## Abstract

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

© 2010 OSA

## 1. Introduction

Diffraction limited imaging systems are sensitive to misfocus condition effects which are highly dependent on wavelengths. The scope of this article is to present a novel approach which provides extended depth of field (DOF) in systems with polychromatic illumination, maintaining good performance over the entire visible wavelength range, and keeping the same response for all spatial orientations.

This work improves the performance of a system based on a model described earlier [1], which was based on optimizing the performance over a narrow wavelength band. Since the extended DOF solution is based on a phase mask, wavelength variations affect the phase shift introduced by the mask, and as such affects the DOF, the minimum contrast value and the cut off spatial frequency of the acquired images.

Misfocus positioning is inherent in practical systems such as barcode readers, where the operator often cannot accurately focus the imaging system on the barcode (the target). Such difficulties may result in poor identifications.

It is possible to mitigate the problem partially by decreasing the lens aperture size, or by softening its edge by a weighting function or an apodizer [2,3]. These techniques may be followed by a digital restoration stage for enhancement purposes. Reducing the aperture size reduces also the light throughput and the resolution of the imaging system. For practical reasons, such solution is often unacceptable.

Comparing the output of out of focus images obtained with clear aperture pupil diffraction limited imaging system, with a same size apodized aperture imaging system, the latter provides an enhanced output image for a certain misfocus conditions range, albeit in the presence of contrast reduction and decreased optical power throughput.

Hybrid, opto-digital approaches have been devised that overcome misfocus problems by utilizing a non-absorptive phase mask that distorts the image in a predetermined way so that, if followed by post processing digital restoration operations, acceptable images are obtained [4–11]. These hybrid approaches use masks that have a cubic phase along two Cartesian coordinates. Such masks may distort the image considerably, and are inherently not circularly symmetric resulting in a non circular Optical Transfer Function (OTF) and relatively low contrast values. The latter also may exhibit regions of image contrast reversals for relatively low spatial frequencies within the DOF range which make the solution unacceptable. In some instances hybrid systems may be used for applications in which the digital post processing step is not required, such as the case of iris recognition [12,13], where post processing is in any event necessary, for pattern recognition rather than correcting image distortion.

Binary phase shifting apodizers that extend the depth of field where the peak of the point spread function maintains a maximum value have been treated earlier [14–16]. However, design of circularly symmetric imaging systems, with a combination of such features as relatively high cut-off frequencies, acceptable contrast values and a wide DOF for a wide range of illumination wavelengths have not been described yet to the best of our knowledge.

Properly designed imaging systems may be useful for surveillance purposes, computer vision and many other applications, where the designer is willing to ease resolution constraints to about half of the cutoff frequency provided by a diffraction limited lens while exhibiting extended DOF. Such systems do not require auto focus mechanisms, or digital restoration steps. Therefore, their complexity as well as their cost is reduced. Developing design tools for imaging optical systems overcoming an inherent tradeoff between high resolution and depth of field for white light illumination was the main goal of our earlier research.

Several solutions for the design and optimization of incoherent imaging systems with extended depth of field which provide high DOF for certain levels of contrast were earlier presented [1,16–22]. First, the calculation of a mask that minimized the mean square image error within a required depth of field was carried, when considering optical field distribution. Then, by using the fact that the phase provides additional degrees of freedom, increased resolution for the same depth of field was achieved. Next, the resolution has been increased even further by applying the minimum mean square error criterion with respect to the optical intensity, and not the optical field. This criterion is more suitable for incoherently illuminated images, where the degradation of image quality due to diffraction is linear with the intensity.

The chromatic response of systems equipped with a phase mask intended to extend the depth of field of an imaging system were only marginally mentioned in earlier publications. In this study we search for the structure of the phase mask distribution, whereby its effect on each one of the primary colors R, G, B was evaluated. As a result of that, it is possible to derive a better binary phase mask which provides contrast values above a set level for three R, G, B wavelengths, while maintaining all other properties such as circular symmetry, and unlike other solutions mentioned above, without the need of post processing. It has been also shown that satisfying the requirements for three wavelengths (R, G and B) was sufficient to maintain high performance over the entire visible range.

## 2. Theory

After summarizing briefly the basic results for imaging systems illuminated by a monochromatic source, the analysis of the misfocus condition and its influence on the resulting image quality is briefly reviewed in section 2.1. The chromatic aspects of the design of the phase masks are covered in section 2.2.

#### 2.1. Misfocus condition

In the following, “coherent illumination” means monochromatic illumination, harmonically time dependent, and “incoherent illumination” means spatially incoherent quasi-monochromatic illumination, harmonically time dependent, in which the phase of an arbitrary point does not provide any information about the phase at any other point.

For coherent illumination, the output image field distribution can be expressed as [22]:

For isoplanatic systems, straight forward calculation leads to:

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

The wave front within the exit pupil defines the Coherent Transfer Function (CTF) of the imaging system, as expressed in Eq. (3) [23]:

*M*, written in Eq. (1) equals to${d}_{img}/{d}_{obj}$. In the presence of misfocus conditions, the impulse response is the Fourier transform of a generalized pupil function, whose function is the following [23]:

*f*is the focal length and ${d}_{obj}$ is the distance between the object and the lens. Therefore, the CTF for misfocus conditions is given by$\widehat{p}(x,y)$, rather than$\tilde{p}(x,y)$.

The misfocus condition is quantified by a parameter, *ψ*, which is the maximum phase error of the spherical wave front at the edge of a circular aperture that has a diameter D:

For incoherent illumination, the output intensity is expressed by:

*κ*is a factor that normalizes the integral. For monochromatically illuminated images the frequency response of the imaging system is the Optical Transfer Function (OTF):

As long as *ψ*≤ 1, the variations in the OTF are acceptable for perfect quality imaging [24]. For clear apertures, as *ψ* increases, the OTF degrades significantly and as a result of that, imaging is distorted and some features even lost.

The goal of this work is to find a mask that enables imaging condition with a contrast above a minimum value for up to a certain value${\psi}_{\mathrm{max}}$. With a mask located in the pupil of the optical system, acceptable image quality should be maintained over the entire visible wavelength spectrum.

#### 2.2 Chromatic aspects of the Depth of Field

As indicated in Eq. (5), the parameter *ψ* in case of misfocus is a function of the wavelength*λ*:

For polychromatic illumination, the parameter *ψ* varies with the wavelength, so that for 2 different wavelengths:

Although the focal distance *f* of a simple lens varies with the wavelength, most imaging systems use lens assemblies that are chromatically corrected. The correction is mostly done by using complex lenses and chosen lens material. We assumed that *f* in the imaging system under consideration is independent of wavelength variations in the visible spectrum.

In order to provide increased DOF for both positive as well as negative values of *ψ*, masks that consist of several rings, each exhibiting a phase shift of π Radians for a center wavelength were considered. The phase shift is achieved by etching (or depositing) a layer of depth *h* into a glass plate with a refractive index of$n(\lambda )$. As taught in [23] the phase shift is related to the layer parameters as:

Accordingly the phase shift of *π* is achieved with the layer thickness of:

Let's assume that a layer with depth *h* is designed with the aid of Eq. (10b) for a wavelength of${\lambda}_{1}$, but exposed to light with different wavelength ${\lambda}_{2}$. Then, Eq. (10a) applied sequentially for different wavelengths ${\lambda}_{1}$ and${\lambda}_{2}$:

The third element that is influenced by the wavelength is the cut off frequency (COF), which for diffraction limited system, is defined by the f number of the system F#:

The variation of the cut off frequency with the wavelength is thus:

Therefore all 3 parameters (*ψ*, φ, COF) that affect the MTF of an imaging system vary inversely with the wavelength.

#### 2.3 Design approach

The goal is to find a mask for which the worst MTF curve for any wavelength within the entire visible range, provides images with a higher than the acceptable minimum limit, both for contrast levels as well as for resolution. The runs carried with the algorithm cover a large range of *ψ* values, thus assuring that the performance will be satisfactory over the extended depth of field under consideration. In order to achieve the same response for all orientations, we will consider binary masks consisting of circular rings. To begin with, the mask under consideration allows the presence of an opaque center of radius ${c}_{1}$and up to two phase rings bound by radii${r}_{1},{r}_{2},{r}_{3},{r}_{4}$. Considerations for using a larger number of phase rings will be discussed later on. It is appropriate to discuss at this point considerations for choosing the phase levels of the planned mask.

Generally speaking, one would like to design a system with a DOF extending on both sides of a nominal position. According to Eq. (5), positioning the object at either one side or the other of the nominal position will correspond respectively to positive or negative values of *ψ*. One notes that a phase mask that contains phase features of value “0 and *π* only” is identical to a mask that has phase features of “0 and -*π* only”. Therefore, in the mask design process, it is sufficient to investigate the imaging system for the green color only for either positive or negative values pf *ψ* thus reducing the computation time in half. As such, we limited ourselves to the design of a system with large DOF, by considering phase masks that have a phase of either 0 or*π*. Nevertheless, when considering other colors (R, B) the phase is no longer *π* and thus the entire range of DOF (negative and positive values of *ψ*) had to be investigated. The desired Polychromatic Phase Mask (PPM) will be the one providing the highest COF with an acceptable contrast level:

The search for the solution summarized in Eq. (14) has been carried over the wavelengths of 450nm, 550nm and 650nm, which are center wavelengths of 100nm-wide spectral bands, and thus can be used to represent the entire visible region. The contrast has been chosen to be above a level of 5%. This level of contrast has been required earlier [1] in order to be above the average noise level of the optical sensor. Moreover, same level (5%) has been required to be the minimum level by other authors as well [5,6]

In view of the fact that the problem at hand consists of a limited number of variables, the fastest results can be obtained by a direct brute force search, by covering all possible solutions with a fine grid of values, rather than using other techniques such as simulated annealing [18] that is an iterative search of the optimal solution. Although the earlier mask design [1] was optimized for monochromatic illumination with a wavelength of 550nm, it was thereafter utilized for imaging over the entire visible spectrum. The earlier mask will be coined in this paper, as monochromatic phase mask (MPM). It was designed for a maximum defocusing range characterized by |*ψ*| ≤ 15 and a minimum contrast of 5%, and resulted in π phase rings, with normalized radii of 0.62, 0.76, 082, and 0.94, and an opaque center with a radius of 0.2.

The new design of a Polychromatic Phase Mask (PPM) carried in this study, takes into account operation over the entire visible spectrum. Variations of the phase values provided by the mask [Eq. (10) as well as for *ψ*
Eq. (5)] for various wavelengths in the entire visible range have been taken into account when searching for the best solution.

## 3. Search algorithm

It may help the reader if we outline in more detail the procedure followed:

**a)** We set 5 dimensional array of parameter values${c}_{1},{r}_{1},{r}_{2},{r}_{3},{r}_{4}$ to cover all possible values between 0 (center of the aperture) and 1(normalized aperture size), with an increment of 0.01. The radii represented by ${c}_{1},{r}_{1},{r}_{2},{r}_{3},{r}_{4}$ have to satisfy the condition${c}_{1}\le {r}_{1}\le {r}_{2}\le {r}_{3}\le {r}_{4}$. We start by assuming all radii to be 0 and increase their values thereafter, up to the entire aperture while maintaining the validity of the inequality just mentioned.

**b)** For each set of ${c}_{1},{r}_{1},{r}_{2},{r}_{3},{r}_{4}$ values, the MTF of the system for a series of values of *ψ* has been calculated. Only positive *ψ* values have been examined due to the arguments presented in section 2.3. For each chosen contrast value, the value of *ψ* was allowed to vary up to${\psi}_{\mathrm{max}}$with an increment of 0.1. The MTF was calculated for the three colors R, G, B characterized by representative wavelengths${\lambda}_{R},\text{\hspace{0.05em} \hspace{0.17em}}{\lambda}_{G},\text{\hspace{0.17em}}{\lambda}_{B}$.

**c)** For each MTF curve the highest frequency up to which the contrast was above 5% was recorded as${\upsilon}_{R},{\upsilon}_{G},{\upsilon}_{B}$. The smallest of those 3 values is retained as it defines the spatial frequency that one associates with the respective *c, *
${r}_{1},{r}_{2},{r}_{3},{r}_{4}$ values.

**d)** One now associates with each choice of values ${c}_{1},{r}_{1},{r}_{2},{r}_{3},{r}_{4}$ a single spatial frequency. The highest of those spatial frequencies that has been obtained when running the radii over the entire set, corresponds to a certain set of${c}_{1},{r}_{1},{r}_{2},{r}_{3},{r}_{4}$. We call this set of values as providing the best obtainable mask. It is thus a mask exhibiting two phase rings [${r}_{1}\text{\hspace{0.17em}}to\text{\hspace{0.17em}}{r}_{2}$] and [${r}_{3}\text{\hspace{0.17em}}to\text{\hspace{0.17em}}{r}_{4}$] and a central stop, which now provides the best response of the system for the range of out of focus conditions $-{\psi}_{\mathrm{min}}$to${\psi}_{\mathrm{max}}$. We chose${\psi}_{\mathrm{min}}={\psi}_{\mathrm{max}}=8$.

The algorithm just described in order to find the PPM is sketched in Fig. 1 :

Covering all possible options, with the increment defined above, a full 5-D array is set. The algorithm doesn't have an interruption mode or exit criteria since it has to go over all those possibilities. The decision about the mask's choice of values occurs at the conclusion of all runs by a simple array oriented code.

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

The same search was repeated for masks with 1 ring, 3 and 4 rings. A mask with a single phase ring provides extended DOF till$\psi =6$and allows imaging with a maximal spatial frequency that is 0.25 of the COF. Masks with 3 and 4 phase rings provide extended DOF till $\psi =8$ and a minimum spatial frequency of 0.28 of the COF, not much different than the results obtained with a 2 rings' mask, but with reduced contrast. Because the resulting masks with more than 2 rings don't provide better performances than the 2 rings' mask, the latter was chosen.

The mask provides indeed a phase shift of π for a wavelength of 550nm; nevertheless the phases for the red (650nm) and blue (450nm) wavelengths are 0.84π and 1.22π respectively. The mask that was optimized [1] for only a single wavelength (i.e. green) provides unacceptable performance for blue and red wavelengths as shown in Fig. 3 . For instance, the minimum spatial frequency of the MPM is only 0.25 of the COF, for the blue channel at$\psi =-8$.

On the other hand, Figs. 4 (a)-(d)
exhibit the MTF curves obtained with the PPM for same defocus positions of *ψ* = 0, −2, −5 and −8 respectively. The contrast value of 5% is marked by a black horizontal line. The normalized value of 2 is the COF evaluated for the green wavelength. The corresponding COF for the Blue wavelength is 2.4 and for Red wavelength 1.6. Using the same scale for all wavelength representations, the COF of the R, G, B respectively occur at 1.6, 2.0 and 2.4, as evidenced in Figs. 3-5
.

The results presented in Fig. 4 clearly show the ability of the polychromatic phase mask to extend the DOF far beyond the DOF range exhibited by a clear aperture. The highest frequency, which is 0.45 of the COF, has a contrast of no less that 5%, and of course, no contrast reversals, for the whole DOF range, as well as for all wavelengths R, G, B.

Since the algorithm of the search was based on 3 wavelengths only, it was necessary to check the validity of the result for the entire visible spectrum (400-700nm). This was done with an increment of 1nm in order to make sure that there are no gaps.

It is instructive to present the MTF curves for a system equipped with a PPM mask (Fig. 5a) for an out of focus condition of $\psi =4$ and compare it to that of a clear aperture. This was done for a large number of wavelengths between 400 and 700 nm, proving once again the consistency of the results. For comparison the MTF for a clear aperture for same out-of-focus condition is shown in Fig. 5b.

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

The PPM provides a simple solution for extending the depth of field of imaging systems while maintaining reliable color rendering. The contrast at each wavelength is above a value of 5% for high frequencies and above 25% for low frequencies.

The PPM provides better results, for color imagery in particular. To visualize the performances along the DOF for the 3 primary colors R, G, B, a new graph was created, with *ψ* along the X-axis and the highest spatial frequency for which the MTF exhibits a contrast above the 5% along Y-axis [Fig. 6
]. Out-of-focus imagery can be analyzed in a general manner by using the parameter *ψ* to identify the amount of deviation away from focus. Trying to express this in terms of the magnification factor *M* eliminates the possibility of treating it in a universal manner, and thus it was not attempted here. One can express the dependence on the magnification *M* only for specific systems, whereby the${d}_{img}$, the pupil *D* and the focal distance *f* are provided. Therefore throughout this paper out-of-focus was analyzed in term of *ψ* only.

Another advantage of the PPM design is that when one does not require large DOF, the mask can be tailored to be efficient for a lower range of *ψ*, say from 0 up to 5, resulting with higher contrast values at low frequencies. One notes that the contrast thus obtained, approaches the one provided by a clear aperture for a low out-of-focus condition

On the other hand, when imaging capability over an extended DOF is desired, requiring large values of *ψ*, there is small difference between the results obtained with the two masks. The advantages of the PPM mask are thus evident when the DOF range is not excessively large. The contrast level for the green color with the PPM is higher than that with the MPM for low values of *ψ*, although the MPM was designed for Green wavelength only. This is due to the fact that the MPM mask was designed to provide improved results for a region extending up to large values of *ψ*. The images provided in Fig. 8
were recorded in a position corresponding to small values of *ψ*. Thus, it is not surprising that the PPM provides better results than those obtained with the MPM even for the green color, when the object is near the in-focus position, i.e., small values of $\psi .$One can easily notice that the colors are distorted in the images obtained with MPM. The reason is due to the fact that the MPM mask was designed for a very low contrast specification of 5%. As such the features appear as a slight modulation over an average background. To the eye this effect appears as a poor contrast image over a “distorted average color.

A visual comparison was carried by simulating images obtained with a “colored -spiral” object [Fig. 7 (a)
], that becomes highly distorted [Fig. 7 (b)] when the object is in an out of focus condition of *ψ* = 5. Not only resolution is limited, but also color distortions occur as it can be readily seen. Simulation results obtained with a system equipped with an MPM [Fig. 7 (c)], or PPM [Fig. 7 (d)] are shown:

## 4. Experiments with extended depth of field imagery

Experiments have been carried in order to verify the ability of a polychromatic phase mask to increase the DOF by either a low-cost all-automatic digital still camera as well as by a high quality video camera. The requirements of the imaging systems as well as the characteristics of the mask that has been used are presented in section 4.1. Section 4.2 is devoted to the presentation of the experimental apparatus as well as the experimental results.

#### 4.1 Experimental Set-Up

A low-cost all-automatic camera [MUSTEK^{®} DV5200], equipped with a CMOS detector with 1200x1600 pixels (2 mega pixels) was utilized. The camera lens consisted of five glass elements with an effective focal length of 8.5mm. A Field of view (FOV) of 50 degrees along the sensor diagonal, and pixel dimension of 4$\mu m$, were measured experimentally. The camera lens was set in an in-focus position, for an object located 30 cm from the lens.

A CCD video camera [UEYE 2210], equipped with a 1/3” CCD detector with resolution of 1280x1024 pixels (1.5 mega pixels), with frame rate of 75fps and an exposure time of 5msec was used as well. The camera lens was Computar^{®} M1614, with a focal length of 16mm. The camera has a field of view (FOV) of 45 degrees along the sensor diagonal and pixel dimension of 4$\mu m$.

The imaging system was required to operate from a distance of 15cm to infinity and provide a minimal contrast value of 5%. A 2mm diameter aperture was chosen, compatible with the aperture of compact imaging systems used in cellular phone cameras. Encouraged by the simulation results, presented earlier, a binary phase mask with two phase rings was considered as a candidate for experimentation. The design of the required mask was carried according to the considerations that have been presented in sections 2.1 and 2.2.

The normalized spatial frequency for the desired contrast value was 0.28 whereby a value of 1 is the diffraction limit cutoff. This COF value matches the Nyquist frequency of the detector (the highest frequency that can be detected by the sensor) for the wavelength under consideration.

#### 4.2 Experimental results

Some experimental qualitative results were obtained, in order to demonstrate the performance improvement when the PPM equipped imaging system is used, instead of the MPM, designed for a monochromatic illumination.

In the field experiments, a Mustek® DV 5200 camera, as well as a UEYE 2210 CCD camera were used. In both cases the mask was mounted externally on the lens so that it can be removed easily for comparison purposes. The experimental apparatus is portable, and can be taken outdoors, or it can be used on an optical bench indoor.

In Fig. 8, the chosen objects are polychromatic (red car, green tree, etc.) and composed of a variety of spatial frequencies. The images were taken outdoors. The image taken with the PPM has more details than the one taken with the MPM. In particular, this is evident when one observes the red car in the foreground. The image is blurred when the MPM is used and pretty sharp with the PPM. The red car is at approximately *ψ* ~-4 and the building in the background is at approximately *ψ* ~-7.

Figure 9
provides the results of color images acquired by an imaging system in an out-of-focus condition (*ψ* = 4) for the two masks under investigation.

In Fig. 10
one can examine and evaluate the improved contrast achieved with the polychromatic mask for all three colors (blue, green, red) by using an object consisting of a chirped Ronchi ruling. The PPM provides better contrast for low spatial frequencies compared to those provided by the MPM. This is true even for the wavelength of 550nm, for which the MPM was optimized. The images displayed in Fig. 10 (b) and (c) were obtained with a gray scale sensor that had the capability of imaging with higher resolution; since the sensor was monochrome the images have a “gray” appearance. This allowed also overcoming the automatic gain control limitation which is governed by the green channel in a normal camera mode and which does not allow direct detection and evaluation of the other color channels. The gray images displayed have thus the same gain control for all wavelengths in the visible spectrum. In order to examine the channel response individually, color filters were used to select the three color bands sequentially; the images in Figs. 10 (b) and (c) were obtained one color at a time and stitched together in the displayed result. The target is located at approximately *ψ* ~5. The exposure time was 5msec. The average contrast at the R, G, B wavelengths obtained with a system equipped with MPM mask was 0.07, 0.1, and 0.1 respectively. When replacing the MPM mask with the PPM one, the contrast values were 0.2, 0.3, and 0.3 respectively. The noise level is below 5%, thus the PPM performance is sufficient since it provides a minimum contrast level of 5%. The last experiment was repeated for other out-of-focus locations up to *ψ* ~7 and other levels of illuminations. In all cases, the PPM provided images with higher contrast.

The results presented in Fig. 7-10 exhibit qualitatively the advantage of the PPM over the MPM. It is not possible to obtain quantitative results from such pictures.

## 5. Conclusions

We have demonstrated the performance of a pupil mask for extended DOF imaging systems that provide high resolution and high light throughput for polychromatic illumination. This mask enables the user to utilize a DOF range for the whole visible spectra, although the mask was designed for providing highest response for three representative wavelengths only. The performance of the imaging systems has been verified experimentally, and the results support the theory.

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