## Pupil coding masks for imaging polychromatic scenes with high resolution and extended depth of field |

Optics Express, Vol. 18, Issue 15, pp. 15569-15584 (2010)

http://dx.doi.org/10.1364/OE.18.015569

Acrobat PDF (2587 KB)

### Abstract

An algorithm for the design of imaging systems with circular symmetry that exhibit high resolution as well as extended depth of field for polychromatic incoherent illumination is presented. The approach provides a significant improvement over a publication [1] where the design was carried for a single wavelength. The approach is based on searching for a binary phase pupil mask that provides imaging with the highest cut-off spatial frequency, while assuring a desired contrast value over a given depth of field. Simulations followed by experimental results are provided.

© 2010 OSA

## 1. Introduction

1. E. Ben-Eliezer, N. Konforti, B. Milgrom, and E. Marom, “An optimal binary amplitude-phase mask for hybrid imaging systems that exhibit high resolution and extended depth of field,” Opt. Express **16**(25), 20540–20561 (2008). [PubMed]

2. J. Ojeda-Castaneda, R. Ramos, and A. Noyola-Isgleas, “High focal depth by apodization and digital restoration,” Appl. Opt. **27**(12), 2583–2586 (1988). [PubMed]

3. J. Ojeda-Castaneda, E. Tepichin, and A. Diaz, “Arbitrary high focal depth with a quasioptimum real and positive transmittance apodizer,” Appl. Opt. **28**(13), 2666–2669 (1989). [PubMed]

4. J. Ojeda-Castaneda and L. R. Berriel-Valdos, “Zone plate for arbitrary high focal depth,” Appl. Opt. **29**(7), 994–997 (1990). [PubMed]

13. D. S. Barwick, “Increasing the information acquisition volume in iris recognition systems,” Appl. Opt. **47**(26), 4684–4691 (2008). [PubMed]

16. X. Gao, Z. Fei, W. Xu, and F. Gan, “Tunable three-dimensional intensity distribution by a pure phase-shifting apodizer,” Appl. Opt. **44**(23), 4870–4873 (2005). [PubMed]

1. E. Ben-Eliezer, N. Konforti, B. Milgrom, and E. Marom, “An optimal binary amplitude-phase mask for hybrid imaging systems that exhibit high resolution and extended depth of field,” Opt. Express **16**(25), 20540–20561 (2008). [PubMed]

16. X. Gao, Z. Fei, W. Xu, and F. Gan, “Tunable three-dimensional intensity distribution by a pure phase-shifting apodizer,” Appl. Opt. **44**(23), 4870–4873 (2005). [PubMed]

22. E. Ben-Eliezer, E. Marom, N. Konforti, and Z. Zalevsky, “Radial mask for imaging systems that exhibit high resolution and extended depths of field,” Appl. Opt. **45**(9), 2001–2013 (2006). [PubMed]

## 2. Theory

### 2.1. Misfocus condition

22. E. Ben-Eliezer, E. Marom, N. Konforti, and Z. Zalevsky, “Radial mask for imaging systems that exhibit high resolution and extended depths of field,” Appl. Opt. **45**(9), 2001–2013 (2006). [PubMed]

*M*is the magnification. The results mean that the output image is a convolution between the geometric output image, and a point spread function which is the Fourier transform of the pupil.

*M*, written in Eq. (1) equals to

*f*is the focal length and

*ψ*, which is the maximum phase error of the spherical wave front at the edge of a circular aperture that has a diameter D:

*κ*is a factor that normalizes the integral. For monochromatically illuminated images the frequency response of the imaging system is the Optical Transfer Function (OTF):The absolute value of the OTF is the well known Modulation Transfer Function (MTF).

*ψ*≤ 1, the variations in the OTF are acceptable for perfect quality imaging [24]. For clear apertures, as

*ψ*increases, the OTF degrades significantly and as a result of that, imaging is distorted and some features even lost.

### 2.2 Chromatic aspects of the Depth of Field

*ψ*varies with the wavelength, so that for 2 different wavelengths:

*f*of a simple lens varies with the wavelength, most imaging systems use lens assemblies that are chromatically corrected. The correction is mostly done by using complex lenses and chosen lens material. We assumed that

*f*in the imaging system under consideration is independent of wavelength variations in the visible spectrum.

*π*is achieved with the layer thickness of:

*h*is designed with the aid of Eq. (10b) for a wavelength of

*ψ*, φ, COF) that affect the MTF of an imaging system vary inversely with the wavelength.

### 2.3 Design approach

*ψ*values, thus assuring that the performance will be satisfactory over the extended depth of field under consideration. In order to achieve the same response for all orientations, we will consider binary masks consisting of circular rings. To begin with, the mask under consideration allows the presence of an opaque center of radius

*ψ*. One notes that a phase mask that contains phase features of value “0 and

*π*only” is identical to a mask that has phase features of “0 and -

*π*only”. Therefore, in the mask design process, it is sufficient to investigate the imaging system for the green color only for either positive or negative values pf

*ψ*thus reducing the computation time in half. As such, we limited ourselves to the design of a system with large DOF, by considering phase masks that have a phase of either 0 or

*π*. Nevertheless, when considering other colors (R, B) the phase is no longer

*π*and thus the entire range of DOF (negative and positive values of

*ψ*) had to be investigated. The desired Polychromatic Phase Mask (PPM) will be the one providing the highest COF with an acceptable contrast level:

18. E. Ben-Eliezer, N. Konforti, and E. Marom, “Super resolution imaging with noise reduction and aberration elimination via random structured illumination and processing,” Opt. Express **15**(7), 3849–3863 (2007). [PubMed]

1. E. Ben-Eliezer, N. Konforti, B. Milgrom, and E. Marom, “An optimal binary amplitude-phase mask for hybrid imaging systems that exhibit high resolution and extended depth of field,” Opt. Express **16**(25), 20540–20561 (2008). [PubMed]

*ψ*| ≤ 15 and a minimum contrast of 5%, and resulted in π phase rings, with normalized radii of 0.62, 0.76, 082, and 0.94, and an opaque center with a radius of 0.2.

*ψ*Eq. (5)] for various wavelengths in the entire visible range have been taken into account when searching for the best solution.

## 3. Search algorithm

**a)**We set 5 dimensional array of parameter values

**b)**For each set of

*ψ*has been calculated. Only positive

*ψ*values have been examined due to the arguments presented in section 2.3. For each chosen contrast value, the value of

*ψ*was allowed to vary up to

**c)**For each MTF curve the highest frequency up to which the contrast was above 5% was recorded as

*c,*

**d)**One now associates with each choice of values

*π*phase for the wavelength of 550nm. The first ring is essentially a circle with an outer normalized radius of 0.04, while the second ring has inner and outer normalized radii of 0.74 and 0.90 respectively (see Fig. 2 ). We considered a central stop when running the search algorithm, but the results obtained indicated that best system behavior is obtained when no central stop is present. Thus the mask consisted of only phase rings.

**16**(25), 20540–20561 (2008). [PubMed]

*ψ*= 0, −2, −5 and −8 respectively. The contrast value of 5% is marked by a black horizontal line. The normalized value of 2 is the COF evaluated for the green wavelength. The corresponding COF for the Blue wavelength is 2.4 and for Red wavelength 1.6. Using the same scale for all wavelength representations, the COF of the R, G, B respectively occur at 1.6, 2.0 and 2.4, as evidenced in Figs. 3-5 .

*ψ*. The chromatic dependence can be clearly seen from the behavior of the MTF curves provided in Fig. 5. Moreover one sees there that the MTF for the clear aperture exhibits contrast reversal at a normalized spatial frequency of 1.1, while the imaging system that uses the mask has no contrast reversal until 1.8. Of course we stop short of reaching such range, since the contrast falls below the design limit, before that.

*ψ*along the X-axis and the highest spatial frequency for which the MTF exhibits a contrast above the 5% along Y-axis [Fig. 6 ]. Out-of-focus imagery can be analyzed in a general manner by using the parameter

*ψ*to identify the amount of deviation away from focus. Trying to express this in terms of the magnification factor

*M*eliminates the possibility of treating it in a universal manner, and thus it was not attempted here. One can express the dependence on the magnification

*M*only for specific systems, whereby the

*D*and the focal distance

*f*are provided. Therefore throughout this paper out-of-focus was analyzed in term of

*ψ*only.

*ψ*, say from 0 up to 5, resulting with higher contrast values at low frequencies. One notes that the contrast thus obtained, approaches the one provided by a clear aperture for a low out-of-focus condition

*ψ*, there is small difference between the results obtained with the two masks. The advantages of the PPM mask are thus evident when the DOF range is not excessively large. The contrast level for the green color with the PPM is higher than that with the MPM for low values of

*ψ*, although the MPM was designed for Green wavelength only. This is due to the fact that the MPM mask was designed to provide improved results for a region extending up to large values of

*ψ*. The images provided in Fig. 8 were recorded in a position corresponding to small values of

*ψ*. Thus, it is not surprising that the PPM provides better results than those obtained with the MPM even for the green color, when the object is near the in-focus position, i.e., small values of

*ψ*= 5. Not only resolution is limited, but also color distortions occur as it can be readily seen. Simulation results obtained with a system equipped with an MPM [Fig. 7 (c)], or PPM [Fig. 7 (d)] are shown:

## 4. Experiments with extended depth of field imagery

### 4.1 Experimental Set-Up

^{®}DV5200], equipped with a CMOS detector with 1200x1600 pixels (2 mega pixels) was utilized. The camera lens consisted of five glass elements with an effective focal length of 8.5mm. A Field of view (FOV) of 50 degrees along the sensor diagonal, and pixel dimension of 4

^{®}M1614, with a focal length of 16mm. The camera has a field of view (FOV) of 45 degrees along the sensor diagonal and pixel dimension of 4

### 4.2 Experimental results

*ψ*~-4 and the building in the background is at approximately

*ψ*~-7.

*ψ*= 4) for the two masks under investigation.

*ψ*~5. The exposure time was 5msec. The average contrast at the R, G, B wavelengths obtained with a system equipped with MPM mask was 0.07, 0.1, and 0.1 respectively. When replacing the MPM mask with the PPM one, the contrast values were 0.2, 0.3, and 0.3 respectively. The noise level is below 5%, thus the PPM performance is sufficient since it provides a minimum contrast level of 5%. The last experiment was repeated for other out-of-focus locations up to

*ψ*~7 and other levels of illuminations. In all cases, the PPM provided images with higher contrast.

## 5. Conclusions

## References and links

1. | E. Ben-Eliezer, N. Konforti, B. Milgrom, and E. Marom, “An optimal binary amplitude-phase mask for hybrid imaging systems that exhibit high resolution and extended depth of field,” Opt. Express |

2. | J. Ojeda-Castaneda, R. Ramos, and A. Noyola-Isgleas, “High focal depth by apodization and digital restoration,” Appl. Opt. |

3. | J. Ojeda-Castaneda, E. Tepichin, and A. Diaz, “Arbitrary high focal depth with a quasioptimum real and positive transmittance apodizer,” Appl. Opt. |

4. | J. Ojeda-Castaneda and L. R. Berriel-Valdos, “Zone plate for arbitrary high focal depth,” Appl. Opt. |

5. | E. R. Dowski Jr and W. T. Cathey, “Extended depth of field through wave-front coding,” Appl. Opt. |

6. | J. van der Gracht, E. R. Dowski Jr, M. G. Taylor, and D. M. Deaver, “Broadband behavior of an optical-digital focus-invariant system,” Opt. Lett. |

7. | 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. |

8. | W. Chi and N. George, “Electronic imaging using a logarithmic asphere,” Opt. Lett. |

9. | W. Chi and N. George, “Computational imaging with the logarithmic asphere: theory,” J. Opt. Soc. Am. A |

10. | S. Prasad, V. P. Pauca, and J. Robert, Plemmons, Todd C. Torgersen and Joseph van der Gracht, “Pupil-phase optimization for extended focus, aberration corrected imaging systems”, Publication: Proc. SPIE |

11. | S. Prasad, T. C. Torgersen, V. P. Pauca, R. J. Plemmons, and J. van der Gracht, “High resolution imaging using integrated optical systems,” Int. J. Imaging Syst. Technol. |

12. | J. van der Gracht, V. P. Pauca, H. Setty, R. Narayanswamy, R. Plemmons, S. Prasad, and T. Torgersen, “Iris recognition with enhanced depth-of-field image acquistion”, Publication: Proc. SPIE |

13. | D. S. Barwick, “Increasing the information acquisition volume in iris recognition systems,” Appl. Opt. |

14. | H. Wang and F. Gan, “High focal depth with a pure-phase apodizer,” Appl. Opt. |

15. | H. Wang and F. Gan, “Phase-shifting apodizers for increasing focal depth,” Appl. Opt. |

16. | X. Gao, Z. Fei, W. Xu, and F. Gan, “Tunable three-dimensional intensity distribution by a pure phase-shifting apodizer,” Appl. Opt. |

17. | E. Ben-Eliezer and E. Marom, “Aberration-free superresolution imaging via binary speckle pattern encoding and processing,” J. Opt. Soc. Am. A |

18. | E. Ben-Eliezer, N. Konforti, and E. Marom, “Super resolution imaging with noise reduction and aberration elimination via random structured illumination and processing,” Opt. Express |

19. | E. Ben-Eliezer, Z. Zalevsky, E. Marom, and N. Konforti, “All-optical extended depth of field imaging system,” J. Opt. A, Pure Appl. Opt. |

20. | E. Ben-Eliezer, Z. Zalevsky, E. Marom, and N. Konforti, “All-optical extended depth of field imaging system,” Proc. SPIE |

21. | E. Ben-Eliezer, E. Marom, N. Konforti, and Z. Zalevsky, “Experimental realization of an imaging system with an extended depth of field,” Appl. Opt. |

22. | E. Ben-Eliezer, E. Marom, N. Konforti, and Z. Zalevsky, “Radial mask for imaging systems that exhibit high resolution and extended depths of field,” Appl. Opt. |

23. | J. W. Goodman, |

24. | H. H. Hopkins, “The Frequency response of a defocus optical system,” Proc. R. Soc. Lond. A Math. Phys. Sci. |

**ToC Category:**

Imaging Systems

**History**

Original Manuscript: March 16, 2010

Revised Manuscript: May 3, 2010

Manuscript Accepted: May 19, 2010

Published: July 8, 2010

**Citation**

Benjamin Milgrom, Naim Konforti, Michael A. Golub, and Emanuel Marom, "Pupil coding masks for imaging polychromatic scenes with high resolution and extended depth of field," Opt. Express **18**, 15569-15584 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-15-15569

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### References

- E. Ben-Eliezer, N. Konforti, B. Milgrom, and E. Marom, “An optimal binary amplitude-phase mask for hybrid imaging systems that exhibit high resolution and extended depth of field,” Opt. Express 16(25), 20540–20561 (2008). [PubMed]
- J. Ojeda-Castaneda, R. Ramos, and A. Noyola-Isgleas, “High focal depth by apodization and digital restoration,” Appl. Opt. 27(12), 2583–2586 (1988). [PubMed]
- J. Ojeda-Castaneda, E. Tepichin, and A. Diaz, “Arbitrary high focal depth with a quasioptimum real and positive transmittance apodizer,” Appl. Opt. 28(13), 2666–2669 (1989). [PubMed]
- J. Ojeda-Castaneda and L. R. Berriel-Valdos, “Zone plate for arbitrary high focal depth,” Appl. Opt. 29(7), 994–997 (1990). [PubMed]
- E. R. Dowski and W. T. Cathey, “Extended depth of field through wave-front coding,” Appl. Opt. 34(11), 1859 (1995). [PubMed]
- J. van der Gracht, E. R. Dowski, M. G. Taylor, and D. M. Deaver, “Broadband behavior of an optical-digital focus-invariant system,” Opt. Lett. 21(13), 919–921 (1996). [PubMed]
- 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(13), 2709–2721 (2004). [PubMed]
- W. Chi and N. George, “Electronic imaging using a logarithmic asphere,” Opt. Lett. 26(12), 875–877 (2001).
- W. Chi and N. George, “Computational imaging with the logarithmic asphere: theory,” J. Opt. Soc. Am. A 20(12), 2260–2273 (2003).
- S. Prasad, V. P. Pauca, and J. Robert, Plemmons, Todd C. Torgersen and Joseph van der Gracht, “Pupil-phase optimization for extended focus, aberration corrected imaging systems”, Proc. SPIE 5559, 335–345 (2004).
- S. Prasad, T. C. Torgersen, V. P. Pauca, R. J. Plemmons, and J. van der Gracht, “High resolution imaging using integrated optical systems,” Int. J. Imaging Syst. Technol. 14(2), 67–74 (2004).
- J. van der Gracht, V. P. Pauca, H. Setty, R. Narayanswamy, R. Plemmons, S. Prasad, and T. Torgersen, "Iris recognition with enhanced depth-of-field image acquistion," Proc. SPIE 5438, 120–129 (2004).
- D. S. Barwick, “Increasing the information acquisition volume in iris recognition systems,” Appl. Opt. 47(26), 4684–4691 (2008). [PubMed]
- H. Wang and F. Gan, “High focal depth with a pure-phase apodizer,” Appl. Opt. 40(31), 5658–5662 (2001).
- H. Wang and F. Gan, “Phase-shifting apodizers for increasing focal depth,” Appl. Opt. 41(25), 5263–5266 (2002). [PubMed]
- X. Gao, Z. Fei, W. Xu, and F. Gan, “Tunable three-dimensional intensity distribution by a pure phase-shifting apodizer,” Appl. Opt. 44(23), 4870–4873 (2005). [PubMed]
- E. Ben-Eliezer and E. Marom, “Aberration-free superresolution imaging via binary speckle pattern encoding and processing,” J. Opt. Soc. Am. A 24(4), 1003–1010 (2007).
- E. Ben-Eliezer, N. Konforti, and E. Marom, “Super resolution imaging with noise reduction and aberration elimination via random structured illumination and processing,” Opt. Express 15(7), 3849–3863 (2007). [PubMed]
- E. Ben-Eliezer, Z. Zalevsky, E. Marom, and N. Konforti, “All-optical extended depth of field imaging system,” J. Opt. A, Pure Appl. Opt. 5(5), S 164– S 169 (2003).
- E. Ben-Eliezer, Z. Zalevsky, E. Marom, and N. Konforti, “All-optical extended depth of field imaging system,” Proc. SPIE 4829, 221 (2002).
- E. Ben-Eliezer, E. Marom, N. Konforti, and Z. Zalevsky, “Experimental realization of an imaging system with an extended depth of field,” Appl. Opt. 44(14), 2792–2798 (2005). [PubMed]
- E. Ben-Eliezer, E. Marom, N. Konforti, and Z. Zalevsky, “Radial mask for imaging systems that exhibit high resolution and extended depths of field,” Appl. Opt. 45(9), 2001–2013 (2006). [PubMed]
- J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996) 126–151.
- H. H. Hopkins, “The Frequency response of a defocus optical system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 231(1184), 91–103 (1955).

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