Abstract

We analyze the signal-to-noise ratio (SNR) of arbitrary imaging systems in the presence of defocus. The modulation transfer function (MTF) and the mean SNR are combined to calculate the spatial-frequency spectrum of the SNR (the spectral SNR). Computational imaging methods are used for extending the depth of field (DOF) of the system. The DOF of a task-specific imaging system is defined as the range of defocus that causes the spectral SNR to drop below a minimum value within a band of spatial frequencies of interest. We introduce the polar-SNR plot as a tool for visualizing the spectral SNR of defocused imaging systems with asymmetric pupil functions. As an example, we perform the analysis of an imaging system used for biometric iris recognition.

© 2006 Optical Society of America

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  1. See, for example, J. W. Goodman, Introduction to Fourier Optics, 1st ed. (McGraw-Hill, 1968), Chap. 6, pp. 122-125.
  2. J. Hall, "F-number, numerical aperture, and depth of focus," in Encyclopedia of Optical Engineering (Marcel Dekker, 2003), pp. 556-559.
  3. W. J. Smith, Modern Optical Engineering, 3rd. ed. (McGraw-Hill, 2000), Chap. 6, pp. 154-157.
  4. R. Narayanswamy, G. E. Johnson, P. E. X. Silveira, and H. B. Wach, "Extending the imaging volume for biometric iris recognition," Appl. Opt. 44, 701-712 (2005).
    [CrossRef] [PubMed]
  5. R. Narayanswamy, P. E. X. Silveira, H. Setty, V. P. Pauca, and J. van der Gracht, "Extended depth-of-field iris recognition system for a workstation environment," in Biometric Technology for Human Identification II, A. K. Jain and N. K. Ratha, eds., Proc. SPIE 5779, 41-50 (2005).
  6. R. Narayanswamy, A. E. Baron, V. Chumachenko, and A. Greengard, "Applications of wavefront coded imaging," in Computational Imaging II, C.A.Bouman and E.L.Miller, eds., Proc. SPIE 5299, 163-174 (2004).
  7. W. T. Cathey and E. Dowski, "New paradigm for imaging systems," Appl. Opt. 41, 6080-6092 (2002).
    [CrossRef] [PubMed]
  8. E. R. Dowski, Jr., and W. T. Cathey, "Extended depth of field through wavefront coding," Appl. Opt. 34, 1859-1866 (1995).
    [CrossRef] [PubMed]
  9. H. Wach, E. Dowski, and W. T. Cathey, "Control of chromatic focal shift through wavefront coding," Appl. Opt. 37, 5359-5367 (1998).
    [CrossRef]
  10. K. Kubala, E. Dowski, J. Kobus, and B. Brown, "Aberration and error invariant space telescope systems," in Novel Optical Systems Design and Optimization VII, J.M.Sasian, R.J.Koshel, P.K.Manhart, and R.C.Juergens, eds., Proc. SPIE 5524, 54-65 (2004).
  11. R. Fiete and T. Tantalo, "Comparison of SNR image quality metrics for remote sensing systems," Opt. Eng. 40, 574-585 (2001).
    [CrossRef]
  12. C. P. Cain, D. Courant, D. A. Freund, B. A. Grossman, P. A. Kennedy, D. J. Lund, M. A. Mainster, A. A. Manenkov, W. J. Marshall, R. McCally, B. A. Rockwell, D. H. Sliney, P. A. Smith, B. E. Stuck, S. A. Tell, M. L. Wolbarsht, and G. I. Zheltov, "Revision of the guidelines on limits of exposure to laser radiation of wavelengths between 400 nm and 1.4 μm," Health Phys. 79, 431-440 (2000).
    [CrossRef]
  13. W. J. Smith, Modern Optical Engineering, 3rd. ed. (McGraw-Hill, 2000), Chap. 8, pp. 219-230.
  14. R. W. Boyd, Radiometry and the Detection of Optical Radiation (Wiley, 1983).
  15. See, for example, E. R. Dougherty, Random Processes for Image and Signal Processing (IEEE, 1998), Chap. 1, pp. 26-28.
  16. J. W. Goodman, "Fan-in and fan-out with optical interconnections," Opt. Acta 32, 1489-1496 (1985).
    [CrossRef]
  17. D. J. Brady, "Multiplex sensors and the constant radiance theorem," Opt. Lett. 27, 16-18 (2002).
    [CrossRef]
  18. J. G. Daugman, "High confidence visual recognition of persons by a test of statistical independence," IEEE Trans. Pattern Anal. Mach. Intell. 15, 1148-1161 (1993).
    [CrossRef]
  19. R. P. Wildes, "Automated iris recognition: an emerging biometric technology," in Proc. IEEE 85, 1348-1363 (1997).
    [CrossRef]
  20. J. G. Daugman, "The importance of being random: statistical principles of iris recognition," Pattern Recogn. 36, 279-291 (2003).
    [CrossRef]

2005

2003

J. G. Daugman, "The importance of being random: statistical principles of iris recognition," Pattern Recogn. 36, 279-291 (2003).
[CrossRef]

2002

2001

R. Fiete and T. Tantalo, "Comparison of SNR image quality metrics for remote sensing systems," Opt. Eng. 40, 574-585 (2001).
[CrossRef]

2000

C. P. Cain, D. Courant, D. A. Freund, B. A. Grossman, P. A. Kennedy, D. J. Lund, M. A. Mainster, A. A. Manenkov, W. J. Marshall, R. McCally, B. A. Rockwell, D. H. Sliney, P. A. Smith, B. E. Stuck, S. A. Tell, M. L. Wolbarsht, and G. I. Zheltov, "Revision of the guidelines on limits of exposure to laser radiation of wavelengths between 400 nm and 1.4 μm," Health Phys. 79, 431-440 (2000).
[CrossRef]

1998

1997

R. P. Wildes, "Automated iris recognition: an emerging biometric technology," in Proc. IEEE 85, 1348-1363 (1997).
[CrossRef]

1995

1993

J. G. Daugman, "High confidence visual recognition of persons by a test of statistical independence," IEEE Trans. Pattern Anal. Mach. Intell. 15, 1148-1161 (1993).
[CrossRef]

1985

J. W. Goodman, "Fan-in and fan-out with optical interconnections," Opt. Acta 32, 1489-1496 (1985).
[CrossRef]

Baron, A. E.

R. Narayanswamy, A. E. Baron, V. Chumachenko, and A. Greengard, "Applications of wavefront coded imaging," in Computational Imaging II, C.A.Bouman and E.L.Miller, eds., Proc. SPIE 5299, 163-174 (2004).

Boyd, R. W.

R. W. Boyd, Radiometry and the Detection of Optical Radiation (Wiley, 1983).

Brady, D. J.

Brown, B.

K. Kubala, E. Dowski, J. Kobus, and B. Brown, "Aberration and error invariant space telescope systems," in Novel Optical Systems Design and Optimization VII, J.M.Sasian, R.J.Koshel, P.K.Manhart, and R.C.Juergens, eds., Proc. SPIE 5524, 54-65 (2004).

Cain, C. P.

C. P. Cain, D. Courant, D. A. Freund, B. A. Grossman, P. A. Kennedy, D. J. Lund, M. A. Mainster, A. A. Manenkov, W. J. Marshall, R. McCally, B. A. Rockwell, D. H. Sliney, P. A. Smith, B. E. Stuck, S. A. Tell, M. L. Wolbarsht, and G. I. Zheltov, "Revision of the guidelines on limits of exposure to laser radiation of wavelengths between 400 nm and 1.4 μm," Health Phys. 79, 431-440 (2000).
[CrossRef]

Cathey, W. T.

Chumachenko, V.

R. Narayanswamy, A. E. Baron, V. Chumachenko, and A. Greengard, "Applications of wavefront coded imaging," in Computational Imaging II, C.A.Bouman and E.L.Miller, eds., Proc. SPIE 5299, 163-174 (2004).

Courant, D.

C. P. Cain, D. Courant, D. A. Freund, B. A. Grossman, P. A. Kennedy, D. J. Lund, M. A. Mainster, A. A. Manenkov, W. J. Marshall, R. McCally, B. A. Rockwell, D. H. Sliney, P. A. Smith, B. E. Stuck, S. A. Tell, M. L. Wolbarsht, and G. I. Zheltov, "Revision of the guidelines on limits of exposure to laser radiation of wavelengths between 400 nm and 1.4 μm," Health Phys. 79, 431-440 (2000).
[CrossRef]

Daugman, J. G.

J. G. Daugman, "The importance of being random: statistical principles of iris recognition," Pattern Recogn. 36, 279-291 (2003).
[CrossRef]

J. G. Daugman, "High confidence visual recognition of persons by a test of statistical independence," IEEE Trans. Pattern Anal. Mach. Intell. 15, 1148-1161 (1993).
[CrossRef]

Dougherty, E. R.

See, for example, E. R. Dougherty, Random Processes for Image and Signal Processing (IEEE, 1998), Chap. 1, pp. 26-28.

Dowski, E.

W. T. Cathey and E. Dowski, "New paradigm for imaging systems," Appl. Opt. 41, 6080-6092 (2002).
[CrossRef] [PubMed]

H. Wach, E. Dowski, and W. T. Cathey, "Control of chromatic focal shift through wavefront coding," Appl. Opt. 37, 5359-5367 (1998).
[CrossRef]

K. Kubala, E. Dowski, J. Kobus, and B. Brown, "Aberration and error invariant space telescope systems," in Novel Optical Systems Design and Optimization VII, J.M.Sasian, R.J.Koshel, P.K.Manhart, and R.C.Juergens, eds., Proc. SPIE 5524, 54-65 (2004).

Dowski, E. R.

Fiete, R.

R. Fiete and T. Tantalo, "Comparison of SNR image quality metrics for remote sensing systems," Opt. Eng. 40, 574-585 (2001).
[CrossRef]

Freund, D. A.

C. P. Cain, D. Courant, D. A. Freund, B. A. Grossman, P. A. Kennedy, D. J. Lund, M. A. Mainster, A. A. Manenkov, W. J. Marshall, R. McCally, B. A. Rockwell, D. H. Sliney, P. A. Smith, B. E. Stuck, S. A. Tell, M. L. Wolbarsht, and G. I. Zheltov, "Revision of the guidelines on limits of exposure to laser radiation of wavelengths between 400 nm and 1.4 μm," Health Phys. 79, 431-440 (2000).
[CrossRef]

Goodman, J. W.

J. W. Goodman, "Fan-in and fan-out with optical interconnections," Opt. Acta 32, 1489-1496 (1985).
[CrossRef]

Greengard, A.

R. Narayanswamy, A. E. Baron, V. Chumachenko, and A. Greengard, "Applications of wavefront coded imaging," in Computational Imaging II, C.A.Bouman and E.L.Miller, eds., Proc. SPIE 5299, 163-174 (2004).

Grossman, B. A.

C. P. Cain, D. Courant, D. A. Freund, B. A. Grossman, P. A. Kennedy, D. J. Lund, M. A. Mainster, A. A. Manenkov, W. J. Marshall, R. McCally, B. A. Rockwell, D. H. Sliney, P. A. Smith, B. E. Stuck, S. A. Tell, M. L. Wolbarsht, and G. I. Zheltov, "Revision of the guidelines on limits of exposure to laser radiation of wavelengths between 400 nm and 1.4 μm," Health Phys. 79, 431-440 (2000).
[CrossRef]

Hall, J.

J. Hall, "F-number, numerical aperture, and depth of focus," in Encyclopedia of Optical Engineering (Marcel Dekker, 2003), pp. 556-559.

Johnson, G. E.

Kennedy, P. A.

C. P. Cain, D. Courant, D. A. Freund, B. A. Grossman, P. A. Kennedy, D. J. Lund, M. A. Mainster, A. A. Manenkov, W. J. Marshall, R. McCally, B. A. Rockwell, D. H. Sliney, P. A. Smith, B. E. Stuck, S. A. Tell, M. L. Wolbarsht, and G. I. Zheltov, "Revision of the guidelines on limits of exposure to laser radiation of wavelengths between 400 nm and 1.4 μm," Health Phys. 79, 431-440 (2000).
[CrossRef]

Kobus, J.

K. Kubala, E. Dowski, J. Kobus, and B. Brown, "Aberration and error invariant space telescope systems," in Novel Optical Systems Design and Optimization VII, J.M.Sasian, R.J.Koshel, P.K.Manhart, and R.C.Juergens, eds., Proc. SPIE 5524, 54-65 (2004).

Kubala, K.

K. Kubala, E. Dowski, J. Kobus, and B. Brown, "Aberration and error invariant space telescope systems," in Novel Optical Systems Design and Optimization VII, J.M.Sasian, R.J.Koshel, P.K.Manhart, and R.C.Juergens, eds., Proc. SPIE 5524, 54-65 (2004).

Lund, D. J.

C. P. Cain, D. Courant, D. A. Freund, B. A. Grossman, P. A. Kennedy, D. J. Lund, M. A. Mainster, A. A. Manenkov, W. J. Marshall, R. McCally, B. A. Rockwell, D. H. Sliney, P. A. Smith, B. E. Stuck, S. A. Tell, M. L. Wolbarsht, and G. I. Zheltov, "Revision of the guidelines on limits of exposure to laser radiation of wavelengths between 400 nm and 1.4 μm," Health Phys. 79, 431-440 (2000).
[CrossRef]

Mainster, M. A.

C. P. Cain, D. Courant, D. A. Freund, B. A. Grossman, P. A. Kennedy, D. J. Lund, M. A. Mainster, A. A. Manenkov, W. J. Marshall, R. McCally, B. A. Rockwell, D. H. Sliney, P. A. Smith, B. E. Stuck, S. A. Tell, M. L. Wolbarsht, and G. I. Zheltov, "Revision of the guidelines on limits of exposure to laser radiation of wavelengths between 400 nm and 1.4 μm," Health Phys. 79, 431-440 (2000).
[CrossRef]

Manenkov, A. A.

C. P. Cain, D. Courant, D. A. Freund, B. A. Grossman, P. A. Kennedy, D. J. Lund, M. A. Mainster, A. A. Manenkov, W. J. Marshall, R. McCally, B. A. Rockwell, D. H. Sliney, P. A. Smith, B. E. Stuck, S. A. Tell, M. L. Wolbarsht, and G. I. Zheltov, "Revision of the guidelines on limits of exposure to laser radiation of wavelengths between 400 nm and 1.4 μm," Health Phys. 79, 431-440 (2000).
[CrossRef]

Marshall, W. J.

C. P. Cain, D. Courant, D. A. Freund, B. A. Grossman, P. A. Kennedy, D. J. Lund, M. A. Mainster, A. A. Manenkov, W. J. Marshall, R. McCally, B. A. Rockwell, D. H. Sliney, P. A. Smith, B. E. Stuck, S. A. Tell, M. L. Wolbarsht, and G. I. Zheltov, "Revision of the guidelines on limits of exposure to laser radiation of wavelengths between 400 nm and 1.4 μm," Health Phys. 79, 431-440 (2000).
[CrossRef]

McCally, R.

C. P. Cain, D. Courant, D. A. Freund, B. A. Grossman, P. A. Kennedy, D. J. Lund, M. A. Mainster, A. A. Manenkov, W. J. Marshall, R. McCally, B. A. Rockwell, D. H. Sliney, P. A. Smith, B. E. Stuck, S. A. Tell, M. L. Wolbarsht, and G. I. Zheltov, "Revision of the guidelines on limits of exposure to laser radiation of wavelengths between 400 nm and 1.4 μm," Health Phys. 79, 431-440 (2000).
[CrossRef]

Narayanswamy, R.

R. Narayanswamy, G. E. Johnson, P. E. X. Silveira, and H. B. Wach, "Extending the imaging volume for biometric iris recognition," Appl. Opt. 44, 701-712 (2005).
[CrossRef] [PubMed]

R. Narayanswamy, A. E. Baron, V. Chumachenko, and A. Greengard, "Applications of wavefront coded imaging," in Computational Imaging II, C.A.Bouman and E.L.Miller, eds., Proc. SPIE 5299, 163-174 (2004).

R. Narayanswamy, P. E. X. Silveira, H. Setty, V. P. Pauca, and J. van der Gracht, "Extended depth-of-field iris recognition system for a workstation environment," in Biometric Technology for Human Identification II, A. K. Jain and N. K. Ratha, eds., Proc. SPIE 5779, 41-50 (2005).

Pauca, V. P.

R. Narayanswamy, P. E. X. Silveira, H. Setty, V. P. Pauca, and J. van der Gracht, "Extended depth-of-field iris recognition system for a workstation environment," in Biometric Technology for Human Identification II, A. K. Jain and N. K. Ratha, eds., Proc. SPIE 5779, 41-50 (2005).

Rockwell, B. A.

C. P. Cain, D. Courant, D. A. Freund, B. A. Grossman, P. A. Kennedy, D. J. Lund, M. A. Mainster, A. A. Manenkov, W. J. Marshall, R. McCally, B. A. Rockwell, D. H. Sliney, P. A. Smith, B. E. Stuck, S. A. Tell, M. L. Wolbarsht, and G. I. Zheltov, "Revision of the guidelines on limits of exposure to laser radiation of wavelengths between 400 nm and 1.4 μm," Health Phys. 79, 431-440 (2000).
[CrossRef]

Setty, H.

R. Narayanswamy, P. E. X. Silveira, H. Setty, V. P. Pauca, and J. van der Gracht, "Extended depth-of-field iris recognition system for a workstation environment," in Biometric Technology for Human Identification II, A. K. Jain and N. K. Ratha, eds., Proc. SPIE 5779, 41-50 (2005).

Silveira, P. E. X.

R. Narayanswamy, G. E. Johnson, P. E. X. Silveira, and H. B. Wach, "Extending the imaging volume for biometric iris recognition," Appl. Opt. 44, 701-712 (2005).
[CrossRef] [PubMed]

R. Narayanswamy, P. E. X. Silveira, H. Setty, V. P. Pauca, and J. van der Gracht, "Extended depth-of-field iris recognition system for a workstation environment," in Biometric Technology for Human Identification II, A. K. Jain and N. K. Ratha, eds., Proc. SPIE 5779, 41-50 (2005).

Sliney, D. H.

C. P. Cain, D. Courant, D. A. Freund, B. A. Grossman, P. A. Kennedy, D. J. Lund, M. A. Mainster, A. A. Manenkov, W. J. Marshall, R. McCally, B. A. Rockwell, D. H. Sliney, P. A. Smith, B. E. Stuck, S. A. Tell, M. L. Wolbarsht, and G. I. Zheltov, "Revision of the guidelines on limits of exposure to laser radiation of wavelengths between 400 nm and 1.4 μm," Health Phys. 79, 431-440 (2000).
[CrossRef]

Smith, P. A.

C. P. Cain, D. Courant, D. A. Freund, B. A. Grossman, P. A. Kennedy, D. J. Lund, M. A. Mainster, A. A. Manenkov, W. J. Marshall, R. McCally, B. A. Rockwell, D. H. Sliney, P. A. Smith, B. E. Stuck, S. A. Tell, M. L. Wolbarsht, and G. I. Zheltov, "Revision of the guidelines on limits of exposure to laser radiation of wavelengths between 400 nm and 1.4 μm," Health Phys. 79, 431-440 (2000).
[CrossRef]

Smith, W. J.

W. J. Smith, Modern Optical Engineering, 3rd. ed. (McGraw-Hill, 2000), Chap. 8, pp. 219-230.

W. J. Smith, Modern Optical Engineering, 3rd. ed. (McGraw-Hill, 2000), Chap. 6, pp. 154-157.

Stuck, B. E.

C. P. Cain, D. Courant, D. A. Freund, B. A. Grossman, P. A. Kennedy, D. J. Lund, M. A. Mainster, A. A. Manenkov, W. J. Marshall, R. McCally, B. A. Rockwell, D. H. Sliney, P. A. Smith, B. E. Stuck, S. A. Tell, M. L. Wolbarsht, and G. I. Zheltov, "Revision of the guidelines on limits of exposure to laser radiation of wavelengths between 400 nm and 1.4 μm," Health Phys. 79, 431-440 (2000).
[CrossRef]

Tantalo, T.

R. Fiete and T. Tantalo, "Comparison of SNR image quality metrics for remote sensing systems," Opt. Eng. 40, 574-585 (2001).
[CrossRef]

Tell, S. A.

C. P. Cain, D. Courant, D. A. Freund, B. A. Grossman, P. A. Kennedy, D. J. Lund, M. A. Mainster, A. A. Manenkov, W. J. Marshall, R. McCally, B. A. Rockwell, D. H. Sliney, P. A. Smith, B. E. Stuck, S. A. Tell, M. L. Wolbarsht, and G. I. Zheltov, "Revision of the guidelines on limits of exposure to laser radiation of wavelengths between 400 nm and 1.4 μm," Health Phys. 79, 431-440 (2000).
[CrossRef]

van der Gracht, J.

R. Narayanswamy, P. E. X. Silveira, H. Setty, V. P. Pauca, and J. van der Gracht, "Extended depth-of-field iris recognition system for a workstation environment," in Biometric Technology for Human Identification II, A. K. Jain and N. K. Ratha, eds., Proc. SPIE 5779, 41-50 (2005).

Wach, H.

Wach, H. B.

Wildes, R. P.

R. P. Wildes, "Automated iris recognition: an emerging biometric technology," in Proc. IEEE 85, 1348-1363 (1997).
[CrossRef]

Wolbarsht, M. L.

C. P. Cain, D. Courant, D. A. Freund, B. A. Grossman, P. A. Kennedy, D. J. Lund, M. A. Mainster, A. A. Manenkov, W. J. Marshall, R. McCally, B. A. Rockwell, D. H. Sliney, P. A. Smith, B. E. Stuck, S. A. Tell, M. L. Wolbarsht, and G. I. Zheltov, "Revision of the guidelines on limits of exposure to laser radiation of wavelengths between 400 nm and 1.4 μm," Health Phys. 79, 431-440 (2000).
[CrossRef]

Zheltov, G. I.

C. P. Cain, D. Courant, D. A. Freund, B. A. Grossman, P. A. Kennedy, D. J. Lund, M. A. Mainster, A. A. Manenkov, W. J. Marshall, R. McCally, B. A. Rockwell, D. H. Sliney, P. A. Smith, B. E. Stuck, S. A. Tell, M. L. Wolbarsht, and G. I. Zheltov, "Revision of the guidelines on limits of exposure to laser radiation of wavelengths between 400 nm and 1.4 μm," Health Phys. 79, 431-440 (2000).
[CrossRef]

Appl. Opt.

Health Phys.

C. P. Cain, D. Courant, D. A. Freund, B. A. Grossman, P. A. Kennedy, D. J. Lund, M. A. Mainster, A. A. Manenkov, W. J. Marshall, R. McCally, B. A. Rockwell, D. H. Sliney, P. A. Smith, B. E. Stuck, S. A. Tell, M. L. Wolbarsht, and G. I. Zheltov, "Revision of the guidelines on limits of exposure to laser radiation of wavelengths between 400 nm and 1.4 μm," Health Phys. 79, 431-440 (2000).
[CrossRef]

IEEE Trans. Pattern Anal. Mach. Intell.

J. G. Daugman, "High confidence visual recognition of persons by a test of statistical independence," IEEE Trans. Pattern Anal. Mach. Intell. 15, 1148-1161 (1993).
[CrossRef]

Opt. Acta

J. W. Goodman, "Fan-in and fan-out with optical interconnections," Opt. Acta 32, 1489-1496 (1985).
[CrossRef]

Opt. Eng.

R. Fiete and T. Tantalo, "Comparison of SNR image quality metrics for remote sensing systems," Opt. Eng. 40, 574-585 (2001).
[CrossRef]

Opt. Lett.

Pattern Recogn.

J. G. Daugman, "The importance of being random: statistical principles of iris recognition," Pattern Recogn. 36, 279-291 (2003).
[CrossRef]

Proc. IEEE

R. P. Wildes, "Automated iris recognition: an emerging biometric technology," in Proc. IEEE 85, 1348-1363 (1997).
[CrossRef]

Other

W. J. Smith, Modern Optical Engineering, 3rd. ed. (McGraw-Hill, 2000), Chap. 8, pp. 219-230.

R. W. Boyd, Radiometry and the Detection of Optical Radiation (Wiley, 1983).

See, for example, E. R. Dougherty, Random Processes for Image and Signal Processing (IEEE, 1998), Chap. 1, pp. 26-28.

See, for example, J. W. Goodman, Introduction to Fourier Optics, 1st ed. (McGraw-Hill, 1968), Chap. 6, pp. 122-125.

J. Hall, "F-number, numerical aperture, and depth of focus," in Encyclopedia of Optical Engineering (Marcel Dekker, 2003), pp. 556-559.

W. J. Smith, Modern Optical Engineering, 3rd. ed. (McGraw-Hill, 2000), Chap. 6, pp. 154-157.

R. Narayanswamy, P. E. X. Silveira, H. Setty, V. P. Pauca, and J. van der Gracht, "Extended depth-of-field iris recognition system for a workstation environment," in Biometric Technology for Human Identification II, A. K. Jain and N. K. Ratha, eds., Proc. SPIE 5779, 41-50 (2005).

R. Narayanswamy, A. E. Baron, V. Chumachenko, and A. Greengard, "Applications of wavefront coded imaging," in Computational Imaging II, C.A.Bouman and E.L.Miller, eds., Proc. SPIE 5299, 163-174 (2004).

K. Kubala, E. Dowski, J. Kobus, and B. Brown, "Aberration and error invariant space telescope systems," in Novel Optical Systems Design and Optimization VII, J.M.Sasian, R.J.Koshel, P.K.Manhart, and R.C.Juergens, eds., Proc. SPIE 5524, 54-65 (2004).

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Figures (10)

Fig. 1
Fig. 1

Radiometric parameters of an arbitrary imaging system at best focus. Light reflected by the object (left) is detected by an electronic sensor array (right).

Fig. 2
Fig. 2

Comparison between a traditional (left) and computational (right) incoherent imaging systems. (a), (b) Pupil functions given by amplitude and phase (x and y axes are in units of normalized pupil coordinates). (c), (d) Two-dimensional MTFs (x and y axes in units of normalized spatial frequency). (e), (f) One-dimensional slices of the two-dimensional MTFs (solid curve, along x and y axes; dashed–dotted curve, along diagonal axes). The circular symmetry of the traditional system implies that (e) represents the one-dimensional slice of (c) over any direction.

Fig. 3
Fig. 3

Comparison between a traditional (left) and a computational (right) incoherent imaging system with +2 waves of defocus. (a), (b) Phase of the pupil functions (x and y axes are in units of normalized pupil coordinates). (c), (d) Two-dimensional MTFs (x and y axes in units of normalized spatial frequency). (e), (f) One-dimensional slices of the two-dimensional MTFs (solid curve, along x and y axes; dashed–dotted curve, along diagonal axes). The circular symmetry of the traditional system implies that (e) represents the one-dimensional slice of (c) over any direction.

Fig. 4
Fig. 4

Comparison between the MTFs of a traditional optical system at (a) full aperture and (b) half-aperture. The MTFs are plotted at four defocus positions (shown in units of waves of defocus). The spatial frequencies are normalized with respect to the optical cutoff frequency of the system when operated at full aperture.

Fig. 5
Fig. 5

Comparison between the spectral SNRs of a traditional optical system at (a) full aperture and (b) half-aperture. The spectral SNRs are plotted at four defocus positions (shown in units of waves). The spatial frequencies are normalized with respect to the optical cutoff frequency of the system when operated at full aperture.

Fig. 6
Fig. 6

MTF variation as a function of defocus. (a) Traditional imaging system with four levels of defocus (shown in waves). (b) Corresponding Wavefront Coded system with the same four levels of defocus, showing a higher tolerance to defocus.

Fig. 7
Fig. 7

Spectral SNR used for determining the DOF. Dashed curve, best focus; solid curve, far object range; dotted curve, near object range (overlapped with the line for the far object range). The vertical dashed line represents the highest spatial frequency of interest. (a) traditional imaging system exhibiting a DOF = 3.3 cm. (b) Wavefront Coded system exhibiting a DOF = 30 cm.

Fig. 8
Fig. 8

HD as a function of SNR for a hypothetical biometric recognition system. Note the existence of three regions of operation: (I) low-SNR region, where the SNR is so low that identification is not possible; (II) mid-SNR region, where a drop in SNR is accompanied by a sharp drop in HD; (III) high-SNR region, where a drop in SNR produces a negligible drop in HD. For a system SNR of 6 dB and a minimum required HD of 0.2, the ESNR in this case is 3 dB.

Fig. 9
Fig. 9

Plots of the polar SNR (in log scale) as functions of the object distance (do ) and direction in the object plane (θ). The SNR is calculated at the highest spatial frequency of interest in the image plane, and the variation of the spatial frequency with object distance is taken into account. (a) Diffraction-limited system and (b) computational imaging system. The strong responses at θ = −π, −π∕2, 0, π∕2, and π rad correspond to the horizontal and vertical image directions.

Fig. 10
Fig. 10

Plots of the mean SNR as a function of object distance, resulting from the average of the polar-SNR plots over all directions. Solid curve, traditional system; dashed–dotted curve, computational imaging system; gray line, 3 dB SNR level.

Equations (16)

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L = E 0 R π ,
E i = L T π ( NA ) 2 ,
E d = E 0 T R sin ( tan 1 D 2 d i ) 2 E 0 T R D 2 4 d i     2 ,
P p = P total n p 2 = E d f f       2 p 2 .
β 2 = 1 A A | r ( x , y ) R | 2 d x d y ,
R = 1 A A | r ( x , y ) | 2  d x d y ,
( u , v ) = 1 A [ r ( x , y ) r * ( x , y ) ] exp [ j 2 π ( x u + y v ) ] d x d y ,
m s = P p e t λ η h c ,
m S N = m s .
m N 2 = m R N 2 + m S N 2 .
S N R m = ( β m s m N ) 2 ( m s <   FWC ) ,
S N R max = β 2 FWC 2 m RN             2 + FWC β 2 FWC .
W 20 λ = D 2 8 λ ( 1 d o + 1 d i 1 f ) .
( u , v ) = ( u o d i d o , v o d i d o ) ,
S N R ( u , v ) = S NR m ( u , v ) MTF ( u , v ) 2 | S ( u , v ) | 2 ,
DOF = 2 p d i D .

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