## 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]. 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 [2–7]. 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], where the resulting DOF is quite limited, or a lens, which is divided into annular rings [9]. 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 [10–14]. Optimizations of the processes of encoding and decoding the wave-front phase in integrated optical-digital imaging systems has been carried as well [13–14]. 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, 16] also contains regions of contrast reversals for relatively low normalized frequencies, at the DOF extreme positions.

The hybrid approach, presented by Dowski *et al* [10–12], 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], where a blurred image is sufficient for further processing.

Recently, a super-resolution technique that allows defocus elimination has been presented [18–19]. 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.

In previous studies, a phase mask that consists of sixteen interlaced Fresnel lenses (FL’s) was proposed [20–22]. It has been proven that this phase mask is optimal in the sense of a minimum mean square error (MMSE) criterion with respect to the optical field, and can be modified to be sub-optimal in case of incoherent illumination [22].

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

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.

This paper is devoted to present the design considerations as well as the optimization results of four different mask structures. The sub-optimal binary phase mask derived in a previous study [23], is illustrated in Fig. 1 for the case of two “*π*” phase rings. We search for optimal designs when considering a mask with a single annular phase ring (providing high contrast and resolution, but reduced DOF range) as well as for a case of two annular phase rings (providing reduced contrast and resolution, but larger DOF range), denoted as “1R” and “2R” respectively.

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].

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], namely:

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

where R is the pupil radius, λ is the wavelength, f, d_{obj} and d_{img} 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:

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

where x and y are the lateral coordinates in the image plain, I_{g}(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.

The Optical Transfer Function (OTF) is the normalized Fourier transform of the intensity impulse response, |h(x,y)|^{2}. It is related to the generalized pupil function, P′(u,v) by an auto-correlation operation as expressed in [1]:

Here, ν_{x} and ν_{y} are the normalized spatial frequencies, Ω defines the integration region and *P*̄(*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

Experimental results, obtained in a previous study with a sub-optimal phase mask (illustrated in Fig. 1) with quasi-monochromatic as well as polychromatic illuminations were encouraging [23]. It is now of high interest to optimize this phase mask in order to derive the conditions that will lead to the highest PCF values. The optimization should provide the radia of the phase transition rings of the optimal mask that maximize a certain design criterion.

When searching for a design criterion, it is worthwhile to remember that the contrast can be enhanced a posteriori by digital means, as long as the information is available, while reduced resolution is associated with information loss and thus cannot be enhanced. Therefore, one is directed to search for a design criterion that gives preferences to resolution over contrast. When a digital restoration stage is used, the contrast can be easily enhanced; therefore the exact value of the contrast at a certain frequency is not very important as long as its value exceeds a minimum, required to ensure a sufficient SNR value. Digital restoration will allow the recovery of the contrast in the acquired image for all frequencies with high enough signal to noise ratio (SNR). Therefore, the shape of the OTF curves at different object planes within the DOF region is not the main desired design feature requirement. Rather, the highest frequency that assures a certain minimum contrast value throughout the DOF range, namely the PCF value is the dominant factor.

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 r_{1}…r_{2N} be the inner and outer radia of N *π*-phase annular rings. Obviously, each choice of radia, {r_{1}…r_{2N}}, results in a certain mask structure that provides a certain DOF range, as well as a PCF value, denoted by ν(r_{1}…r_{2N}). 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:

Here, C_{d} is the desired minimum contrast. We now search for a mask to be ultimately used with incoherently illuminated scenes, such that Eq. (6) is maximized. We expect the image to display high resolution, up to pixel size resolution, for the whole DOF range.

#### 2.2 Optimization considerations.

The optimization has been achieved by a direct search over all radia values. We used a direct search method and not an optimization technique such as simulated annealing, since the structures under consideration are well defined, and therefore the search for the optimal masks is restricted to no more than five degrees of freedom. Five degrees are needed for the 2Rc mask: two annular rings radia and opaque center. Therefore, a direct search over all possibilities provides the global maximum, and is faster than the simulated annealing method for example, which is stochastic in nature.

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]. 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.

Throughout the direct search, the normalized radia increments were chosen to be 2%, i.e., the normalized radius was divided into 50 increments, and the search has been done over all possibilities. In view of the circular symmetry, it was sufficient to calculate only a radial cross section of the MTF.

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

$$\mathrm{where}:\rho =\sqrt{{u}^{2}+{v}^{2}};\phantom{\rule{.2em}{0ex}}\psi =a\u20442$$

$$p\left(\rho \right)=\{\begin{array}{cc}1;& \rho <1\\ 0;& \mathrm{else}\end{array}$$

The PSF in the presence of a circular, defocus aberrated aperture P′(ρ), is represented by a combination of two Lommel functions, U_{1}(a,w) and U_{2}(a,w), which are defined by the following series [2]:

$${U}_{2}(a,w)=\sum _{n=0}^{\infty}{\left(-1\right)}^{n}{\left(\frac{a}{w}\right)}^{2n+2}{J}_{2n+2}\left(w\right)$$

where J_{m}(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]:

$$\mathrm{where}:w=\frac{2\pi R}{\lambda {d}_{\mathrm{img}}}r\prime $$

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:

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**
_{1}…**r**
_{2N}, where smaller index indicates smaller radius value. Therefore, in Eq. (9) the sub-apertures normalized pupil radia are defined as ρ_{n}=r/r_{n}. P′(ρ_{n},a) has been defined in Eq. (7) and ψ is the maximal phase shift at the pupil edge. The reader should note that P′(ρ_{n}, a)=0 for ρ_{n}>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 [1–2], of the intensity impulse response:

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.

Since the direct search optimization saves the configuration that provides the highest PCF achieved “so far” during the process, the optimization output is the required result. The contrast achieved for frequencies lower than the PCF value, is higher than the desired contrast, by definition.

#### 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]. 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.

On the other hand, the optimal circular symmetric phase mask, examined in the previous section extends the DOF considerably, allowing at the same time 100% light throughput, albeit with lower ACF. Moreover, the robustness of the OTF shape is reduced, meaning that the OTF shape varies for different object positions.

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.

The direct search has been modified to fit this mask family as well. Now the first radius, r_{1}, defines an opaque centered circle, while the rest of the radiuses, r_{2}…r_{2N+1}, are the radial positions of the phase transitions. Similar to Eq. (9), the expression for a generalized pupil mask of this family is provided as follows:

Thereafter, radial cross-sections of the PSF as well as the OTF curves are calculated using Lommel functions; the rest of the direct search procedure remained unchanged. Schematic of the combined binary amplitude and phase mask with a notation of the variables used for optimization is presented in Fig. 2.

## 3. Results

#### 3.1 Optimization results

In this section we summarize the results of the direct search optimization, carried along the lines described in the previous section. Performance comparison is then being made between the “optimal phase masks” and the “optimal binary amplitude and phase masks”, when used for extended depth of field imaging systems with same characteristics.

For a certain required depth of field (DOF) range, defined by the maximal aberration phase ψ_{max} 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, ν_{max}, which assures a certain desired minimal contrast, denoted by C_{d} 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, C_{d}, 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.

Similar results, but for the case where a central opaque circle is included as well, are provided in Tables 3 and 4 respectively. Here, r_{1} 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.

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 ψ_{max}=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.

It is interesting to compare the increase in the frequency ν_{max} 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 C_{d}=5%, while Table 10 shows the same, but for a contrast value of C_{d}=10%.

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 C_{d}=5% and Table 12 for a desired contrast value of 10%.

Comparing Tables 9–10 to Tables 11–12, it is interesting to note that when one increases the desired DOF range (i.e., higher ψ_{max} value), the size of the opaque center radius *increases* in case of the annular amplitude ring (Tables 11–12). Therefore, the light throughput is lower for a larger ψ_{max}. However, in the case of the amplitude *and phase* masks (1Rc and 2Rc), up to a certain ψ_{max} value the size of the opaque center *decreases* with ψ_{max}, due to the presence of the phase rings. Therefore, the light throughput increases with ψ_{max}, up to a certain ψ_{max} 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 C_{d} 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 ψ_{max }value (ψ_{max}=10 for one phase ring and ψ_{max}=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 ψ_{max} 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 C_{d} 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 9–10 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 C_{d}=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.

One notes from Fig. 3 that an optimal phase mask, designed for a low desired contrast (C_{d}) 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 Ψ_{max}=14 are presented. Defocus conditions of ψ=0, 7 and 14 are shown in Figs. 4 (a)–4(c) respectively.

As a final remark, one should note that the design according to the MMSE criterion [23] 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 Ψ_{max}=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.

One notes from Figs. 5(a)–5(c) that the optimal mask with the two phase rings provides lower contrast values. Therefore, for moderate defocus, there is no advantage in using more than one phase ring. The need of more phase rings will be more apparent in the next section, where experimental results that test the predictions of the computer simulations for the various masks are presented, for much larger DOF range. Nevertheless, even for the case presented in Fig. 5, where the DOF requirement is moderate, an electronic restoration filter is beneficial to increase contrast so that a high quality image is displayed.

## 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 (Ψ_{max}@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.

The results in Fig. 6 clearly show the ability of the mask to extend the DOF far beyond the DOF range that is provided by a simple, unprotected aperture. The highest frequency that is observed by the sensor (two pixels period) has a contrast of no less that 5%, and of course, no contrast reversals, for the whole DOF range, contrary to the performance provided by the open aperture where such reversals occur. However, those results assume quasimonochromatic illumination. The performance of the mask for polychromatic illumination is presented in the next section.

#### 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.

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.

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]. 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 [25–27] 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.

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

The MMSE design criterion has been abandoned, since a more convenient criterion, which provides higher resolution (albeit with reduced contrast) has been used. This criterion allowed the search of optimal circularly symmetric binary phase masks.

The OTF curves, obtained by an annular ring aperture applied to the pupil of an imaging system, do not change much throughout the DOF range. This behavior lead us to investigate binary phase structures with an added opaque on-axis circle, and to search for high resolution extended DOF imaging systems with relatively high light throughput and robust OTF behavior. Such OTF robust behavior allows the use of a single digital restoration filter for the whole DOF range.

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.

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