperture,” ACM Trans. Graph. 26(3), 70 (2007).
[CrossRef]

Fienup, J. R.

J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21(15), 2758–2769 (1982).
[CrossRef] [PubMed]

J. R. Fienup, “Iterative method applied to image reconstruction and to computer generated holograms,” Opt. Eng. 19, 297–305 (1980).

Freeman, W. T.

A. Levin, R. Fergus, F. Durand, and W. T. Freeman, “Image and depth from a conventional camera with a coded aperture,” ACM Trans. Graph. 26(3), 70 (2007).
[CrossRef]

Furlan, W. D.

W. D. Furlan, D. Zalvidea, and G. Saavedra, “Synthesis of filters for specified axial irradiance by use of phase-space tomography,” Opt. Commun. 189(1-3), 15–19 (2001).
[CrossRef]

George, N.

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane figures,” Optik (Stuttg.) 35, 237–246 (1972).

Greengard, A.

Kutay, M. A.

Levin, A.

A. Levin, R. Fergus, F. Durand, and W. T. Freeman, “Image and depth from a conventional camera with a coded aperture,” ACM Trans. Graph. 26(3), 70 (2007).
[CrossRef]

Levy, U.

Li, Y.

Liu, X.

X. Liu and K. H. Brenner, “Reconstruction of two-dimensional complex amplitudes from intensity measurements,” Opt. Commun. 225(1-3), 19–30 (2003).
[CrossRef]

Lohmann, A.

K. H. Brenner, A. Lohmann, and J. Ojeda-Castaneda, “The ambiguity function as a polar display of the OTF,” Opt. Commun. 44(5), 323–326 (1983).
[CrossRef]

Lohmann, A. W.

A. W. Lohmann, “Three-dimensional properties of wave fields,” Optik (Stuttg.) 51, 105–117 (1978).

McAlister, D. F.

M. G. Raymer, M. Beck, and D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72(8), 1137–1140 (1994).
[CrossRef] [PubMed]

Mendlovic, D.

Montes, E.

Neifeld, M.

Ojeda-Castaneda, J.

K. H. Brenner, A. Lohmann, and J. Ojeda-Castaneda, “The ambiguity function as a polar display of the OTF,” Opt. Commun. 44(5), 323–326 (1983).
[CrossRef]

Ojeda-Castañeda, J.

Ozaktas, H. M.

Papoulis, A.

Piestun, R.

A. Greengard, Y. Y. Schechner, and R. Piestun, “Depth from diffracted rotation,” Opt. Lett. 31(2), 181–183 (2006).
[CrossRef] [PubMed]

R. Piestun and J. Shamir, “Synthesis of three-dimensional light fields and applications,” Proc. IEEE 90(2), 222–244 (2002).
[CrossRef]

Raymer, M. G.

M. G. Raymer, M. Beck, and D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72(8), 1137–1140 (1994).
[CrossRef] [PubMed]

Saavedra, G.

W. D. Furlan, D. Zalvidea, and G. Saavedra, “Synthesis of filters for specified axial irradiance by use of phase-space tomography,” Opt. Commun. 189(1-3), 15–19 (2001).
[CrossRef]

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane figures,” Optik (Stuttg.) 35, 237–246 (1972).

Schechner, Y. Y.

Shamir, J.

R. Piestun and J. Shamir, “Synthesis of three-dimensional light fields and applications,” Proc. IEEE 90(2), 222–244 (2002).
[CrossRef]

Tamura, S.

J. Tu and S. Tamura, “Wave-field determination using tomography of the ambiguity function,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 55(2), 1946–1949 (1997).
[CrossRef]

Teague, M. R.

Tian, L.

Tu, J.

J. Tu and S. Tamura, “Wave-field determination using tomography of the ambiguity function,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 55(2), 1946–1949 (1997).
[CrossRef]

Waller, L.

Wolf, E.

Yüksel, S.

Zalevsky, Z.

Zalvidea, D.

W. D. Furlan, D. Zalvidea, and G. Saavedra, “Synthesis of filters for specified axial irradiance by use of phase-space tomography,” Opt. Commun. 189(1-3), 15–19 (2001).
[CrossRef]

Zhao, H.

ACM Trans. Graph. (1)

A. Levin, R. Fergus, F. Durand, and W. T. Freeman, “Image and depth from a conventional camera with a coded aperture,” ACM Trans. Graph. 26(3), 70 (2007).
[CrossRef]

Appl. Opt. (5)

J. Opt. Soc. Am. (3)

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

Opt. Commun. (3)

W. D. Furlan, D. Zalvidea, and G. Saavedra, “Synthesis of filters for specified axial irradiance by use of phase-space tomography,” Opt. Commun. 189(1-3), 15–19 (2001).
[CrossRef]

X. Liu and K. H. Brenner, “Reconstruction of two-dimensional complex amplitudes from intensity measurements,” Opt. Commun. 225(1-3), 19–30 (2003).
[CrossRef]

K. H. Brenner, A. Lohmann, and J. Ojeda-Castaneda, “The ambiguity function as a polar display of the OTF,” Opt. Commun. 44(5), 323–326 (1983).
[CrossRef]

Opt. Eng. (1)

J. R. Fienup, “Iterative method applied to image reconstruction and to computer generated holograms,” Opt. Eng. 19, 297–305 (1980).

Opt. Express (1)

Opt. Lett. (3)

Optik (Stuttg.) (2)

A. W. Lohmann, “Three-dimensional properties of wave fields,” Optik (Stuttg.) 51, 105–117 (1978).

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane figures,” Optik (Stuttg.) 35, 237–246 (1972).

Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics (1)

J. Tu and S. Tamura, “Wave-field determination using tomography of the ambiguity function,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 55(2), 1946–1949 (1997).
[CrossRef]

Phys. Rev. Lett. (1)

M. G. Raymer, M. Beck, and D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72(8), 1137–1140 (1994).
[CrossRef] [PubMed]

Proc. IEEE (1)

R. Piestun and J. Shamir, “Synthesis of three-dimensional light fields and applications,” Proc. IEEE 90(2), 222–244 (2002).
[CrossRef]

Other (4)

M. D. Stenner, D. J. Townsend, and M. E. Gehm, “Static Architecture for Compressive Motion Detection in Persistent, Pervasive Surveillance Applications,” in Imaging Systems, OSA technical Digest (CD) (Optical Society of America, 2010), paper IMB2. http://www.opticsinfobase.org/abstract.cfm?URI=IS-2010-IMB2

M. Testorf, B. Hennelly, and J. Ojeda-Castaneda, Phase Space Optics: Fundamentals and Applications, (McGraw-Hill, (2010).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1982), Chap. 4.

L. Hogben, Handbook of Linear Algebra (Chapman & Hall, 2007) Chap. 5.

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Figures (11)

Fig. 1
Fig. 1

A schematic comparison of current mask design techniques with the proposed method. All can determine the amplitude and phase at one plane from 2 or more intensity distributions. (a) Phase retrieval techniques iterate between a few planes and the mask plane, one at a time. (b) Phase-space tomography techniques do not iterate, but require intensity distributions from many planes. (c) Our method iterates between the mask plane and a few planes viewed simultaneously, using a global rank-1 constraint on the mutual intensity function.

Fig. 2
Fig. 2

(a) Simplified 1D diagram of a camera setup with an aperture mask in the pupil plane. The mask will generate different OTFs at different planes of defocus. (b) The OTFs of an open aperture in focus (at z0) and defocused (at z1) are given as slices of the AF of an open aperture from Eq. (6). Note that the AF is complex – diagrams will show its absolute value.

Fig. 3
Fig. 3

A schematic diagram of the proposed algorithm operating in 1D. (a) A set of n desired OTFs (here n = 3 for an open aperture), which are determined from desired PSF responses, are used as input. (b) Each OTF populates a slice of the AF from Eq. (6). (c) A one-time interpolation from Eq. (7) is used to fill in zeros between desired slices. (d) The mutual intensity (MI) can be constrained by taking the first singular value shown in (e). Details of this constraint are in Fig. 4. (f) An optimized AF is now obtained, which is re-populated with the desired OTF values in (a) along the specific slices in (b). Iteration is stopped at a specified error value, and Eq. (3) is then used to invert the AF into the optimal 1D aperture mask function.

Fig. 4
Fig. 4

The decomposition of a mutual intensity function. An initial mutual intensity “guess” of a wavefront incident upon an open aperture (left) is decomposed into multiple modes using an SVD (right), with weights given by their singular values. For example, λ1 = 1, λ2 = 0.21, and λ3 = 0.13 after the algorithm’s first iteration, but quickly approach a single large value. Our constraint simply takes the first mode of this decomposition.

Fig. 5
Fig. 5

A binary amplitude mask comprised of five slits is used as a known input to test algorithm performance. (b) Three OTFs generated from the aperture mask at different focal depths are used as input. (c) They generate an MI and AF guess, which improve upon iteration. (d) Output OTFs after 25 iterations closely match input OTFs.

Fig. 6
Fig. 6

Recovery of the CPM from 3 shift-invariant OTFs. (a) Three ground-truth OTFs (G, green) and algorithm reconstructions (R, blue) from a CPM (α = 40) at 0mm, 0.2mm and 0.4mm of defocus, using the example f/5 setup after 15 iterations. The ground-truth AF (b) and reconstructed AF (c) exhibit a large degree of similarity. (d) The output phase mask is comprised of the expected cubic phase profile.

Fig. 7
Fig. 7

Convergence analysis plots for the above two mask examples. (a) With increased iteration, the MSE between ground truth and reconstructed OTFs approaches zero. (b) Singular values, representing modes of partial coherence, approach a single mode with increased iteration n (shown for the binary mask example). This single mode implies spatially coherent light, which follows from our assumption of modeling a camera’s response to a point source.

Fig. 8
Fig. 8

A demonstration of algorithm performance as a function of two free parameters: maximum input defocus parameter and number of input slices. (a) MSE between all input and output OTFs decreases as the maximum input defocus parameter is increased for both example masks. Each MSE value is an average MSE for 3 to 8 equally spaced input planes, each after 15 iterations. (b) MSE of input vs. output OTFs also decreases as the number of pre-determined equally spaced input planes is increased. Here, each MSE value is an average over a maximum W20 value of 2λ-7/2λ, also after 15 iterations.

Fig. 9
Fig. 9

The design procedure for a desired 3D PSF response. (a) Three desired PSFs of one, two and three sinc functions of 5μm width at three planes of focus yield three OTFs in (b) to populate an AF guess. (c) Iterative mode selection is applied to converge to an approximate solution after 50 iterations. (d) Three output PSFs at the same planes of defocus as the PSFs in (a) show the same 1, 2, and 3 peak pattern, but are not exact solutions.

Fig. 10
Fig. 10

Different masks can be used to approximate the desired PSF set in Fig. 9. (a) The optimized amplitude and phase distribution for an aperture mask that creates the desired PSF set in Fig. 9(d). (b) MSE drops with number of iterations for an amplitude and phase mask (A + P), amplitude-only mask (A-only), and phase-only mask (P-only), but at different rates. (c) The optimized binary amplitude mask and (d) phase-only mask (in radians) in 2D.

Fig. 11
Fig. 11

Three simulated PSFs (a) in focus, (b) at a defocus plane of 0.1mm, and (c) at a defocus plane of 0.2mm. (d)-(f) PSF measurements obtained with the experimental setup described above for the same amounts of defocus as each PSF in (a)-(c) above. The scale bar in the lower right represents 50μm.

Equations (10)

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A ( x , u ) = U ( x + x 2 ) U * ( x x 2 ) e 2 π i x u d x ,
A ( x ' , u ) e 2 π i x u d u = U ( x + x ' 2 ) U * ( x x ' 2 ) = U ( x 1 ) U * ( x 2 ) .
U ( x 1 ) U * ( 0 ) = A ( x 1 , u ) e π i u x 1 d u ,
H ( x , W 20 ) = U ( x + x 2 ) e i k W 20 ( x + x 2 ) 2 U * ( x x 2 ) e i k W 20 ( x x 2 ) 2 d x ,
W 20 = r 2 2 ( ± Δ z f 2 ± f Δ z ) .
H ( x , W 20 ) = A ( x , x W 20 k / π ) .
A ( x , u ) = A ( x , u = 0 ) + 2 W 20 x ' A u ( x , u = 0 ) + ( 2 W 20 x ' ) 2 2 ! 2 A u 2 ( x , u = 0 ) + ... ,
Γ ( x 1 , x 2 ) = U C ( x 1 ) U C * ( x 2 ) .
Γ = S Λ V T U C ( x 1 ) U C * ( x 2 ) = λ 1 | s 1 v 1 | .
A c ( x , u ) = x [ L x 1 [ R 45 [ U C ( x 1 ) U C * ( x 2 ) ] ] ] ,

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