Abstract

A performance scaling formulation for flat form-factor cameras is introduced. The analysis follows from basic geometric and sensitivity constraints found in low-profile imaging sensors. A capacity metric is proposed and used to estimate performance cost scaling as a function of the width-to-height aspect ratio in the optics of thin imagers. Two basic flat imaging sensor classes are considered—one folds the optical path of an annular telescope within the volume of a central obscuration, and the other uses spatial multiplexing and filtering across an array of low-resolution small cameras to generate an estimate of the high-resolution image. Scaling trends are highlighted that enable general performance comparisons at the optical signal collection level, thereby providing conclusions that are independent of the computational aspects of any particular approach. The results indicate that thin imagers face significant costs in physical size and sampling requirements if they are to match the performance of conventional cameras in the basic parameters of field of view, resolution, dynamic range, and sensitivity.

© 2006 Optical Society of America

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References

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  1. J. Tanida, T. Kumagai, K. Yamada, S. Miyatake, K. Ishida, T. Morimoto, N. Kondou, D. Miyazaki, and Y. Ichioka, "Thin observation module by bound optics (TOMBO): concept and experimental verification," Appl. Opt. 40, 1806-1813 (2001).
    [CrossRef]
  2. E. J. Tremblay, J. Rutkowski, I. Tamay, R. A. Stack, R. L. Morrison, D. Combs, J. Mader, H. Kim, U. Levy, Y. Fainman, and J. E. Ford, "Ultra-thin folded imager," presented at the OSA Topical Meeting on Computational Optical Sensing and Imaging, Charlotte, N.C., 6-8 June 2005.
  3. D. J. Brady, M. A. Fiddy, U. Shahid, and T. J. Suleski, "Compressive optical MONTAGE photography initiative: noise and error analysis," presented at the OSA Topical Meeting on Computational Optical Sensing and Imaging, Charlotte, N.C., 6-8 June 2005.
  4. M. W. Haney, M. P. Christensen, D. Rajan, S. C. Douglas, and S. L. Wood, "Adaptive flat micro-mirror-array-based computational imaging architecture," presented at the OSA Topical Meeting on Computational Optical Sensing and Imaging, Charlotte, N.C., 6-8 June 2005.
  5. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999), p. 275.

2001

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999), p. 275.

Brady, D. J.

D. J. Brady, M. A. Fiddy, U. Shahid, and T. J. Suleski, "Compressive optical MONTAGE photography initiative: noise and error analysis," presented at the OSA Topical Meeting on Computational Optical Sensing and Imaging, Charlotte, N.C., 6-8 June 2005.

Christensen, M. P.

M. W. Haney, M. P. Christensen, D. Rajan, S. C. Douglas, and S. L. Wood, "Adaptive flat micro-mirror-array-based computational imaging architecture," presented at the OSA Topical Meeting on Computational Optical Sensing and Imaging, Charlotte, N.C., 6-8 June 2005.

Combs, D.

E. J. Tremblay, J. Rutkowski, I. Tamay, R. A. Stack, R. L. Morrison, D. Combs, J. Mader, H. Kim, U. Levy, Y. Fainman, and J. E. Ford, "Ultra-thin folded imager," presented at the OSA Topical Meeting on Computational Optical Sensing and Imaging, Charlotte, N.C., 6-8 June 2005.

Douglas, S. C.

M. W. Haney, M. P. Christensen, D. Rajan, S. C. Douglas, and S. L. Wood, "Adaptive flat micro-mirror-array-based computational imaging architecture," presented at the OSA Topical Meeting on Computational Optical Sensing and Imaging, Charlotte, N.C., 6-8 June 2005.

Fainman, Y.

E. J. Tremblay, J. Rutkowski, I. Tamay, R. A. Stack, R. L. Morrison, D. Combs, J. Mader, H. Kim, U. Levy, Y. Fainman, and J. E. Ford, "Ultra-thin folded imager," presented at the OSA Topical Meeting on Computational Optical Sensing and Imaging, Charlotte, N.C., 6-8 June 2005.

Fiddy, M. A.

D. J. Brady, M. A. Fiddy, U. Shahid, and T. J. Suleski, "Compressive optical MONTAGE photography initiative: noise and error analysis," presented at the OSA Topical Meeting on Computational Optical Sensing and Imaging, Charlotte, N.C., 6-8 June 2005.

Ford, J. E.

E. J. Tremblay, J. Rutkowski, I. Tamay, R. A. Stack, R. L. Morrison, D. Combs, J. Mader, H. Kim, U. Levy, Y. Fainman, and J. E. Ford, "Ultra-thin folded imager," presented at the OSA Topical Meeting on Computational Optical Sensing and Imaging, Charlotte, N.C., 6-8 June 2005.

Haney, M. W.

M. W. Haney, M. P. Christensen, D. Rajan, S. C. Douglas, and S. L. Wood, "Adaptive flat micro-mirror-array-based computational imaging architecture," presented at the OSA Topical Meeting on Computational Optical Sensing and Imaging, Charlotte, N.C., 6-8 June 2005.

Ichioka, Y.

Ishida, K.

Kim, H.

E. J. Tremblay, J. Rutkowski, I. Tamay, R. A. Stack, R. L. Morrison, D. Combs, J. Mader, H. Kim, U. Levy, Y. Fainman, and J. E. Ford, "Ultra-thin folded imager," presented at the OSA Topical Meeting on Computational Optical Sensing and Imaging, Charlotte, N.C., 6-8 June 2005.

Kondou, N.

Kumagai, T.

Levy, U.

E. J. Tremblay, J. Rutkowski, I. Tamay, R. A. Stack, R. L. Morrison, D. Combs, J. Mader, H. Kim, U. Levy, Y. Fainman, and J. E. Ford, "Ultra-thin folded imager," presented at the OSA Topical Meeting on Computational Optical Sensing and Imaging, Charlotte, N.C., 6-8 June 2005.

Mader, J.

E. J. Tremblay, J. Rutkowski, I. Tamay, R. A. Stack, R. L. Morrison, D. Combs, J. Mader, H. Kim, U. Levy, Y. Fainman, and J. E. Ford, "Ultra-thin folded imager," presented at the OSA Topical Meeting on Computational Optical Sensing and Imaging, Charlotte, N.C., 6-8 June 2005.

Miyatake, S.

Miyazaki, D.

Morimoto, T.

Morrison, R. L.

E. J. Tremblay, J. Rutkowski, I. Tamay, R. A. Stack, R. L. Morrison, D. Combs, J. Mader, H. Kim, U. Levy, Y. Fainman, and J. E. Ford, "Ultra-thin folded imager," presented at the OSA Topical Meeting on Computational Optical Sensing and Imaging, Charlotte, N.C., 6-8 June 2005.

Rajan, D.

M. W. Haney, M. P. Christensen, D. Rajan, S. C. Douglas, and S. L. Wood, "Adaptive flat micro-mirror-array-based computational imaging architecture," presented at the OSA Topical Meeting on Computational Optical Sensing and Imaging, Charlotte, N.C., 6-8 June 2005.

Rutkowski, J.

E. J. Tremblay, J. Rutkowski, I. Tamay, R. A. Stack, R. L. Morrison, D. Combs, J. Mader, H. Kim, U. Levy, Y. Fainman, and J. E. Ford, "Ultra-thin folded imager," presented at the OSA Topical Meeting on Computational Optical Sensing and Imaging, Charlotte, N.C., 6-8 June 2005.

Shahid, U.

D. J. Brady, M. A. Fiddy, U. Shahid, and T. J. Suleski, "Compressive optical MONTAGE photography initiative: noise and error analysis," presented at the OSA Topical Meeting on Computational Optical Sensing and Imaging, Charlotte, N.C., 6-8 June 2005.

Stack, R. A.

E. J. Tremblay, J. Rutkowski, I. Tamay, R. A. Stack, R. L. Morrison, D. Combs, J. Mader, H. Kim, U. Levy, Y. Fainman, and J. E. Ford, "Ultra-thin folded imager," presented at the OSA Topical Meeting on Computational Optical Sensing and Imaging, Charlotte, N.C., 6-8 June 2005.

Suleski, T. J.

D. J. Brady, M. A. Fiddy, U. Shahid, and T. J. Suleski, "Compressive optical MONTAGE photography initiative: noise and error analysis," presented at the OSA Topical Meeting on Computational Optical Sensing and Imaging, Charlotte, N.C., 6-8 June 2005.

Tamay, I.

E. J. Tremblay, J. Rutkowski, I. Tamay, R. A. Stack, R. L. Morrison, D. Combs, J. Mader, H. Kim, U. Levy, Y. Fainman, and J. E. Ford, "Ultra-thin folded imager," presented at the OSA Topical Meeting on Computational Optical Sensing and Imaging, Charlotte, N.C., 6-8 June 2005.

Tanida, J.

Tremblay, E. J.

E. J. Tremblay, J. Rutkowski, I. Tamay, R. A. Stack, R. L. Morrison, D. Combs, J. Mader, H. Kim, U. Levy, Y. Fainman, and J. E. Ford, "Ultra-thin folded imager," presented at the OSA Topical Meeting on Computational Optical Sensing and Imaging, Charlotte, N.C., 6-8 June 2005.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999), p. 275.

Wood, S. L.

M. W. Haney, M. P. Christensen, D. Rajan, S. C. Douglas, and S. L. Wood, "Adaptive flat micro-mirror-array-based computational imaging architecture," presented at the OSA Topical Meeting on Computational Optical Sensing and Imaging, Charlotte, N.C., 6-8 June 2005.

Yamada, K.

Appl. Opt.

Other

E. J. Tremblay, J. Rutkowski, I. Tamay, R. A. Stack, R. L. Morrison, D. Combs, J. Mader, H. Kim, U. Levy, Y. Fainman, and J. E. Ford, "Ultra-thin folded imager," presented at the OSA Topical Meeting on Computational Optical Sensing and Imaging, Charlotte, N.C., 6-8 June 2005.

D. J. Brady, M. A. Fiddy, U. Shahid, and T. J. Suleski, "Compressive optical MONTAGE photography initiative: noise and error analysis," presented at the OSA Topical Meeting on Computational Optical Sensing and Imaging, Charlotte, N.C., 6-8 June 2005.

M. W. Haney, M. P. Christensen, D. Rajan, S. C. Douglas, and S. L. Wood, "Adaptive flat micro-mirror-array-based computational imaging architecture," presented at the OSA Topical Meeting on Computational Optical Sensing and Imaging, Charlotte, N.C., 6-8 June 2005.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999), p. 275.

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Figures (8)

Fig. 1
Fig. 1

Baseline f∕1 camera parameters.

Fig. 2
Fig. 2

Depiction of baseline and folded annular telescope collectionaperture geometries.

Fig. 3
Fig. 3

Folded annular-camera parameters showing that wd aha for f∕1 optics and the resultant limitation on the achievable solid-angle FOV (θ a 2).

Fig. 4
Fig. 4

Multiplexed camera geometry (side view) achieves a ×n reduction in height (and therefore volume) because f∕1 optics are maintained. The total number and size of sensor pixels are assumed to be the same in both cameras.

Fig. 5
Fig. 5

Performance metric scaling for the two classes of thin imagers relative to a baseline conventional f∕1 camera with FOV = 0.25 sr.

Fig. 6
Fig. 6

Sensor volume scaling for the two classes of thin imagers relative to a baseline conventional f∕1 camera with FOV = 0.25 sr.

Fig. 7
Fig. 7

Sensor footprint-area scaling for the two classes of thin imagers relative to a baseline conventional f∕1 camera with FOV = 0.25 sr.

Fig. 8
Fig. 8

Number of frames required for the two classes of thin imagers relative to a baseline conventional f∕1 camera with FOV = 0.25 sr.

Equations (18)

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M = Ω V S ( d μ ) ( d ν ) ,
n = D / h a D / w 2 ( 1 + 2 N f ) 1 / 2 ,
θ a 2 tan 1 [ d a 2 ( w + D a ) ] 2 tan 1 [ h a 2 ( h a + D a ) ] .
θ a 2 tan 1 [ h a 2 [ ( D 2 / 4 h a ) + h a ] ] h a [ ( D 2 / 4 h a ) + h a ] ,
θ a 1 1 + ( n 2 / 4 ) .
M a M o = θ a 2 D 3 V a Ω o = D 3 V a Ω o ( 1 + n 2 / 4 ) 2 .
M a ( M o / Ω o ) n 3 ( 1 + n 2 / 4 ) 4 .
N ̃ p ( μ , ν ) [ W k ( μ , ν ) × ( η N ( μ , ν ) / n 2 ) d μ d ν ] W k - 1 ( μ , ν ) ,
σ 2 = n 2 σ t     2 + σ s     2 ,
σ 2 n 2 σ t     2 + η o n 2 η n = n 2 ( σ t     2 + η o η n ) .
M s M o = V o Ω o Ω o ( V o / n ) [ n 2 ( 1 + σ s     2 / σ t     2 ) ] [ n 2 ( 1 + σ s     2 / σ t     2 ) ] ,
M s M o n 3 ( 1 + σ s     2 / σ t     2 ) 2 .
V a = V o Ω o ( 1 + n 2 / 4 ) 4 / n 3 ( cm 3 ) ,
A a = A o Ω o ( 1 + n 2 / 4 ) 4 / n 2 ( cm 2 ) ,
N a = 1.
V s = V o n ( 1 + σ s     2 / σ t     2 ) ( cm 3 ) ,
A s = A o n 2 ( 1 + σ s     2 / σ t     2 ) ( cm 2 ) ,
N s = n 2 ( 1 + σ s     2 / σ t     2 ) ,

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