Abstract

Sparse apertures find imaging applications in diverse fields such as astronomy and medicine. We are motivated by the design of a wide-area imaging system where sparse apertures can be used to construct novel and efficient optical designs. Specifically, we investigate the use of sparse apertures for off-axis imaging at infrared wavelengths while combating the effects of chromaticity to preserve resolution. In principle, several such sparse apertures can be interleaved within a common aperture to simultaneously image in multiple directions. This can ultimately lead to the design of wide-area imaging systems that require considerably less optical and electronic hardware. The resolution achievable using a sparse aperture is the same as that of a fully open aperture. In the case of off-axis imaging, however, the point spread function (PSF) introduces a blur due to chromaticity that degrades the resolution of the system. Of course, the blur can be eliminated by imaging at a single wavelength. However the signal-to-noise ratio (SNR) is poor, which ultimately degrades image quality. To improve SNR, it is necessary to widen the band of wavelengths, which of course degrades resolution due to chromaticity. Hence there is a fundamental trade between the SNR and the resolution as a function of bandwidth. We show that by using a combination of microprisms and phase optimized micropistons it is possible to reduce the chromatic blur over a band of wavelengths and improve the PSF considerably to restore the resolution of the image. The concepts are validated by means of simulations and verified with experimental data to demonstrate the advantages of phase optimized micropistons in off-axis sparse aperture imaging systems.

© 2009 Optical Society of America

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References

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  1. M. Dana, “Establishment of air defense sensor requirements for automatic aircraft tracking,” in AGARD Strategies for Automatic Track Initiation (1979), pp. 21-32.
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    [CrossRef] [PubMed]
  5. S. Basty, M. A. Neifeld, D. Brady, and S. Kraut, “Nonlinear estimation for interferometric imaging,” Opt. Commun. 228, 249-261 (2003).
    [CrossRef]
  6. 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, 2965-2974 (2006).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  8. C.-Y. Chen, T.-T. Yang, and Wen--Shing Sun, “Optics system design applying a microprism array of a single lens stereo image pair,” Opt. Express 1615495-15505 (2008).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  19. S. H. Goodwin-Johansson, M. R. Davidson, D. E. Dausch, P. H. Holloway, and G. McGuire, “Reduced voltage artificial eyelid for protection of optical sensors,” Proc. SPIE 4695, 451-458 (2002).
    [CrossRef]
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2008 (3)

S. Shikhar, N. A. Goodman, M. A. Neifeld, C. Kim, J. Kim, and D. J.Brady, “Optically multiplexed imaging with superposition space tracking,” Proc. SPIE 7096709607 (2008).

R. Muise and A. Mahalanobis, “Computational sensing algorithms for image reconstruction and the detection of moving objects in multiplexed imaging systems,” Proc. SPIE 6977, 69770M (2008).
[CrossRef]

C.-Y. Chen, T.-T. Yang, and Wen--Shing Sun, “Optics system design applying a microprism array of a single lens stereo image pair,” Opt. Express 1615495-15505 (2008).
[CrossRef] [PubMed]

2007 (1)

2006 (1)

2004 (1)

R. J. Plemmons, M. Horvath, E. Leonhardt, V. P. Pauca, S. Prasad, S. B. Robinson, H. Setty, T. C. Torgersen, J. van der Gracht, E. Dowski, R. Narayanswamy, and P. E. X. Silveira, “Computational imaging systems for iris recognition,” Proc. SPIE 5559, 346-357 (2004).
[CrossRef]

2003 (2)

S. Basty, M. A. Neifeld, D. Brady, and S. Kraut, “Nonlinear estimation for interferometric imaging,” Opt. Commun. 228, 249-261 (2003).
[CrossRef]

S. Prasad, T. Torgersen, V. P. Pauca, R. Plemmons, and J. van der Gracht, “Engineering the pupil phase to improve image quality,” Proc. SPIE 5108, 1-12 (2003).
[CrossRef]

2002 (3)

2000 (1)

1999 (2)

D. Marks, R. Stack, D. Brady, D. Munson, and R. Brady, “Visible cone-beam tomography with a lensless camera,” Science 284, 2164-6166 (1999).
[CrossRef] [PubMed]

E. R. Dowski, Jr., and G. E. Johnson, “Wavefront coding: a modern method of achieving high performance and/or low cost imaging systems,” Proc. SPIE 3779, 137-145 (1999).
[CrossRef]

1998 (1)

1995 (1)

Andrews, H. C.

H. C. Andrews and B. R. Hunt, Digital Image Restoration (Prentice-Hall, 1977).

Ashok, A.

Athale, R. A.

Basty, S.

S. Basty, M. A. Neifeld, D. Brady, and S. Kraut, “Nonlinear estimation for interferometric imaging,” Opt. Commun. 228, 249-261 (2003).
[CrossRef]

Brady, D.

S. Basty, M. A. Neifeld, D. Brady, and S. Kraut, “Nonlinear estimation for interferometric imaging,” Opt. Commun. 228, 249-261 (2003).
[CrossRef]

D. Marks, R. Stack, D. Brady, D. Munson, and R. Brady, “Visible cone-beam tomography with a lensless camera,” Science 284, 2164-6166 (1999).
[CrossRef] [PubMed]

Brady, D. J.

Brady, R.

D. Marks, R. Stack, D. Brady, D. Munson, and R. Brady, “Visible cone-beam tomography with a lensless camera,” Science 284, 2164-6166 (1999).
[CrossRef] [PubMed]

Cathey, W. T.

Chen, C.-Y.

Christensen, M. P.

Coyle, K. M.

Dana, M.

M. Dana, “Establishment of air defense sensor requirements for automatic aircraft tracking,” in AGARD Strategies for Automatic Track Initiation (1979), pp. 21-32.

Dausch, D. E.

S. H. Goodwin-Johansson, M. R. Davidson, D. E. Dausch, P. H. Holloway, and G. McGuire, “Reduced voltage artificial eyelid for protection of optical sensors,” Proc. SPIE 4695, 451-458 (2002).
[CrossRef]

Davidson, M. R.

S. H. Goodwin-Johansson, M. R. Davidson, D. E. Dausch, P. H. Holloway, and G. McGuire, “Reduced voltage artificial eyelid for protection of optical sensors,” Proc. SPIE 4695, 451-458 (2002).
[CrossRef]

Dowski, E.

R. J. Plemmons, M. Horvath, E. Leonhardt, V. P. Pauca, S. Prasad, S. B. Robinson, H. Setty, T. C. Torgersen, J. van der Gracht, E. Dowski, R. Narayanswamy, and P. E. X. Silveira, “Computational imaging systems for iris recognition,” Proc. SPIE 5559, 346-357 (2004).
[CrossRef]

Dowski, E. R.

Dowski, Jr., E. R.

Euliss, G. W.

Gehm, M. E.

Goodman, N. A.

S. Shikhar, N. A. Goodman, M. A. Neifeld, C. Kim, J. Kim, and D. J.Brady, “Optically multiplexed imaging with superposition space tracking,” Proc. SPIE 7096709607 (2008).

Goodwin-Johansson, S. H.

S. H. Goodwin-Johansson, M. R. Davidson, D. E. Dausch, P. H. Holloway, and G. McGuire, “Reduced voltage artificial eyelid for protection of optical sensors,” Proc. SPIE 4695, 451-458 (2002).
[CrossRef]

Haney, M. W.

Holloway, P. H.

S. H. Goodwin-Johansson, M. R. Davidson, D. E. Dausch, P. H. Holloway, and G. McGuire, “Reduced voltage artificial eyelid for protection of optical sensors,” Proc. SPIE 4695, 451-458 (2002).
[CrossRef]

Horvath, M.

R. J. Plemmons, M. Horvath, E. Leonhardt, V. P. Pauca, S. Prasad, S. B. Robinson, H. Setty, T. C. Torgersen, J. van der Gracht, E. Dowski, R. Narayanswamy, and P. E. X. Silveira, “Computational imaging systems for iris recognition,” Proc. SPIE 5559, 346-357 (2004).
[CrossRef]

Hunt, B. R.

H. C. Andrews and B. R. Hunt, Digital Image Restoration (Prentice-Hall, 1977).

Johnson, G. E.

E. R. Dowski, Jr., and G. E. Johnson, “Wavefront coding: a modern method of achieving high performance and/or low cost imaging systems,” Proc. SPIE 3779, 137-145 (1999).
[CrossRef]

Kim, C.

S. Shikhar, N. A. Goodman, M. A. Neifeld, C. Kim, J. Kim, and D. J.Brady, “Optically multiplexed imaging with superposition space tracking,” Proc. SPIE 7096709607 (2008).

Kim, J.

S. Shikhar, N. A. Goodman, M. A. Neifeld, C. Kim, J. Kim, and D. J.Brady, “Optically multiplexed imaging with superposition space tracking,” Proc. SPIE 7096709607 (2008).

Kraut, S.

S. Basty, M. A. Neifeld, D. Brady, and S. Kraut, “Nonlinear estimation for interferometric imaging,” Opt. Commun. 228, 249-261 (2003).
[CrossRef]

Leonhardt, E.

R. J. Plemmons, M. Horvath, E. Leonhardt, V. P. Pauca, S. Prasad, S. B. Robinson, H. Setty, T. C. Torgersen, J. van der Gracht, E. Dowski, R. Narayanswamy, and P. E. X. Silveira, “Computational imaging systems for iris recognition,” Proc. SPIE 5559, 346-357 (2004).
[CrossRef]

Mahalanobis, A.

R. Muise and A. Mahalanobis, “Computational sensing algorithms for image reconstruction and the detection of moving objects in multiplexed imaging systems,” Proc. SPIE 6977, 69770M (2008).
[CrossRef]

Marks, D.

D. Marks, R. Stack, D. Brady, D. Munson, and R. Brady, “Visible cone-beam tomography with a lensless camera,” Science 284, 2164-6166 (1999).
[CrossRef] [PubMed]

Marks, D. L.

McCain, S. T.

McFadden, M. J.

McGuire, G.

S. H. Goodwin-Johansson, M. R. Davidson, D. E. Dausch, P. H. Holloway, and G. McGuire, “Reduced voltage artificial eyelid for protection of optical sensors,” Proc. SPIE 4695, 451-458 (2002).
[CrossRef]

Milojkovic, P.

Muise, R.

R. Muise and A. Mahalanobis, “Computational sensing algorithms for image reconstruction and the detection of moving objects in multiplexed imaging systems,” Proc. SPIE 6977, 69770M (2008).
[CrossRef]

Munson, D.

D. Marks, R. Stack, D. Brady, D. Munson, and R. Brady, “Visible cone-beam tomography with a lensless camera,” Science 284, 2164-6166 (1999).
[CrossRef] [PubMed]

Narayanswamy, R.

R. J. Plemmons, M. Horvath, E. Leonhardt, V. P. Pauca, S. Prasad, S. B. Robinson, H. Setty, T. C. Torgersen, J. van der Gracht, E. Dowski, R. Narayanswamy, and P. E. X. Silveira, “Computational imaging systems for iris recognition,” Proc. SPIE 5559, 346-357 (2004).
[CrossRef]

Neifeld, M. A.

S. Shikhar, N. A. Goodman, M. A. Neifeld, C. Kim, J. Kim, and D. J.Brady, “Optically multiplexed imaging with superposition space tracking,” Proc. SPIE 7096709607 (2008).

A. Ashok and M. A. Neifeld, “Pseudorandom phase masks for superresolution imaging from subpixel shifting,” Appl. Opt. 46, 2256-2268 (2007).
[CrossRef] [PubMed]

S. Basty, M. A. Neifeld, D. Brady, and S. Kraut, “Nonlinear estimation for interferometric imaging,” Opt. Commun. 228, 249-261 (2003).
[CrossRef]

M. D. Stenner, P. Shankar, and M. A. Neifeld, “Wide-field feature-specific imaging,” in Frontiers in Optics (Optical Society of America, 2007), paper FMJ2.

Pauca, V. P.

R. J. Plemmons, M. Horvath, E. Leonhardt, V. P. Pauca, S. Prasad, S. B. Robinson, H. Setty, T. C. Torgersen, J. van der Gracht, E. Dowski, R. Narayanswamy, and P. E. X. Silveira, “Computational imaging systems for iris recognition,” Proc. SPIE 5559, 346-357 (2004).
[CrossRef]

S. Prasad, T. Torgersen, V. P. Pauca, R. Plemmons, and J. van der Gracht, “Engineering the pupil phase to improve image quality,” Proc. SPIE 5108, 1-12 (2003).
[CrossRef]

Pitsianis, N. P.

Plemmons, R.

S. Prasad, T. Torgersen, V. P. Pauca, R. Plemmons, and J. van der Gracht, “Engineering the pupil phase to improve image quality,” Proc. SPIE 5108, 1-12 (2003).
[CrossRef]

Plemmons, R. J.

R. J. Plemmons, M. Horvath, E. Leonhardt, V. P. Pauca, S. Prasad, S. B. Robinson, H. Setty, T. C. Torgersen, J. van der Gracht, E. Dowski, R. Narayanswamy, and P. E. X. Silveira, “Computational imaging systems for iris recognition,” Proc. SPIE 5559, 346-357 (2004).
[CrossRef]

Potuluri, P.

Prasad, S.

R. J. Plemmons, M. Horvath, E. Leonhardt, V. P. Pauca, S. Prasad, S. B. Robinson, H. Setty, T. C. Torgersen, J. van der Gracht, E. Dowski, R. Narayanswamy, and P. E. X. Silveira, “Computational imaging systems for iris recognition,” Proc. SPIE 5559, 346-357 (2004).
[CrossRef]

S. Prasad, T. Torgersen, V. P. Pauca, R. Plemmons, and J. van der Gracht, “Engineering the pupil phase to improve image quality,” Proc. SPIE 5108, 1-12 (2003).
[CrossRef]

Robinson, D.

D. Stork and D. Robinson, “Joint digital-optical design of multi-frame imaging systems,” in Computational Sensing and Imaging (Optical Society of America, 2007).

Robinson, S. B.

R. J. Plemmons, M. Horvath, E. Leonhardt, V. P. Pauca, S. Prasad, S. B. Robinson, H. Setty, T. C. Torgersen, J. van der Gracht, E. Dowski, R. Narayanswamy, and P. E. X. Silveira, “Computational imaging systems for iris recognition,” Proc. SPIE 5559, 346-357 (2004).
[CrossRef]

Setty, H.

R. J. Plemmons, M. Horvath, E. Leonhardt, V. P. Pauca, S. Prasad, S. B. Robinson, H. Setty, T. C. Torgersen, J. van der Gracht, E. Dowski, R. Narayanswamy, and P. E. X. Silveira, “Computational imaging systems for iris recognition,” Proc. SPIE 5559, 346-357 (2004).
[CrossRef]

Shankar, P.

M. D. Stenner, P. Shankar, and M. A. Neifeld, “Wide-field feature-specific imaging,” in Frontiers in Optics (Optical Society of America, 2007), paper FMJ2.

Shikhar, S.

S. Shikhar, N. A. Goodman, M. A. Neifeld, C. Kim, J. Kim, and D. J.Brady, “Optically multiplexed imaging with superposition space tracking,” Proc. SPIE 7096709607 (2008).

Silveira, P. E. X.

R. J. Plemmons, M. Horvath, E. Leonhardt, V. P. Pauca, S. Prasad, S. B. Robinson, H. Setty, T. C. Torgersen, J. van der Gracht, E. Dowski, R. Narayanswamy, and P. E. X. Silveira, “Computational imaging systems for iris recognition,” Proc. SPIE 5559, 346-357 (2004).
[CrossRef]

Stack, R.

D. Marks, R. Stack, D. Brady, D. Munson, and R. Brady, “Visible cone-beam tomography with a lensless camera,” Science 284, 2164-6166 (1999).
[CrossRef] [PubMed]

Stack, R. A.

Stenner, M. D.

M. D. Stenner, P. Shankar, and M. A. Neifeld, “Wide-field feature-specific imaging,” in Frontiers in Optics (Optical Society of America, 2007), paper FMJ2.

Stork, D.

D. Stork and D. Robinson, “Joint digital-optical design of multi-frame imaging systems,” in Computational Sensing and Imaging (Optical Society of America, 2007).

Sullivan, M. E.

Sun, Wen--Shing

Torgersen, T.

S. Prasad, T. Torgersen, V. P. Pauca, R. Plemmons, and J. van der Gracht, “Engineering the pupil phase to improve image quality,” Proc. SPIE 5108, 1-12 (2003).
[CrossRef]

Torgersen, T. C.

R. J. Plemmons, M. Horvath, E. Leonhardt, V. P. Pauca, S. Prasad, S. B. Robinson, H. Setty, T. C. Torgersen, J. van der Gracht, E. Dowski, R. Narayanswamy, and P. E. X. Silveira, “Computational imaging systems for iris recognition,” Proc. SPIE 5559, 346-357 (2004).
[CrossRef]

van der Gracht, J.

R. J. Plemmons, M. Horvath, E. Leonhardt, V. P. Pauca, S. Prasad, S. B. Robinson, H. Setty, T. C. Torgersen, J. van der Gracht, E. Dowski, R. Narayanswamy, and P. E. X. Silveira, “Computational imaging systems for iris recognition,” Proc. SPIE 5559, 346-357 (2004).
[CrossRef]

S. Prasad, T. Torgersen, V. P. Pauca, R. Plemmons, and J. van der Gracht, “Engineering the pupil phase to improve image quality,” Proc. SPIE 5108, 1-12 (2003).
[CrossRef]

M. P. Christensen, G. W. Euliss, M. J. McFadden, K. M. Coyle, P. Milojkovic, M. W. Haney, J. van der Gracht, and R. A. Athale, “Active-eyes: an adaptive pixel-by-pixel image-segmentation sensor architecture for high-dynamic-range hyperspectral imaging,” Appl. Opt. 41, 6093-6103 (2002).
[CrossRef] [PubMed]

Wach, H. B.

Yang, T.-T.

Appl. Opt. (6)

Opt. Commun. (1)

S. Basty, M. A. Neifeld, D. Brady, and S. Kraut, “Nonlinear estimation for interferometric imaging,” Opt. Commun. 228, 249-261 (2003).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Proc. SPIE (5)

R. Muise and A. Mahalanobis, “Computational sensing algorithms for image reconstruction and the detection of moving objects in multiplexed imaging systems,” Proc. SPIE 6977, 69770M (2008).
[CrossRef]

E. R. Dowski, Jr., and G. E. Johnson, “Wavefront coding: a modern method of achieving high performance and/or low cost imaging systems,” Proc. SPIE 3779, 137-145 (1999).
[CrossRef]

S. Prasad, T. Torgersen, V. P. Pauca, R. Plemmons, and J. van der Gracht, “Engineering the pupil phase to improve image quality,” Proc. SPIE 5108, 1-12 (2003).
[CrossRef]

R. J. Plemmons, M. Horvath, E. Leonhardt, V. P. Pauca, S. Prasad, S. B. Robinson, H. Setty, T. C. Torgersen, J. van der Gracht, E. Dowski, R. Narayanswamy, and P. E. X. Silveira, “Computational imaging systems for iris recognition,” Proc. SPIE 5559, 346-357 (2004).
[CrossRef]

S. H. Goodwin-Johansson, M. R. Davidson, D. E. Dausch, P. H. Holloway, and G. McGuire, “Reduced voltage artificial eyelid for protection of optical sensors,” Proc. SPIE 4695, 451-458 (2002).
[CrossRef]

Science (1)

D. Marks, R. Stack, D. Brady, D. Munson, and R. Brady, “Visible cone-beam tomography with a lensless camera,” Science 284, 2164-6166 (1999).
[CrossRef] [PubMed]

Other (5)

H. C. Andrews and B. R. Hunt, Digital Image Restoration (Prentice-Hall, 1977).

S. Shikhar, N. A. Goodman, M. A. Neifeld, C. Kim, J. Kim, and D. J.Brady, “Optically multiplexed imaging with superposition space tracking,” Proc. SPIE 7096709607 (2008).

M. Dana, “Establishment of air defense sensor requirements for automatic aircraft tracking,” in AGARD Strategies for Automatic Track Initiation (1979), pp. 21-32.

M. D. Stenner, P. Shankar, and M. A. Neifeld, “Wide-field feature-specific imaging,” in Frontiers in Optics (Optical Society of America, 2007), paper FMJ2.

D. Stork and D. Robinson, “Joint digital-optical design of multi-frame imaging systems,” in Computational Sensing and Imaging (Optical Society of America, 2007).

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Figures (17)

Fig. 1
Fig. 1

Point sources distributed over a large spatial area can be imaged through (a) a conventional imager using one very large FPA, (b) several smaller conventional imagers working in parallel, (c) separate conventional apertures sharing a common FPA, and (d) several sparse apertures interleaved within a common aperture imaged on a common FPA. Both (c) and (d) also require shutters combining different look directions.

Fig. 2
Fig. 2

Eyelid comprises a transmissive surface and a electrostatic flap that unfolds when a voltage is applied to close the aperture. An array of individually controlled eyelids can be used to implement a sparse aperture.

Fig. 3
Fig. 3

Sparse aperture PSFs in (b) and (c) have the same main lobe width as the PSF of the fully open aperture in (a). Regular spacing between eyelids introduces replicas of the main lobe at intervals of λ F / K Δ as shown in (b), whereas random patterns of open eyelids suppress the replicas but introduce a small bias as shown in (c). The parameters used are λ = 4.65 μm , F = 1.5 m , Δ = 50 μm , K = 8 , and N e = 200 .

Fig. 4
Fig. 4

Normalized MTF of the sparse aperture ( p = 0.25 ) drops rapidly compared to that of the full aperture, and thus reduces the contrast of the image.

Fig. 5
Fig. 5

Aperture can be made to look separately in different directions using bulk prisms (a). The interleaved arrangement in (b) allows the aperture to look in both directions simultaneously. The sections of the bulk prisms in (b) can be thought of as a small microprism on top of a shim (also referred to as a micropiston). The microshutters allow different combinations of look directions to be opened or closed.

Fig. 6
Fig. 6

Sections of the bulk prism on the left are replaced by microprisms and smaller shims (micropistons) that are optimized to achieve the best possible PSF over a range of wavelengths.

Fig. 7
Fig. 7

In the 2D imaging system, a prism tilts the optical axis with respect to the z axis while rotating the prism moves the optical axis along a circle in the object plane.

Fig. 8
Fig. 8

Experiments were performed with (a) a small array of eyelids that are used to dynamically shutter the sparse aperture and (b) an array of microprisms that determine its look direction.

Fig. 9
Fig. 9

Random pattern in (a) was used to implement the sparse aperture. The optimum phase values (at the nominal wavelength of 4.65 μm ) shown in (b) yield the improved PSF in Fig. 10.

Fig. 10
Fig. 10

PSF without phase optimization is expected to produce a 12 pixel smear in the look direction, whereas after optimization, the PSFs in (b) and with quantization in (c) are much narrower and better behaved.

Fig. 11
Fig. 11

Collimated source (pinhole) was imaged through the microprism. The main features in the image were closely matched by the simulation, and chromatic blur, the sinc function due to the eyelid aperture, as well as the spacing between the array term created by the spots were all validated.

Fig. 12
Fig. 12

Restoration filter models the inverse of an unknown PSF and can be applied to the measured data to recover the ideal scene being imaged.

Fig. 13
Fig. 13

Experimental setup used for (a) bar pattern evaluation and (b) PSF evaluation.

Fig. 14
Fig. 14

Ideal bar pattern was imaged through the microprism array only and restored. The bars and the spacing between them are recovered and resolved (proving that resolution is achieved), but the edges are still smeared by chromaticity.

Fig. 15
Fig. 15

Image on the left is obtained using only microprisms and eyelids and shows the effect of chromatic blurring at the edges. The image on the right was obtained using phase correcting micropistons (in addition to microprisms and eyelids) and has sharp edges that preserve the resolution in the horizontal direction.

Fig. 16
Fig. 16

Compensated MTF is dramatically improved when the optimized micropistons are used.

Fig. 17
Fig. 17

Micropistons improve the quality of reconstructed image as the range of wavelength increases. When a single wavelength is used (toward the left), image quality remains low even though there is no chromatic blurring due to poor SNR. At a wider range of wavelengths, image quality improves but is hampered by chromatic blurring. The micropistons compensate for the blur, thereby making the image quality significantly better when a larger range of wavelengths is used.

Tables (1)

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Table 1 Values of Key Parameters Used in Experiments

Equations (22)

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p ( x ) = n = 0 N e 1 t n · rect ( x n Δ a ) = rect ( x a ) * ( n = 0 N e 1 t n · δ ( x n Δ ) ) ,
P ( u ) = | a sinc ( a u λ F ) n = 0 N e 1 t n e j 2 π Δ n u λ F | 2 ,
I ( u ) = G ( u ) * P ( u ) .
MTF = FT { P ( u ) } p ( x ) p ( x ) ,
P ( u ) = | a sinc ( a u λ F ) · [ 1 e j 2 π Δ N e λ F u 1 e j 2 π Δ λ F u ] | 2 = | a sinc ( a u λ F ) | 2 · | sin ( N e π Δ u λ F ) sin ( π Δ u λ F ) | 2 .
t n = { 1 if     n = i * K , i is a nonnegative integer 0 otherwise ,
P ( u ) = | a sinc ( a u λ F ) i = 0 N e K 1 e j 2 π Δ i K u λ F | 2 = | a sinc ( a u λ F ) | 2 · | sin ( N e π Δ u λ F ) sin ( π Δ K u λ F ) | 2 .
t n = { 1 with probability p 0 with probability ( 1 p ) , 1 n N e .
R ˜ t ( j ) = E { t n t n + j } = { p j = 0 p 2 j 0 .
R t ( j ) = { p · N e j = 0 p 2 · ( N e | j | ) j 0 , | j | N e .
F T { R t } = N e p ( 1 p ) + p 2 sin 2 ( π N e f ) sin 2 ( π f ) ,
P ( u ) = | a sinc ( a u λ F ) | 2 | N e p ( 1 p ) + p 2 sin 2 ( π N e Δ u λ F ) sin 2 ( π Δ u λ F ) | .
α j = 2 π ( n 1 ) d j ,
p ( x ) = j = 1 N t j · e i α j λ · rect ( x j Δ a ) e i 2 π λ ( n 1 ) η ( x j Δ ) .
P ( u ) = λ min λ max [ sinc [ a λ F ( u [ n 1 ] η F ) ] j = 1 N t j · e i α j λ · e i 2 π ( j Δ λ F ) u ] 2 d λ ,
f = u | P ( u ) D ( u ) | 2 d u
λ F N e Δ = ( 4.65 × 10 6 ) ( 100 × 10 3 ) ( 20 ) ( 1 × 10 3 ) = 23.3 × 10 6 m .
e = m , n | ( y ( m , n ) + ν ( m , n ) ) * h ( m , n ) x ( m , n ) | 2
e = k , l | [ Y ( k , l ) + N ( k , l ) ] H ( k , l ) X ( k , l ) | 2 .
MSE = E { e } = k , l | H ( k , l ) | 2 [ | Y ( k , l ) | 2 + S ν ( k , l ) ] 2 H ( k , l ) Y ( k , l ) X ( k , l ) + | X ( k , l ) | 2 ,
H mse = 2 H ( k , l ) [ | Y ( k , l ) | 2 + S ν ( k , l ) ] 2 Y ( k , l ) X ( k , l ) .
H ( k , l ) = Y ( k , l ) X ( k , l ) [ | Y ( k , l ) | 2 + S ν ( k , l ) ] .

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