Abstract

In two papers [Proc. SPIE 4471, 272–280 (2001) and Appl. Opt. 43, 2709–2721 (2004)], a logarithmic phase mask was proposed and proved to be effective in extending the depth of field; however, according to our research, this mask is not that perfect because the corresponding defocused modulation transfer function has large oscillations in the low-frequency region, even when the mask is optimized. So, in a previously published paper [Opt. Lett. 33, 1171–1173 (2008)], we proposed an improved logarithmic phase mask by making a small modification. The new mask can not only eliminate the drawbacks to a certain extent but can also be even less sensitive to focus errors according to Fisher information criteria. However, the performance comparison was carried out with the modified mask not being optimized, which was not reasonable. In this manuscript, we optimize the modified logarithmic phase mask first before analyzing its performance and more convincing results have been obtained based on the analysis of several frequently used metrics.

© 2010 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. S. S. Sherif, E. R. Dowski, and W. T. Cathey, “Logarithmic phase filter to extend the depth of field of incoherent hybrid imaging systems,” Proc. SPIE 4471, 272-280 (2001).
    [CrossRef]
  2. S. S. Sherif, W. T. Cathey, and E. R. Dowski, “Phase plate to extend the depth of field of incoherent hybrid imaging systems,” Appl. Opt. 43, 2709-2721 (2004).
    [CrossRef]
  3. E. R. Dowski, Jr., and W. T. Cathey, “Extended depth of field through wave-front coding,” Appl. Opt. 34, 1859-1866 (1995).
    [CrossRef]
  4. Q. Yang, L. Liu, and J. Sun, “Optimized phase pupil masks for extended depth of field,” Opt. Commun. 272, 56-66 (2007).
    [CrossRef]
  5. S. Prasad, T. C. Torgersen, V. P. Pauca, R. J. Plemmons, and J. van der Gracht, “Engineering the pupil phase to improve image quality,” Proc. SPIE 5108, 1-12 (2003).
    [CrossRef]
  6. A. Sauceda and J. Ojeda-Castañeda, “High focal depth with fractional-power wave fronts,” Opt. Lett. 29, 560-562 (2004).
    [CrossRef]
  7. W. Chi and N. George, “Computational Imaging with the logarithmic asphere: theory,” J. Opt. Soc. Am. A 20, 2260-2273(2003).
    [CrossRef]
  8. N. Caron and Y. Sheng, “Polynomial phase mask for extending the depth of field optimized by simulated annealing,” Proc. SPIE 6832, 68321G (2007).
    [CrossRef]
  9. H. Zhao, Q. Li, and H. Feng, “Improved logarithmic phase mask to extend the depth of field of an incoherent imaging system,” Opt. Lett. 33, 1171-1173 (2008).
    [CrossRef]
  10. L. L. Scharf, Statistical Signal Processing (Addison-Wesley, 1991), Chap. 6, pp. 209-276.
  11. H. H. Hopkins, “The frequency response of a defocused optical system,” Proc. R. Soc. London Ser. A 231, 91-103 (1955).
    [CrossRef]
  12. R. C. Gonzales and R. E. Woods, Digital Image Processing (Addison-Wesley, 1992), pp. 147-213.

2008 (1)

2007 (2)

Q. Yang, L. Liu, and J. Sun, “Optimized phase pupil masks for extended depth of field,” Opt. Commun. 272, 56-66 (2007).
[CrossRef]

N. Caron and Y. Sheng, “Polynomial phase mask for extending the depth of field optimized by simulated annealing,” Proc. SPIE 6832, 68321G (2007).
[CrossRef]

2004 (2)

2003 (2)

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

W. Chi and N. George, “Computational Imaging with the logarithmic asphere: theory,” J. Opt. Soc. Am. A 20, 2260-2273(2003).
[CrossRef]

2001 (1)

S. S. Sherif, E. R. Dowski, and W. T. Cathey, “Logarithmic phase filter to extend the depth of field of incoherent hybrid imaging systems,” Proc. SPIE 4471, 272-280 (2001).
[CrossRef]

1995 (1)

1992 (1)

R. C. Gonzales and R. E. Woods, Digital Image Processing (Addison-Wesley, 1992), pp. 147-213.

1991 (1)

L. L. Scharf, Statistical Signal Processing (Addison-Wesley, 1991), Chap. 6, pp. 209-276.

1955 (1)

H. H. Hopkins, “The frequency response of a defocused optical system,” Proc. R. Soc. London Ser. A 231, 91-103 (1955).
[CrossRef]

Caron, N.

N. Caron and Y. Sheng, “Polynomial phase mask for extending the depth of field optimized by simulated annealing,” Proc. SPIE 6832, 68321G (2007).
[CrossRef]

Cathey, W. T.

Chi, W.

Dowski, E. R.

Feng, H.

George, N.

Gonzales, R. C.

R. C. Gonzales and R. E. Woods, Digital Image Processing (Addison-Wesley, 1992), pp. 147-213.

Hopkins, H. H.

H. H. Hopkins, “The frequency response of a defocused optical system,” Proc. R. Soc. London Ser. A 231, 91-103 (1955).
[CrossRef]

Li, Q.

Liu, L.

Q. Yang, L. Liu, and J. Sun, “Optimized phase pupil masks for extended depth of field,” Opt. Commun. 272, 56-66 (2007).
[CrossRef]

Ojeda-Castañeda, J.

Pauca, V. P.

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

Plemmons, R. J.

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

Prasad, S.

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

Sauceda, A.

Scharf, L. L.

L. L. Scharf, Statistical Signal Processing (Addison-Wesley, 1991), Chap. 6, pp. 209-276.

Sheng, Y.

N. Caron and Y. Sheng, “Polynomial phase mask for extending the depth of field optimized by simulated annealing,” Proc. SPIE 6832, 68321G (2007).
[CrossRef]

Sherif, S. S.

S. S. Sherif, W. T. Cathey, and E. R. Dowski, “Phase plate to extend the depth of field of incoherent hybrid imaging systems,” Appl. Opt. 43, 2709-2721 (2004).
[CrossRef]

S. S. Sherif, E. R. Dowski, and W. T. Cathey, “Logarithmic phase filter to extend the depth of field of incoherent hybrid imaging systems,” Proc. SPIE 4471, 272-280 (2001).
[CrossRef]

Sun, J.

Q. Yang, L. Liu, and J. Sun, “Optimized phase pupil masks for extended depth of field,” Opt. Commun. 272, 56-66 (2007).
[CrossRef]

Torgersen, T. C.

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

van der Gracht, J.

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

Woods, R. E.

R. C. Gonzales and R. E. Woods, Digital Image Processing (Addison-Wesley, 1992), pp. 147-213.

Yang, Q.

Q. Yang, L. Liu, and J. Sun, “Optimized phase pupil masks for extended depth of field,” Opt. Commun. 272, 56-66 (2007).
[CrossRef]

Zhao, H.

Appl. Opt. (2)

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

Opt. Commun. (1)

Q. Yang, L. Liu, and J. Sun, “Optimized phase pupil masks for extended depth of field,” Opt. Commun. 272, 56-66 (2007).
[CrossRef]

Opt. Lett. (2)

Proc. R. Soc. London Ser. A (1)

H. H. Hopkins, “The frequency response of a defocused optical system,” Proc. R. Soc. London Ser. A 231, 91-103 (1955).
[CrossRef]

Proc. SPIE (3)

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

N. Caron and Y. Sheng, “Polynomial phase mask for extending the depth of field optimized by simulated annealing,” Proc. SPIE 6832, 68321G (2007).
[CrossRef]

S. S. Sherif, E. R. Dowski, and W. T. Cathey, “Logarithmic phase filter to extend the depth of field of incoherent hybrid imaging systems,” Proc. SPIE 4471, 272-280 (2001).
[CrossRef]

Other (2)

L. L. Scharf, Statistical Signal Processing (Addison-Wesley, 1991), Chap. 6, pp. 209-276.

R. C. Gonzales and R. E. Woods, Digital Image Processing (Addison-Wesley, 1992), pp. 147-213.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (13)

Fig. 1
Fig. 1

Defocused MTF with different focus errors denoted by W 20 corresponding to the original logarithmic phase mask with [ α , β ] equaling [113.085, 0.57].

Fig. 2
Fig. 2

Comparison between one-dimensional phase profiles of the original, where [ α , β ] is [113.085, 0.57], and the improved logarithmic phase masks, where [ α , β ] is [113.086, 1.115 ].

Fig. 3
Fig. 3

Defocused MTF with different defocus corresponding to the modified phase mask, which is not optimized; the frequency u is the reduced spatial frequency [11].

Fig. 4
Fig. 4

Comparison between FI of the original and the improved logarithmic phase masks when only the original mask was optimized.

Fig. 5
Fig. 5

Comparison between the stability of defocused MTF curves when (a)–(c) the original and (d)–(f) the improved logarithmic phase masks are optimized with same Th criterion and same defocus range (from 0 to 10 π ); the symbols alpha and beta denote the parameters α and β, respectively.

Fig. 6
Fig. 6

Comparison between FI curves of the original and the improved logarithmic phase masks when they are both optimized on the condition that the minimum-acceptable MTF represented by Th is the same; the terms “Old” and “New” denote the original and the modified logarithmic phase masks, respectively.

Fig. 7
Fig. 7

AF plots corresponding to (a) the original and (b) the modified mask, with Th set to 0.27 for example.

Fig. 8
Fig. 8

One-dimensional phase profiles of several different phase masks: the cubic [3], exponential [4], logarithmic/modified logarithmic [1, 9], high-order [6], and polynomial [5, 8] types.

Fig. 9
Fig. 9

PSF displacement effect of (a) the original phase mask and (b) the modified phase mask when defocus pa rameter W 20 equals 5; the cross denotes the ideal PSF position.

Fig. 10
Fig. 10

PSF displacement effect of (a) the original phase mask and (b) the modified phase mask when defocus pa rameter W 20 equals 15; the cross denotes the ideal PSF position.

Fig. 11
Fig. 11

Intermediate blurred and inverse filtered images corresponding to the original logarithmic phase mask (left) and the modified one (right); different filters computed by Eq. (5), which are the function of the defocus parameter, are used for restoration and the defocus parameter equals π, 3 π , 6 π , and 10 π , from top to bottom.

Fig. 12
Fig. 12

Intermediate blurred and inverse filtered images corresponding to the original logarithmic phase mask (left) and the modified one (right); the defocus parameter equals π, 3 π , 6 π , and 10 π from top to bottom; only one filter computed using the one π defocused OTF for each mask is used for restoration.

Fig. 13
Fig. 13

Intermediate blurred and inverse filtered images corresponding to the original logarithmic phase mask (left) and the modified one (right); the defocus parameter equals π, 3 π , 6 π , and 10 π from top to bottom; only one filter computed using the least-mean-square method for each mask is used for restoration.

Tables (1)

Tables Icon

Table 1 Optimum Parameters of the Original and the Improved Logarithmic Phase Mask with Different Acceptable MTF Degradation Criterion a

Equations (5)

Equations on this page are rendered with MathJax. Learn more.

f _ original ( x , y ) = α · sgn ( x ) · x 2 ( log | x | + β ) + α · sgn ( y ) · y 2 ( log | y | + β ) α 0 , β > 0 f _ modified ( x , y ) = α · sgn ( x ) · x 2 ( log | | x | + β | ) + α · sgn ( y ) · y 2 ( log | | y | + β | ) α 0 , β < 0 ,
[ α β ] = { arg min ( ψ 0 ψ 0 J ( W 20 , α , β ) d W 20 ) subject to { ( LB ) [ α β ] ( UB ) MAMTF ( α , β ) = | H ( u , W 20 = 0 , α , β ) | d u T h ,
{ J ( W 20 , α , β ) = | W 20 H ( u , W 20 , α , β ) | 2 d u H ( u , W 20 , α , β ) = 1 2 · e j · ( 2 W 20 u x + f ( x + u / 2 ) f ( x u / 2 ) ) d x .
H ( u , W 20 ) = A ( u , u W 20 / π ) ,
I = IFFT { FFT { I } · F ( W 20 ) } where     F ( W 20 ) = H o H c ( W 20 ) ,

Metrics