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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 25 — Dec. 8, 2008
  • pp: 20540–20561
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An optimal binary amplitude-phase mask for hybrid imaging systems that exhibit high resolution and extended depth of field

Eyal Ben-Eliezer, Naim Konforti, Benjamin Milgrom, and Emanuel Marom  »View Author Affiliations


Optics Express, Vol. 16, Issue 25, pp. 20540-20561 (2008)
http://dx.doi.org/10.1364/OE.16.020540


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Abstract

The design of a circularly symmetric hybrid imaging system that exhibits high resolution as well as extended depth of field is presented. The design, which assumes spatially incoherent illumination, searches for an optimal “binary amplitude and phase” pupil mask, which for a certain desired depth of field, provides the largest spatial frequency band that assures a certain desired contrast value. The captured images are electronically processed by an off-line Wiener filter, to finally obtain high quality output images. Simulations as well as experimental results are provided.

© 2008 Optical Society of America

1. Introduction

Imaging systems are known to require accurate focus alignment [1

1. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1996).

]. It is well known that conventional imaging systems are very sensitive to defocus. When the object and image planes are not in conjugate positions, the resultant image is severely degraded. Nevertheless, there are applications that require high image quality along with extended depth of field (DOF). Extended DOF is also important in some conservative applications, e.g. barcode reading, computer or machine vision, surveillance cameras etc, where reduced contrast and resolution are not detrimental.

Digital imaging systems are sometimes undersampled due to the need to open the pupil for light gathering. In such cases, the optical PSF is smaller than the pixel dimension, and one gets an increased DOF by virtue of allowing the PSF to increase in spread due to the out of focus condition. However, such “Geometrical Optics” effects are very limited, and do not provide large DOF range.

Other conventional solutions for imaging systems with an extended DOF involve pupil stopping and apodization [2

2. A. Papoulis, Systems and Transforms with Applications in Optics (McGraw- Hill, New York, 1968).

7

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

]. The main disadvantages of such solutions are reduced resolution and low light throughput.

Other approaches make use of the multi-focal concept, where the imaging lens has several foci, thus extending the depth of field. Examples to such approach are a quasi bifocus birefringent lens [8

8. S. Sanyal and A. Ghosh, “High focal depth with quasi-bifocus birefringent lens,” Appl. Opt. 39, 2321–2325 (2000). [CrossRef]

], where the resulting DOF is quite limited, or a lens, which is divided into annular rings [9

9. E. Peli and A. Lang, “Appearance of images through a multifocal intraocular lens,” J. Opt. Soc. Am. A 18, 302–309 (2001). [CrossRef]

]. The latter approach suffers from relatively low resolution if a wide DOF range is required. Hybrid, opto-digital, approaches that overcome these deficiencies use a non-absorptive phase mask and require post processing digital restoration operations [10

10. E.R Dowski and W.T. Cathey, “Extended depth of field through wave-front coding,” Appl. Opt 34, 1859–1866 (1995). [CrossRef] [PubMed]

14

14. 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, 67–74 (2004). [CrossRef]

]. Optimizations of the processes of encoding and decoding the wave-front phase in integrated optical-digital imaging systems has been carried as well [13

13. S. Prasad, V. Paul Pauca, R. J. Plemmons, T. C. Torgersen, and J. van der Gracht, “Pupil-phase optimization for extended focus, aberration corrected imaging systems,” Proc. SPIE 5559, 335–345 (2004). [CrossRef]

14

14. 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, 67–74 (2004). [CrossRef]

]. These optimizations involve the use of information metrics, such as the Strehl ratio and Fisher information, for determining the optimal pupil-phase distribution for which the resulting image is less sensitive to certain aberrations, such as focus errors. However, if the post-processing step is omitted, these hybrid approaches provide distorted images, with low contrast values. The Optical Transfer Function (OTF), provided by a logarithmic asphere approach [15

15. W. Chi and N. George, “Electronic imaging using a logarithmic asphere,” Opt. Lett. 26, 875–877 (2001). [CrossRef]

, 16

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

] also contains regions of contrast reversals for relatively low normalized frequencies, at the DOF extreme positions.

The hybrid approach, presented by Dowski et al [10

10. E.R Dowski and W.T. Cathey, “Extended depth of field through wave-front coding,” Appl. Opt 34, 1859–1866 (1995). [CrossRef] [PubMed]

12

12. 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] [PubMed]

], is the only approach that deals directly with incoherent illumination sources. However, its main drawback is the requirement to use post-processing steps in order to achieve a high quality image. Without such post-processing step, the image is often incomprehensible. There are however hybrid systems that may have applications in which the digital step is not required, such as for the case of iris recognition [17

17. 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 acquisition,” Proc. SPIE 5438, 120–129 (2004). [CrossRef]

], where a blurred image is sufficient for further processing.

Recently, a super-resolution technique that allows defocus elimination has been presented [18

18. E. Ben-Eliezer and E. Marom, “Aberration-free superresolution imaging via binary speckle pattern encoding and processing,” J. Opt. Soc. Am. A 24, 1003–1010 (2007). [CrossRef]

19

19. E. Ben-Eliezer, N. Konfori, and E. Marom, “Superresolution imaging with noise reduction and aberration elimination via random structured illumination and processing,” Opt. Express 15, 3849–3863 (2007). [CrossRef] [PubMed]

]. There, the image is encoded and decoded by binary speckle patterns. This method, however, requires successive image capturing of an encoded object, as well as registration of the captured images in order to carry the image restoration step; therefore, it is impractical for consumer optics applications where processing speed and simplicity are of prime value. All the practical applications, mentioned above, require a passive wave-front encoding mask in the optical aperture, which may be followed by a digital restoration stage.

A mask that is optimal according to the MMSE criterion with respect to the object intensity has been suggested as well [23

23. E. Ben-Eliezer, E. Marom, N. Konforti, and Z. Zalevsky, “A Radial Mask for Systems that exhibit High Resolution and Extended Depth of Field,” Appl. Opt. 45, 2001–2013 (2006). [CrossRef] [PubMed]

]. The resulting imaging system, for which the design assumed incoherent illumination, provided relatively high resolution for the whole DOF range, but with relatively low light throughput.

The MMSE criterion applied on the intensity distribution [23

23. E. Ben-Eliezer, E. Marom, N. Konforti, and Z. Zalevsky, “A Radial Mask for Systems that exhibit High Resolution and Extended Depth of Field,” Appl. Opt. 45, 2001–2013 (2006). [CrossRef] [PubMed]

] has some disadvantages if a hybrid, optical-digital imaging system, is under consideration. The reason is that if an electronic restoration step is to be used, the exact shape of the optical transfer function (OTF) is not very important, as long as it possesses the following characteristics: first, the OTF should exhibit a relatively large useful spatial frequency band (i.e. no contrast reversals) for the whole DOF range, preferably up to the pixel-resolution; second, high enough contrast values should be achieved for the entire usable spatial frequency band, so that acceptable signal to noise ratio (SNR) is achieved; for such a case, electronic restoration can be performed to finally obtain an image displaying high contrast and resolution; third, a robust OTF shape throughout the DOF range is preferred. This last item is not a strict requirement, but rather a “nice to have” one.

When the MMSE between the actual OTF and a desired OTF has been considered as a quality criterion [23

23. E. Ben-Eliezer, E. Marom, N. Konforti, and Z. Zalevsky, “A Radial Mask for Systems that exhibit High Resolution and Extended Depth of Field,” Appl. Opt. 45, 2001–2013 (2006). [CrossRef] [PubMed]

], we found out that the obtained OTF curves tend to possess relatively high contrast values for low frequencies. However, when high contrast values at low spatial frequencies are sought, the actual cutoff frequency, i.e. the promised resolution for the whole DOF range, is reduced. Therefore, several desired OTF curves were studied, in order to find the best curve that provides the highest resolution with acceptable contrast. A better design approach is to find a criterion that searches for the highest resolution directly, contrary to the design according to the MMSE criterion, where high resolution throughout the DOF range has been achieved indirectly, by making educated guesses on the shape of the desired OTF curve. Moreover, the main disadvantage of those optimal masks is that they are absorptive, and as a result, provide low light throughput [23

23. E. Ben-Eliezer, E. Marom, N. Konforti, and Z. Zalevsky, “A Radial Mask for Systems that exhibit High Resolution and Extended Depth of Field,” Appl. Opt. 45, 2001–2013 (2006). [CrossRef] [PubMed]

]. However, high light throughput is essential for commercial and industrial applications. Since complex valued masks are complicated for fabrication, a simple sub-optimal binary phase mask, illustrated in Fig. 1, was studied as well.

Fig. 1. An illustration of a binary phase mask, with two annular phase rings in red. The rings introduce a π phase difference for a central wavelength within the illumination light bandwidth.

In this article we present simple and easy-to-manufacture mask designs with optimal performance, designed especially for incoherently illuminated scenes. These designs provide high light throughput as well as high resolution throughout the entire DOF range, along with circularly symmetric OTF behavior. To the best of our knowledge, such designs have not been presented yet.

Systems with extended DOF ranges, as well as high resolution, usually come along with low (but acceptable) contrast values. In case the OTF shape remains relatively unchanged throughout the DOF range, a universal restoration filter can be designed and used to enhance the contrast, independent of the position of the object. As a result, high quality hybrid optical-digital imaging systems with extended DOF, which provide high resolution and contrast for quasi monochromatic as well as for outdoor illumination, is obtained.

Thereafter, optimization for similar circular symmetric structures (with one and two annular phase rings) that also incorporate an opaque on-axis circle, denoted as “1Rc” and “2Rc” is carried. Simulations as well as experimental results are presented. The interest in adding a central stop is due to the fact that conventional optical systems that are equipped with a pupil having a center stop, exhibit high resolution for increased depth of field capabilities, albeit with a low contrast, for most spatial frequencies [4

4. T. C. Poon and M. Motamedi, “Optical Digital Incoherent Image Processing for Extended Depth of Field,” Appl. Opt. 26, 4612–4615 (1987). [CrossRef] [PubMed]

].

The paper is organized as follows: Section 2 presents the theory as well as design and optimization considerations. Optimization results are presented in section 3, followed by experimental results, presented in section 4. The conclusion follows in section 5.

2. Theory

It is well known that defocus aberration manifests itself by a quadratic phase at the imaging system pupil [1

1. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1996).

], namely:

G(u,v;ψ)=exp[jψ(u2+v2)],
(1)

where (u,v) are the normalized coordinates of the pupil plane. The defocus parameter, Ψ, is defined by the following expression [1

1. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1996).

]:

ψ=πR2λ(1dobj+1dimg1f)
(2)

where R is the pupil radius, λ is the wavelength, f, dobj and dimg are the lens focal length, and the distances between the object and the image to the lens respectively. Clearly, for an in-focus position, ψ=0. When defocus occurs, the phase factor, given in Eq. (1), multiplies the pupil of the imaging system, resulting in a generalized pupil, expressed in the following expression:

P(u,v)=P(u,v)exp[jψ(u2+v2)]
(3)

where P(u,v) is the pupil function. In the presence of such generalized pupil, the object is not imaged in the plane where the detector is nominally located, but in a different one. Thus, the detector acquires a degraded image.

Most practical imaging systems use incoherent illumination as a light source. Such systems are linear with intensity, contrary to the linearity with respect to the field distribution, as is the case for coherent source illumination [1

1. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1996).

]. The output intensity is provided by the convolution integral:

Iout(x,y)=h(xx,yy)2Ig(x,y)dxdy
(4)

where x and y are the lateral coordinates in the image plain, Ig(x,y) is the intensity of the image, in a system without diffraction (i.e. geometrical optics considerations), and h(x,y) is the coherent point spread function. Eq. (4) reveals that the phase of the coherent point spread function is irrelevant when considering incoherently illuminated imaging systems.

OTF(νx,νy)=ΩPˉ(uνx2;vνy2)P(u+νx2;v+νy2)dudvΩPˉ(u;v)P(u;v)dudv
(5)

Here, νx and νy are the normalized spatial frequencies, Ω defines the integration region and P̄(u, v) is the complex conjugate of P(u,v).

Let us define by the term “actual cutoff frequency” (ACF) the lowest spatial frequency that exhibits zero contrast value for the whole DOF range. Clearly, the contrast value, associated with all spatial frequencies that are lower than the ACF, is positive. This means that the ACF is the highest frequency, for which no zeros occur in the OTF for the whole DOF range. Likewise, by the term “practical cutoff frequency” (PCF) we define the lowest spatial frequency that assures a certain desired contrast value for the whole DOF range, so that the contrast associated with every spatial frequency, which is lower than the PCF, will be equal or higher to that desired contrast value. Obviously, the ACF is a special case of the PCF, for a desired contrast value of zero.

When digital restoration is applied on the acquired image, one may prefer to define as a quality criterion, the highest PCF value, defined for a certain desired contrast that is high enough to assure acceptable signal to noise ratio (SNR). When the PCF is designed to be equal to the pixel-resolution or higher, the restoration filter can enhance only the contrast. This is possible since the OTF does not possess regions of very low contrast, close to zero, and does not exhibit contrast reversals for the whole DOF range. Moreover, the robustness of the OTF shape allows the use of the same “universal” restoration filter for the entire DOF range.

2.1 Considerations for choosing the design criterion

A criterion that maximizes the PCF for a certain desired minimal contrast value, described in the following, has been used in a direct optimization algorithm. Let r1…r2N be the inner and outer radia of N π-phase annular rings. Obviously, each choice of radia, {r1…r2N}, results in a certain mask structure that provides a certain DOF range, as well as a PCF value, denoted by ν(r1…r2N). This PCF was obtained by examining the corresponding MTF curves obtained for several defocus positions. Then, the radia that provide the highest PCF value were chosen. Mathematically, the direct optimization process is described by the following expression:

νmax=maxr1r2NminψDOF[ν(r1r2N):MTF(ν,ψ)=Cd]
(6)

2.2 Optimization considerations.

We restricted the number of annular phase rings to be no more than two, i.e. N=2, since this was the result obtained when the design for an imaging system with an extended DOF, up to ψ=15, was considered [23

23. E. Ben-Eliezer, E. Marom, N. Konforti, and Z. Zalevsky, “A Radial Mask for Systems that exhibit High Resolution and Extended Depth of Field,” Appl. Opt. 45, 2001–2013 (2006). [CrossRef] [PubMed]

]. Of course, more phase rings may provide wider DOF range, but with the penalty of lower contrast values in the output image, and even worse, with lower cutoff frequency PCF value and thus poorer resolution.

A well known representation for the PSF of systems with circular symmetry is by using Lommel functions [2

2. A. Papoulis, Systems and Transforms with Applications in Optics (McGraw- Hill, New York, 1968).

], since the circular symmetry allows one to deal only with vectors, rather than with two dimensional matrices. As such, a significant reduction in computation time is achieved.

Let a generalized, circularly symmetric aperture P′(ρ) be defined for a normalized radius ρ, and for a certain defocus condition of ψ=a/2 by the following expression:

P(ρ;a)=p(ρ)eja2ρ2
where:ρ=u2+v2;ψ=a2
p(ρ)={1;ρ<10;else
(7)

The PSF in the presence of a circular, defocus aberrated aperture P′(ρ), is represented by a combination of two Lommel functions, U1(a,w) and U2(a,w), which are defined by the following series [2

2. A. Papoulis, Systems and Transforms with Applications in Optics (McGraw- Hill, New York, 1968).

]:

U1(a,w)=n=0(1)n(aw)2n+1J2n+1(w)
U2(a,w)=n=0(1)n(aw)2n+2J2n+2(w)
(8a)

where Jm(w) stands for a Bessel function of the first kind and order m.

The radial cross section of h(r’), represents the field distribution PSF. For a certain defocus condition ψ=a/2, the field distribution in the image plane, is proportional to the following expression [2

2. A. Papoulis, Systems and Transforms with Applications in Optics (McGraw- Hill, New York, 1968).

]:

h(r)=eja2a[U1(a,w)jU2(a,w)],
where:w=2πRλdimgr
(8b)

and R is the outer radius of the aperture.

We first start with optimizations for a binary phase masks, i.e. 1R, 2R masks. The following expression provides a generalization for the cases of defocus aberrated binary phase-only mask made of N annular rings, where the outer mask radius is R:

Pph(ρ)eiψρ2=P(ρ;2ψ)+2n=12N(1)n+1P(ρn;2ψrn2R2)
(9)

Here, ρ=r/R is the normalized pupil radius. Note that the aperture is composed of sub-apertures with different radia, denoted by rn. Those are the radia transitions of the annular phase rings, r 1r 2N, where smaller index indicates smaller radius value. Therefore, in Eq. (9) the sub-apertures normalized pupil radia are defined as ρn=r/rn. P′(ρn,a) has been defined in Eq. (7) and ψ is the maximal phase shift at the pupil edge. The reader should note that P′(ρn, a)=0 for ρn>1. Applying Eq. (8b) to each one of the sub-pupils, one obtains the radial cross-section of the total PSF field distribution. Since we deal with incoherent illumination, we are interested in the square of the absolute value of the PSF radial cross section vectors, to express the intensity impulse response. Thereafter, the OTF radial cross sections are readily calculated by the Bessel-Fourier transform [1

1. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1996).

2

2. A. Papoulis, Systems and Transforms with Applications in Optics (McGraw- Hill, New York, 1968).

], of the intensity impulse response:

OTF=0h(r)2J0(2πρr)rdr0h(r)2rdr
(10)

The optimization procedure is carried along the following lines:

• Set the phase transitions radia values {r1..r2N}, to obtain a pupil according to Eq. (9).

• Calculate the PSF with Lommel functions for a set of defocus positions within the required DOF.

• Evaluate the square of the absolute values of the PSFs for all defocus conditions within the DOF region.

• Calculate the MTF curves according to Bessel-Fourier transform by using Eq. (10).

• Find the current PCF for this radia configuration.

• If the current configuration provides a PCF that is higher than an earlier calculation of the highest PCF, save the current configuration as well as the current PCF as the new “so far” best result.

• If not all radia configurations have been examined yet, go back to the first step. Otherwise, finish.

2.3 Optimized binary amplitude-phase circular mask.

It is well known that a sufficiently thin annular ring, placed at the pupil of a conventional imaging system provides an extended DOF and preserves diffraction-limit resolution, albeit with reduced contrast and low light throughput [4

4. T. C. Poon and M. Motamedi, “Optical Digital Incoherent Image Processing for Extended Depth of Field,” Appl. Opt. 26, 4612–4615 (1987). [CrossRef] [PubMed]

]. The main characteristic of such system is the resulting OTF which has robust shape exhibiting a constant pedestal for moderate and high frequencies for all object positions within the DOF range. Moreover, the OTF is circular symmetric, and thus, independent of the object orientation.

It is, therefore, worthwhile to combine both approaches, and as such a binary amplitude and phase mask is considered. Placing an opaque central circle in the middle of a phase mask indeed reduces somewhat the light throughput; however the presence of the phase rings results in a reduced opaque area, thus achieving relatively high light throughput. The OTF of the combined binary amplitude and phase mask is more robust and provide higher ACF and PCF values, as shown below.

Pamp_ph(ρ)eiψρ2=P(ρ;2ψ)P(ρ1;2ψr12R2)+2n=22N+1(1)nP(ρn;2ψrn2R2)
(11)

Fig. 2. Schematic of the combined binary amplitude and phase mask (2Rc case), as well as an illustration of the optimization process (green arrows).

3. Results

3.1 Optimization results

For a certain required depth of field (DOF) range, defined by the maximal aberration phase ψmax at the aperture edge, we seek the locations of one or two π-phase rings, with or without an amplitude opaque center that provide the maximal normalized spatial frequency, νmax, which assures a certain desired minimal contrast, denoted by Cd within the whole DOF range. Since we observed that the contrast has a tendency to fluctuate around the minimal acceptable contrast value, in our calculations we accepted variations of +/-10% from the nominal one. As a result of this, a minimal defined contrast of 10% was essentially allowed to drop as low as 9% and a 5% contrast was allowed to reach 4.5%.

The results are summarized in the following tables. Table 1 and Table 2 summarize the normalized radial locations (outer radius R=1) of a single phase ring, i.e. two phase transitions, for the case of an optimal binary phase mask. The results were derived for three different DOF ranges. The optimizations assumed desired contrast values, Cd, of five and nine percents respectively. PCF values in all the tables were normalized assuming that the diffraction-limit cutoff for spatially incoherent illumination is 2.

Table 1. Optimization results for a phase mask with only one annular phase ring for different DOF ranges, defined by ψmax. The normalized radial phase transition locations are r1 and r2 and the normalized PCF, denoted by νmax, is provided for a desired contrast value of Cd=5%.

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Table 2. Optimization results for a phase mask with only one annular phase ring for different DOF ranges, defined by ψmax. The normalized radial phase transition locations are r1 and r2 and the normalized PCF, denoted by νmax, is provided for a desired contrast value of Cd=10%.

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Similar results, but for the case where a central opaque circle is included as well, are provided in Tables 3 and 4 respectively. Here, r1 is the normalized radius of the opaque circle and the other radia identify the radial locations of the phase transition. Note an additional column that provides the resulting light throughput.

Table 3. Optimization results for a mask with only one annular phase ring as well as an opaque central circle with a normalized radius of r1. Different DOF ranges were considered, defined by the ψmax values. The normalized radial phase transition locations are r2 and r3 and the normalized PCF, denoted by νmax, is provided for a desired contrast value of Cd=5%.

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Table 4. Optimization results for a mask with only one annular phase ring as well as an opaque central circle with a normalized radius of r1. Different DOF ranges were considered, defined by the ψmax values. The normalized radial phase transition locations are r2 and r3 and the normalized PCF, denoted by νmax, is provided for a desired contrast value of Cd=10%.

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One notes from Tables 1–4 that a mask with only one phase ring has the ability to protect only against moderate defocus conditions, up to ψmax=10, where for larger DOF ranges, the size of the central opaque circle increases. In order to have high performance in the presence of severe defocus conditions, i.e., large DOF, the optimization was carried out for the case of a binary phase mask with two phase rings, as well as a mask with two phase rings and an opaque central circle. The results are summarized in Tables 5–8 respectively.

Table 5. Optimization results for a binary phase mask with two annular phase rings designed for a desired contrast value of Cd=5%. Different DOF ranges were considered, defined by their ψmax values. The normalized radial phase transition locations are r1 … r4 and the normalized PCF, is denoted by νmax.

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Table 6. Optimization results for a binary phase mask with two annular phase rings designed for a desired contrast value of Cd=10%. Different DOF ranges were considered, defined by their ψmax values. The normalized radial phase transition locations are r1 … r4 and the normalized PCF, is denoted by νmax.

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Table 7. Optimization results for a binary phase mask with central opaque circle and two annular phase rings designed for a desired contrast value of Cd=5%. Different DOF ranges were considered, defined by their ψmax values. The normalized radius of the opaque center is r1 while the normalized radial phase transition locations are r2 … r5. The normalized PCF is denoted by νmax.

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Table 8. Optimization results for a binary phase mask with central opaque circle and two annular phase rings designed for a desired contrast value of Cd=10%. Different DOF ranges were considered, defined by their ψmax values. The normalized radius of the opaque center is r1 while the normalized radial phase transition locations are r2 … r5. The normalized PCF is denoted by νmax.

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It is interesting to compare the increase in the frequency νmax as well as the amount of reduction in the optical light throughput. Tables 9 and 10 show these parameters. The notation 1R means “one ring”. The notation 1RC means “one ring and opaque center”. Similar notations are used for two rings. Table 9 presents the maximal frequency in bold letters in each left sub-column, as well as the light throughput (right sub-column) for an assumed acceptable contrast value of Cd=5%, while Table 10 shows the same, but for a contrast value of Cd=10%.

Table 9. Performance comparison when design for a contrast of Cd=5% is assumed. The normalized PCF (bold), as well as the light throughput for several defocus conditions, denoted by ψmax are shown. The notation 1R means “one ring.” The notation 1RC means “one ring and opaque center”. Similar notations are used for two rings

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Table 10. Performance comparison when design for a contrast of Cd=10% is assumed. The normalized PCF (bold), as well as the light throughput for several defocus conditions, denoted by ψmax are shown. The notation 1R means “one ring”. The notation 1RC means “one ring and opaque center”. Similar notations are used for two rings.

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The performance achieved with the masks mentioned above, should be compared to the performance achievable with an optimal binary amplitude mask (no phase rings), obtained with an opaque center for the same DOF range. The comparison is made for an optical system with same aperture size and according to the same criterion, defined in Eq. (6). The results are summarized in Table 11 for a contrast of an acceptable value of Cd=5% and Table 12 for a desired contrast value of 10%.

Table 11. Optimization results for an amplitude annular ring, defined by an inner normalized radius of r1. Different DOF ranges were considered, defined by the ψmax values. The normalized PCF, denoted by νmax, is provided for a desired contrast value of Cd=5%.

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Table 12. Optimization results for an amplitude annular ring, defined by an inner normalized radius of r1. Different DOF ranges were considered, defined by the ψmax values. The normalized PCF, denoted by νmax, is provided for a desired contrast value of Cd=10%.

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Comparing Tables 910 to Tables 1112, it is interesting to note that when one increases the desired DOF range (i.e., higher ψmax value), the size of the opaque center radius increases in case of the annular amplitude ring (Tables 1112). Therefore, the light throughput is lower for a larger ψmax. However, in the case of the amplitude and phase masks (1Rc and 2Rc), up to a certain ψmax value the size of the opaque center decreases with ψmax, due to the presence of the phase rings. Therefore, the light throughput increases with ψmax, up to a certain ψmax value.

Comparison between Tables 1 and 3 and Tables 5 and 7 with Tables 2 and 4 and Tables 6 and 8 respectively, reveal that for lower Cd values, the central opaque circle is bigger, and higher resolution is achieved. However, the light throughput for those cases is relatively low. One further notes that up to a certain ψmax value (ψmax=10 for one phase ring and ψmax=16 for two phase rings), the 1RC mask tends to coincide with the 1R mask, and similarly the 2RC mask tends to coincide with the 2R mask. For higher ψmax values the opaque center size tends to increase since it tries to reduce the aperture size and thus the phase only masks and the amplitude-phase masks do not coincide anymore. This phenomenon is clearer for low Cd values, since if small Cd values are allowed, the resulting OTF tends to resemble an OTF of an amplitude annular ring.

Increasing the number of phase rings provides the same PCF or higher, with high light throughput. However, contrast values that support such high PCF values for large DOF ranges are expected to be smaller than the values that are being used here. Tables 910 reveal that when using the opaque central circle, improvement with respect to the phase-only-masks is achieved only if one is allowed to use low desired contrast values, as is the case when digital restoration is considered. Thus, the improvement in resolution, achieved for Cd=5%, is higher than the one achieved for 10%. Moreover, the higher the resolution improvement is, the lower the obtained light throughput.

Tables 11 and 12 reveal that an amplitude ring cannot maintain contrast values of 10% for wide DOF regions. As the contrast requirement is lower, the obtained frequency is higher, as seen for the case of 5%. However, even 5% is too high if one desires extremely wide DOF ranges. For obtaining large DOF ranges, operating at lower desired contrast values are required. Moreover, the light throughput is extremely deteriorated with such amplitude masks. Therefore, masks containing phase rings only are the preferred solution if high frequency and relatively high contrast performance are required for an extended DOF. We conclude that the phase rings are more important than the central opaque circle when large DOF as well as high resolution are required, along with a relatively high desired contrast value. In the following section, OTF results for several optimized designs are presented, in order to gain a deeper understanding of behavior of the family of masks under investigation.

3.2 Performance of the binary phase mask

A comparison between the OTF curves obtained for systems equipped with the different masks under investigation, for selected defocus conditions, is presented in Fig. 3. The red and the blue curves denote the OTF of the optimal phase masks with one phase ring, which were designed for desired contrast values of 5% and 10% respectively, while the black curve stands for the OTF, provided by a clear aperture. Defocus conditions, selected for ψ=0, 5 and 10, and are shown in Figs. 3(a)–3(c) respectively.

Fig. 3. OTF radial cross-sections. The red and the blue curves denote the OTF of the optimal phase masks with one phase ring, which were designed for ψmax=10 and desired contrast values of 5% and 10% respectively, while the black curve stands for the OTF, provided by the clear aperture. (a)- in-focus position (b)- defocus parameter of ψ=5. (c)- defocus parameter of ψ=10.

One notes from Fig. 3 that an optimal phase mask, designed for a low desired contrast (Cd) tends to provide higher ACF value (red curve). This is seen in Fig. 4 as well, where the OTF curves of the optimal phase masks with two annular phase rings, which were designed for desired contrast values of 5% (red curve) and 10% (blue curve), for a DOF range of Ψmax=14 are presented. Defocus conditions of ψ=0, 7 and 14 are shown in Figs. 4 (a)–4(c) respectively.

Fig. 4. OTF radial cross-sections. The red and the blue curves denote the OTF of the optimal phase mask with two annular phase rings, which were designed for ψmax=14 and desired contrast values of 5% and 10% respectively. (a)- in-focus position (b)- defocus parameter of ψ=7. (c)- defocus parameter of ψ=14.

As a final remark, one should note that the design according to the MMSE criterion [23

23. E. Ben-Eliezer, E. Marom, N. Konforti, and Z. Zalevsky, “A Radial Mask for Systems that exhibit High Resolution and Extended Depth of Field,” Appl. Opt. 45, 2001–2013 (2006). [CrossRef] [PubMed]

] was carried on a complex valued mask, and the sub-optimal phase mask, was defined by using the phase of that complex valued optimal mask. Therefore, one could not predict the exact PCF of the sub-optimal phase mask from the PCF of the optimal complex valued mask. Contrary, the present design is carried directly on the phase mask, and the PCF is one of the optimization outputs; therefore, control on design parameters is eased.

3.3 Performance of the binary phase and amplitude mask

In this section, the OTF curves obtained for the circular symmetric phase and amplitude masks, in which a central opaque circle blocks the light around the optical axis, are provided. As shown in Tables 9 and 10, a higher desired contrast results in lower PCF, while a lower light throughput, results in higher PCF, as expected. Moreover, the performance of the designs with and without the opaque center is almost the same when the desired DOF range increases.

A comparison between radial OTF cross sections, calculated for the optimal binary phase and amplitude masks with one (red curve) and two (blue curve) annular phase rings respectively, which were designed for desired contrast values of 5%, for a DOF range of Ψmax=8 is shown in Fig. 5. The improvement in the PCF values in both cases are clearly seen, when the above results are compared to the OTF of a clear aperture (black curve) that provides the same light throughput, thus having a lateral dimension of about 80% of the lateral dimension of the optimal mask aperture (64% throughput). Note that the frequency range of all the OTF curves shown in Fig. 5 correspond to a full size aperture (100% lateral dimension size). Defocus conditions of ψ=0, 4 and 8 are shown in Figs. 5(a)–5(c) respectively.

Fig. 5. OTF radial cross-sections. The red and the blue curves denote the OTF of the optimal amplitude-phase binary masks, which were designed for desired contrast values of 5%, with one and two phase rings respectively, while the black curve stands for the OTF, provided by the clear aperture with lateral dimension of 80% of the full aperture size, which provides the same light throughput. (a)- in-focus position (b)- defocus parameter of ψ=4. (c)-defocus parameter of ψ=8.

4. Experimental realizations

An experiment has been carried in order to verify the ability of an optimal binary phase and amplitude mask to increase the DOF that is provided by a low-cost camera lens module. In section 4.1 we present the requirements of the imaging system as well as the optimal mask that has been selected. Some OTF curves are presented as well to allow a comparison between the mask performance versus an open aperture case. Section 4.2 describes the experimental setup as well as the experimental results. The mask performance for white illumination and conclusive remarks are presented in section 4.3.

4.1 Imaging system specifications

In this section, an experimental imaging system that can be considered as a preliminary vehicle for testing extended DOF imager concepts is presented. The chosen apparatus for this experiment was a MUSTEK® DV5200 camera, equipped with a CMOS detector having a resolution of 1200x1600 pixels (2 mega pixels). The camera lens was composed of five glass elements with effective focal length of 8.5mm, and was considered throughout the experiment as an equivalent achromatic thin lens. Field of view (FOV) of 50 degrees along the sensor diagonal and pixel dimension of 4µm were measured experimentally by counting the number of pixels in an image of an object with known dimensions that was located at a known distance from the camera. The camera lens was positioned in an in-focus condition, where object was located 30 cm from the lens. The required specifications of the imaging system are working distance from 15cm to infinity for wavelength of 532 nm. The required desired contrast value is 5%.

An aperture diameter of 2mm was chosen, to mimic imaging systems used for instance in cellular phone cameras. For this aperture size and for an in-focus object distance of 30 cm, the maximal defocus phase at the aperture edge (Ψmax@λ=532nm) was found to be 20 radians. The optimization results, presented earlier, reveal that the binary amplitude and phase mask with two phase rings provided the highest PCF, and therefore it has been used throughout the experiment.

The optimal mask was found to have an opaque central circle with a radius of 0.2 mm, and two annular π-phase rings (for green light). The first ring has an inner radius of 0.62 mm and an outer radius of 0.76 mm, while the second ring has inner and outer radia of 0.86 mm and 0.94 mm respectively. The aperture radius is 1mm. The light throughput reduction of 4% with respect to a same size open aperture is negligible and the normalized PCF for the desired contrast value was 0.56, where a value of 2 is the diffraction limit cutoff. This PCF value matches the Nyquist frequency of the detector (the highest frequency that can be detected by the sensor) for the wavelength under consideration.

Figs. 6(a)–6(e) show several MTF curves of the optimal mask for defocus positions of ψ=0, 5, 10, 15 and 20 respectively (blue curve), as well as the corresponding MTF curves obtainable with a full size open aperture in the same conditions (black curve). The contrast values of 5% and 10% are indicated with green and red horizontal lines respectively.

Fig. 6. MTF curves of the optimal mask (blue curve) for several defocus positions, along with the corresponding MTF curves provided by a full size open aperture in the same defocus conditions (black curve). (a)- defocus parameter ψ=0; (b)- ψ=5; (c)- ψ=10; (d)- ψ=15; (e)- ψ=20; The contrast values of 5% and 10% are marked with green and red horizontal lines respectively.

4.2 Mask behavior under polychromatic illumination

The mask has been designed for a nominal green wavelength of 532 nm. Therefore, when using the mask in an imaging system that is being illuminated by polychromatic light some performance degradation due to lower PCF values for the same desired contrast of 5% is expected. Since most practical applications use polychromatic light sources, it is important to study the mask behavior with natural light sources.

The etching depth of the mask provides correct phase value of π for the nominal wavelength only; thus other wavelengths suffer from different phase values. Therefore, the OTF contrast values at certain spatial frequencies and for certain object positions within the DOF range is expected to be affected. The phase transitions of the mask are designed for a π phase value, for the nominal wavelength only, and thus, the mask is not optimal for the other wavelengths in the frequency band. The polychromatic PCF value, defined as the minimal value among all the PCF values, obtained for each one of the wavelengths under consideration is now lower. For simplicity the term PCF is used in the rest of the paper instead of the term “polychromatic PCF”.

Figs. 7 (a)–7(c) show the obtained theoretical contrast values with respect to the object distance from the lens as a function of distance (in-focus position at d=0.3 m), for the red, green and blue channels respectively, for open aperture (blue curve) as well as for mask-equipped aperture (dash-green curve). The plotted curves of Fig. 7 were calculated for a frequency value of 0.1 in the image plane. This normalized frequency value of 0.1, is 5% of the diffraction limit cutoff, which we defined to have the value of 2. The normalized frequency value of 0.1 translated into actual frequency values, provides spatial frequencies of 16.8, 22.11 and 26.86 line-pairs/mm for the red (λ=700 nm) green (λ=532 nm) and blue (λ=438 nm) wavelengths respectively. Similar results plotted for a normalized frequency value of 0.2 are presented in Fig. 8.

Fig. 7. Obtained theoretical contrast values with respect to the object distance from the lens (in-focus position in d=0.3 m), for (a)- red, (b)- green and (c)-blue channels, for open aperture (blue curve) as well as for the mask (dash-green curve), assuming normalized frequency value of 0.1 in the image plane.
Fig. 8. Obtained theoretical contrast values with respect to the object distance from the lens (in-focus position in d=0.3 m), for (a)- red, (b)- green and (c)-blue channels. Open aperture (blue curve) as well as for the mask-equipped aperture (dash-green curve), are plotted for a normalized frequency value of 0.2 in the image plane.

One should recall that Fig. 7 and Fig. 8 assume that the camera lens is corrected for axial chromatic aberration, so that all wavelengths are assumed to focus on the detector plane for an in-focus position. Examining the curves one clearly recognizes the in-focus position of 0.3 m that provides the highest contrast value for the open aperture. The mask results (green dotted curve) provide lower contrast values for the whole DOF range; however, those curves are more robust and do not change rapidly for various object positions as expected. The designed DOF was from 15 cm to infinity. Assuming an achromatic primary camera lens, the blue wavelength provides the highest defocus parameter value; it thus determines the PCF. Examining Fig. 7(c), one notices that for object distance of d=15 cm, the imaging system, equipped with the mask, provides a contrast value of 0.01. Further examination shows that the normalized PCF is 0.18, for the blue wavelength; this corresponds approximately to 52 line-pairs/mm in the image plane, or equivalently, five sensor pixels per period. Note that examining the performance of a same size open aperture imaging system for the same DOF range, the obtained PCF is half of that obtained with a mask equipped imager, or about 0.08 for the blue wavelength. Therefore, for the specified DOF range, the performance improvement provided by the mask is clearly seen.

4.3 Experimental results

Experiments have been carried in order to measure the performance improvement when the mask-equipped imaging system is used, instead of a same size open aperture. The details of the experimental apparatus were presented in section 4.1

The mask-equipped lens is to be used in consumer optics equipments, such as low cost cameras, as well as for other applications such as barcode readers. Some typical objects, such as a business card as well as a spoke target that contains spatial frequencies encountered in barcodes with finest feature of 7.5 mil were captured.

The experiments were carried with a Mustek® DV 5200 camera. A metal structure that contains the pupil mask was externally mounted on the lens. The experimental apparatus is portable, and can be taken outdoors, or it can be used on an optical table indoor, thus providing the flexibility of checking the behavior for indoor and outdoor illumination with ease. The output images that were acquired with the mask were processed after capture by a digital restoration filter, to achieve improved resolution and contrast. A block diagram of the digital restoration filter that we used is shown in Fig. 9.

Fig. 9. A block diagram of a digital restoration filter for color images.

Since a commercial camera is used in this experiment, the restoration was carried only on the final output JPEG image file it provides. The entire data rendering is done by the camera [24

24. R. Ramanath, W. E. Snyder, Y. Yoo, and M. S. Drew, “Color image processing pipeline,” IEEE Sig. Proc. Mag. 22, 34–43 (2005). [CrossRef]

]. It is being treated as a black box and was not modified. Since each scene was captured with a clear aperture as well as with a mask in the same conditions, one may compare the output images qualitatively.

Each one of the three channels that the camera captures suffers from a different blur kernel, since the aberration is wavelength dependent. Therefore, the deblurring is carried by a different restoration filter for each channel. We used a single Wiener filter [25

25. R. C. Gonzalez and R. E. Woods, Digital Image Processing (Addison-Wesley, New York, 1993).

27

27. E. Marom, E. Ben-Eliezer, and N. Knoforti, PCT/IL2008/000527 “Optical imaging system with an extended depth-of-field and a method for designing an optical imaging system.”

] to restore each one of the channels for the whole DOF; the OTF curve for in-focus position for red, green and blue channels has been used to estimate the blur of each channel for the whole DOF range. Note that all the restoration filters are normalized, and therefore colors are preserved.

Figure 10(a) shows a typical outdoor scene. Since the in-focus position in our configuration was 30 cm from the lens, distant objects are blurred, as seen in the magnified portion, while the grass in the foreground is well resolved. However, when the mask is incorporated, the details are retained, as observed in Fig. 10(b). A Wiener filter improves the contrast even further, as seen in Fig. 10(c).

Result obtained for a written text as an object, located 15 cm from the lens, at the edge of the DOF range, is shown in Fig. 11(a). Once again, objects not in focus are blurred. The image that is obtained with the mask only, as well as the image that is obtained with the mask and electronic restoration, are shown in Figs. 11(b) and 11(c) respectively. Magnified portion of text is shown in the right. Figure 12 exhibits results obtained for a spoke target object in case of indoor illumination. For ease of interpretation this targets contains circles that identify certain commonly used spatial frequencies used for barcodes: the smallest circle corresponds to 7.5 mil, and the following ones are at 10, 13, 20 and 30 mil.

Since all the images were captured in the same conditions and by the same camera, one can easily inspect the quality of the resulting images. An improvement of Figs. 10(b), 11(b) and 12(b) with respect to Figs. 10(a), 11(a) and 12(a) respectively, is clearly seen in resolution as well as in contrast. An off-line digital contrast enhancement stage improves the image quality even further, as shown in Figs. 10(c), 11(c) and 12(c) respectively.

All those images reveal the improved performance of the mask-equipped imaging system with respect to the same size clear aperture imager and support the theoretical results. Such imaging systems may have many practical applications, where extended DOF is required along with high resolution and contrast.

Fig. 10. A natural outdoor scene (in-focus position in d=0.3 m from the lens). (a)- Clear aperture, (b) - Mask only (c) - Output with mask and restoration. Areas in black rectangle are magnified in the right.
Fig. 11. A personal card, located 15 cm from the lens at the DOF edge (in-focus position in d=0.3 m from the lens). (a)- Clear aperture, (b) - Mask only (c) - Output with mask and restoration. Areas in black rectangle are magnified in the right.
Fig. 12. A spoke target, located 15 cm from the lens at the DOF edge (in-focus position in d=0.3 m from the lens). (a)- Clear aperture, (b) - Mask only (c) - Output with mask and restoration. The finest line width in the middle is 7.5 mil, while widths of 10, 13, 20 and 30 mil on successive annular black rings are presented

5. Conclusions

This paper was devoted to describe and demonstrate the performance of an optimal extended DOF imaging systems that provide high resolution and high light throughput. The simple circularly symmetric binary phase structure of the optimal complex valued mask, obtained by the MMSE design criterion [23

23. E. Ben-Eliezer, E. Marom, N. Konforti, and Z. Zalevsky, “A Radial Mask for Systems that exhibit High Resolution and Extended Depth of Field,” Appl. Opt. 45, 2001–2013 (2006). [CrossRef] [PubMed]

], lead us to carry optimizations on such simple structures that have the potential to be incorporated in consumer optics lens modules.

The fact that the OTF curves of circularly symmetric imaging systems possess phase values of only 0 or π, and the fact that the design, presented above, has no contrast reversals up to the pixel resolution, for the whole DOF range, allow one not to use digital restoration when machine vision applications such as barcodes readers, are considered. In such cases, extended DOF all-optical imaging systems are obtained. The performance of the imaging systems has been verified experimentally, and the results support the theory.

References and links

1.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1996).

2.

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw- Hill, New York, 1968).

3.

M. Mino and Y. Okano, “Improvement in the OTF of a Defocused Optical System Trough the Use of Shaded apertures,” Appl. Opt. 10, 2219–2225 (1971). [CrossRef] [PubMed]

4.

T. C. Poon and M. Motamedi, “Optical Digital Incoherent Image Processing for Extended Depth of Field,” Appl. Opt. 26, 4612–4615 (1987). [CrossRef] [PubMed]

5.

J. O. Castaneda, R. Ramos, and A. Noyola-Isgleas, “High focal depth by apodization and digital restoration,” Appl. Opt. 272583–2586 (1988). [CrossRef]

6.

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

7.

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

8.

S. Sanyal and A. Ghosh, “High focal depth with quasi-bifocus birefringent lens,” Appl. Opt. 39, 2321–2325 (2000). [CrossRef]

9.

E. Peli and A. Lang, “Appearance of images through a multifocal intraocular lens,” J. Opt. Soc. Am. A 18, 302–309 (2001). [CrossRef]

10.

E.R Dowski and W.T. Cathey, “Extended depth of field through wave-front coding,” Appl. Opt 34, 1859–1866 (1995). [CrossRef] [PubMed]

11.

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, 919–921 (1996). [CrossRef] [PubMed]

12.

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] [PubMed]

13.

S. Prasad, V. Paul Pauca, R. J. Plemmons, T. C. Torgersen, and J. van der Gracht, “Pupil-phase optimization for extended focus, aberration corrected imaging systems,” Proc. SPIE 5559, 335–345 (2004). [CrossRef]

14.

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, 67–74 (2004). [CrossRef]

15.

W. Chi and N. George, “Electronic imaging using a logarithmic asphere,” Opt. Lett. 26, 875–877 (2001). [CrossRef]

16.

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

17.

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 acquisition,” Proc. SPIE 5438, 120–129 (2004). [CrossRef]

18.

E. Ben-Eliezer and E. Marom, “Aberration-free superresolution imaging via binary speckle pattern encoding and processing,” J. Opt. Soc. Am. A 24, 1003–1010 (2007). [CrossRef]

19.

E. Ben-Eliezer, N. Konfori, and E. Marom, “Superresolution imaging with noise reduction and aberration elimination via random structured illumination and processing,” Opt. Express 15, 3849–3863 (2007). [CrossRef] [PubMed]

20.

E. Ben-Eliezer, Z. Zalevsky, E. Marom, and N. Konforti, “All-Optical Extended Depth of Field Imaging System,” Pure Appl. Opt. 5, S164–S169 (2003). [CrossRef]

21.

E. Ben-Eliezer, Z. Zalevsky, E. Marom, and N. Konforti, “All-Optical Extended Depth of Field Imaging System,” Proc. SPIE 4829, 221–222 (2002).

22.

E. Ben-Eliezer, Z. Zalevsky, E. Marom, and N. Konforti, “Experimental Realization of an Imaging System with an Extended Depth of Field,” Appl. Opt. 44, 2792–2798 (2005). [CrossRef] [PubMed]

23.

E. Ben-Eliezer, E. Marom, N. Konforti, and Z. Zalevsky, “A Radial Mask for Systems that exhibit High Resolution and Extended Depth of Field,” Appl. Opt. 45, 2001–2013 (2006). [CrossRef] [PubMed]

24.

R. Ramanath, W. E. Snyder, Y. Yoo, and M. S. Drew, “Color image processing pipeline,” IEEE Sig. Proc. Mag. 22, 34–43 (2005). [CrossRef]

25.

R. C. Gonzalez and R. E. Woods, Digital Image Processing (Addison-Wesley, New York, 1993).

26.

B. R. Hunt and O. Kubler, “Karhunen-Loeve Multispectral Image Restoration, Part I: Theory,” ASSP 32, 592–600 (1984). [CrossRef]

27.

E. Marom, E. Ben-Eliezer, and N. Knoforti, PCT/IL2008/000527 “Optical imaging system with an extended depth-of-field and a method for designing an optical imaging system.”

OCIS Codes
(080.1010) Geometric optics : Aberrations (global)
(080.3620) Geometric optics : Lens system design
(110.0110) Imaging systems : Imaging systems

ToC Category:
Imaging Systems

History
Original Manuscript: September 23, 2008
Revised Manuscript: October 26, 2008
Manuscript Accepted: October 29, 2008
Published: November 26, 2008

Citation
Eyal Ben-Eliezer, Naim Konforti, Benjamin Milgrom, and Emanuel Marom, "An optimal binary amplitude-phase mask for hybrid imaging systems that exhibit high resolution and extended depth of field," Opt. Express 16, 20540-20561 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-25-20540


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References

  1. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1996).
  2. A. Papoulis, Systems and Transforms with Applications in Optics (McGraw- Hill, New York, 1968).
  3. M. Mino and Y. Okano, "Improvement in the OTF of a Defocused Optical System Trough the Use of Shaded apertures," Appl. Opt. 10, 2219-2225 (1971). [CrossRef] [PubMed]
  4. T. C. Poon and M. Motamedi, "Optical Digital Incoherent Image Processing for Extended Depth of Field," Appl. Opt. 26, 4612-4615 (1987). [CrossRef] [PubMed]
  5. J. O. Castaneda, R. Ramos, and A. Noyola-Isgleas, "High focal depth by apodization and digital restoration," Appl. Opt. 272583-2586 (1988). [CrossRef]
  6. J. O. Castaneda, E. Tepichin, and A. Diaz, "Arbitrary high focal depth with a quasioptimum real and positive transmittance apodizer," Appl. Opt. 28, 2666-2669 (1989). [CrossRef]
  7. J. O. Castaneda and L. R. Berriel-Valdos, "Zone plate for arbitrary high focal depth," Appl. Opt. 29, 994-997 (1990). [CrossRef]
  8. S. Sanyal and A. Ghosh, "High focal depth with quasi-bifocus birefringent lens," Appl. Opt. 39, 2321-2325 (2000). [CrossRef]
  9. E. Peli and A. Lang, "Appearance of images through a multifocal intraocular lens," J. Opt. Soc. Am. A 18, 302-309 (2001). [CrossRef]
  10. E. R Dowski Jr and W. T. Cathey, "Extended depth of field through wave-front coding," Appl. Opt 34, 1859-1866 (1995). [CrossRef] [PubMed]
  11. J. van der Gracht, E. R. DowskiJr, M. G. Taylor, and D. M. Deaver, "Broadband behavior of an optical-digital focus-invariant system," Opt. Lett. 21, 919-921 (1996). [CrossRef] [PubMed]
  12. 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] [PubMed]
  13. S. Prasad, V. Paul Pauca, R. J. Plemmons, T. C. Torgersen and J. van der Gracht, "Pupil-phase optimization for extended focus, aberration corrected imaging systems," Proc. SPIE 5559,335-345 (2004). [CrossRef]
  14. 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, 67-74 (2004). [CrossRef]
  15. W. Chi and N. George, "Electronic imaging using a logarithmic asphere," Opt. Lett. 26, 875-877 (2001). [CrossRef]
  16. N George and W. Chi, "Computational imaging with the logarithmic asphere: theory," J. Opt. Soc. Am. A 20, 2260-2273 (2003). [CrossRef]
  17. 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 acquisition," Proc. SPIE 5438, 120-129 (2004). [CrossRef]
  18. E. Ben-Eliezer and E. Marom, "Aberration-free superresolution imaging via binary speckle pattern encoding and processing," J. Opt. Soc. Am. A 24, 1003-1010 (2007). [CrossRef]
  19. E. Ben-Eliezer, N. Konfori, and E. Marom, "Superresolution imaging with noise reduction and aberration elimination via random structured illumination and processing," Opt. Express 15, 3849-3863 (2007). [CrossRef] [PubMed]
  20. E. Ben-Eliezer, Z. Zalevsky, E. Marom, and N. Konforti, "All-Optical Extended Depth of Field Imaging System," Pure Appl. Opt. 5, S164-S169 (2003). [CrossRef]
  21. E. Ben-Eliezer, Z. Zalevsky, E. Marom, and N. Konforti, "All-Optical Extended Depth of Field Imaging System," Proc. SPIE 4829, 221-222 (2002).
  22. E. Ben-Eliezer, Z. Zalevsky, E. Marom, and N. Konforti, "Experimental Realization of an Imaging System with an Extended Depth of Field," Appl. Opt. 44, 2792-2798 (2005). [CrossRef] [PubMed]
  23. E. Ben-Eliezer, E. Marom, N. Konforti, and Z. Zalevsky, "A Radial Mask for Systems that exhibit High Resolution and Extended Depth of Field," Appl. Opt. 45, 2001-2013 (2006). [CrossRef] [PubMed]
  24. R. Ramanath, W. E. Snyder, Y. Yoo, and M. S. Drew, "Color image processing pipeline," IEEE Sig. Proc. Mag. 22, 34-43 (2005). [CrossRef]
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