The optical transfer function of a cubic phase mask wavefront coding imaging system is experimentally measured across the entire range of defocus values encompassing the system’s functional limits. The results are compared against mathematical expressions describing the spatial frequency response of these computational imagers. Experimental data shows that the observed modulation and phase transfer functions, available spatial frequency bandwidth and design range of this imaging system strongly agree with previously published mathematical analyses. An imaging system characterization application is also presented wherein it is shown that the phase transfer function is more robust than the modulation transfer function in estimating the strength of the cubic phase mask.
©2012 Optical Society of America
The emergence of computational imaging has enabled visual information extraction in ways that were previously thought to be infeasible. The underlying fundamental principle of this field calls for synergistic co-design of optical and digital subsystems of an imaging architecture to achieve data collection that would otherwise be unrealizable by traditional methods. Among a multitude of domains within this field, the concept of wavefront coding has attracted much attention within the computational imaging community [1–18]. Pioneering work in this area by Dowski and Cathey was motivated by a desire to extend the depth of field of incoherent imaging systems . This extension is accomplished by inserting a phase mask into the pupil plane of an imaging system to render a modulation transfer function (MTF) which, while reducing the dynamic range, remains virtually unchanged over a much larger depth of field than its traditional counterpart. In addition, this MTF contains no regions of nulls within its passband. These special traits allow for subsequent digital reconstruction of captured images via a simple linear restoration filter to yield final images with greatly enhanced depths of focus. Wavefront coding has since been investigated for use in myriad applications such as aberration correction [7,19], form factor enhancement [8–10,20], passive ranging , athermalization , reduction in optical design complexity , iris recognition , super-resolution  and zoom lens miniaturization  to name a few.
In their seminal paper , Dowski and Cathey presented an approximate solution for the MTF of a wavefront coding imaging system containing a cubic phase mask (CPM), based on a stationary phase approximation approach. In this approximation, the MTF was shown to be independent of defocus, thereby enabling the desired extended depth of field. However, subsequent simulations  and experimental evaluations [28,29] revealed that the MTF of CPM imagers is not entirely unrelated to the magnitude of defocus present in the system. On one hand, the observed MTF variations in these cases were quite small and limited to the highest spatial frequencies within the optical passband. These deviations were hence reasoned to be of minor consequence to the applications at hand, given that attendant defocus values were rather modest in magnitude. On the other hand, when a need arises to operate under large defocus magnitudes, the ensuing MTF degradation would then have to be taken into consideration during system design in order to assess its impact on imaging performance.
In previous work [8–10], we mathematically examined the spatial frequency response of a CPM wavefront coding imager as a function of defocus and derived from first principles, an exact expression for the optical transfer function (OTF) that applied to all defocus magnitudes. It was shown via this expression that defocus serves to reduce the usable spatial frequency bandwidth, a behavior independently predicted by Muyo and Harvey . It was further shown that this bandwidth reduction limited the range of defocus values to a “design range” within which the imaging system could usefully operate .
The aforementioned theoretical results are an important step in understanding the behavior of CPM wavefront coding imaging systems. In the spirit of the scientific method however, theoretical formulae must undoubtedly withstand the scrutiny of practical experimentation if the former is to be established beyond question. While the MTF of CPM imagers has been experimentally measured for zero and low defocus magnitudes [28–30], to our knowledge, comprehensive experimental corroboration of the spatial frequency response across its entire design range as given in  has not been previously undertaken. In , we described efforts to characterize the MTF and presented initial results for the lower half of the spatial frequency spectrum that showed a trend similar to theoretically predicted behavior. In this paper, we present detailed experimental evidence of the exact OTF of CPM wavefront coding imaging systems for defocus values spanning the entire design range and encompassing all spatial frequencies up to the diffraction limit. It is shown in this work that various facets of the theoretically predicted behavior of CPM imaging systems are substantiated by experimental data, the aggregate of which validates the previously published theoretical findings presented in . Additionally, we herein propose a method to estimate a design parameter of the cubic phase mask, namely its strength, via the phase transfer function (PTF) of this wavefront coding system.
Original contributions of this work include experimental proof that (1) the system MTF of a CPM imaging system across the operating range of defocus values strongly agrees with the mathematical expression for the exact MTF as a function of defocus; (2) defocus-induced reduction in the available spatial frequency bandwidth closely follows theoretically predicted behavior; (3) the measured design range of a CPM imager agrees with the mathematical analysis; (4) the measured PTF has close agreement with theoretical calculations when the MTF remains above the noise floor; and (5) the measured PTF is more robust than the measured MTF in estimating the strength of the cubic phase mask. It is expected that these laboratory results would bolster application-specific design and characterization efforts in CPM wavefront coding imaging systems.
The remainder of this paper is organized as follows: Section 2 highlights key results from previous theoretical analysis of the OTF of cubic phase mask imaging systems as previously presented in . Section 3 elaborates upon the experimental setup used to measure the spatial frequency response of a CPM imaging system and provides a rationale for the choice of components used in the process. Section 4 discusses experimental results and the evidence in support of the analytically predicted behavior. Section 5 concludes the paper with a summary of our findings.
2. Overview of the frequency response of cubic phase mask wavefront coding systems
This section provides a concise overview of the key findings of  and highlights important properties whose experimental proofs are presented in this paper. We begin with a brief review of the pupil function of a CPM wavefront coding imager followed by its MTF as a function of defocus.
2.1. The pupil function
A CPM imaging system is realized by introducing a phase mask or function θ(x, y), into a rectangular clear pupil to modify the incoming wavefront (and hence the PSF). Here, (x, y) represent the normalized pupil plane coordinates. For a cubic phase mask, this function is given by θ(x, y) = α(x3 + y3), where α is a positive constant. When considered in conjunction with the wavefront error introduced by defocus, the unit-power, generalized pupil function of a CPM imaging system may be expressed along one dimension (1D) as 
A one-dimensional analysis of the imaging system suffices since rectangular separability applies to the aperture in this case. The parameter α denotes the strength of the cubic phase mask. It is calculated as [10,32]Eq. (1) is explained in . The term ψ is the defocus parameter, which quantifies the extent of defocus experienced by the imager and is given by [1,10]
2.2. The modulation transfer function
The approximate OTF of CPM wavefront coding imaging systems as obtained using the stationary phase method is given by Eq. (4), the MTF appears to be invariant to the defocus parameter ψ. However, as was subsequently shown in [8–11], this MTF invariance in real systems is not unqualified; rather, the effect of increasing the magnitude of ψ serves to reduce the available spatial frequency bandwidth of the imager. In previous work, we presented an analytical expression for the exact OTF of CPM wavefront coding imaging systems from first principles, which was obtained as 
In the above equation, the terms C() and S() denote the Fresnel cosine integral and Fresnel sine integral respectively, whose operands are 
Simultaneously inspecting Eq. (4) and Eq. (5) reveals that the latter contains an extra term, namely a complex quantity made up of Fresnel integrals. This term is a function of defocus and therefore affects the magnitude as well as the phase of the OTF. In section 4.1, experimental results that corroborate the theoretical MTF described by Eq. (7) are presented. It was further shown in  that increasing the magnitude of defocus caused a reduction in spatial frequency bandwidth of these systems. A quantification of this reduction is briefly reviewed next.
2.3. Available spatial frequency bandwidth
For a given defocus magnitude, there exists a location along the spatial frequency axis within the diffraction-limited passband where the exact MTF Mt(u,ψ) drops to one-half the defocus-independent approximate MTF Ma(u,ψ) = |Ha(u,ψ)|. Defocus thus has a low-pass filtering effect on the frequency response by reducing the extent of the MTF to a cutoff frequency whose value is [10, 11]
In section 4.2, it is shown that the estimated value of the cutoff frequency uc obtained from experimental data closely follows the trend predicted by the theoretical description of Eq. (8). It is also clear from Eq. (8) that uc must obviously remain within the interval [0, 1] which means that two conditions must be satisfied, namely (a) |ψ| ≤ 3α and (b) α > 0. In practical scenarios however, the utility of an imaging system is typically exhausted before uc becomes zero, that is, before |ψ| reaches 3α. A more practical definition of the limit of the defocus magnitude at which the system ceases to usefully operate is hence reviewed next.
2.4. The ambiguity function and the design range
The ambiguity function (AF) of an incoherent imaging system is defined as a polar display of the OTF with the defocus parameter as a variable . The AF is related to the OTF as
For a specified value of ψ, the OTF is thus the projection onto the horizontal u axis of the radial line in the AF plot whose slope is 2ψ/π and intercept is zero. The 2D AF hence contains information about the 1D OTF for all defocus values. The AF is a useful tool to illustrate the design range, i.e., the range of defocus values within which an imaging system can effectively operate. This design range encompasses all defocus values ψ that satisfy the condition –|ψm| ≤ ψ ≤ + |ψm|, where |ψm| is a maximum defocus magnitude given by 
The variable t in the above expression is a threshold parameter whose magnitude is typically taken to be between zero and unity. The value of t is used to test the maximum magnitude of either a(u) or b(u) of Eq. (6) depending on whether ψ is positive or negative respectively, to identify values of ψ under which the Fresnel integrals in Eq. (7) fail to achieve a stationary value of ±½. The significance of t has been further elaborated in . In section 4.3, the validity of the above expression is established by analyzing the theoretical and measured moduli of the ambiguity functions and observing the distribution of power within the bounds set by Eq. (10). It is therein shown that the design range defined by Eq. (10) accurately describes the operating limits in terms of the percentage of power contained within these limits as a fraction of the total power in the AF magnitude plots.
2.5. The phase transfer function
In addition to the MTF, Eq. (5), permits a straightforward evaluation of the PTF of CPM wavefront coding imaging systems as
The first two terms in the above equation are contained within Eq. (4), whereas Eq. (5) reveals a third contribution to the PTF, namely the arctangent term in Eq. (11). Given that both the numerator and the denominator of the argument of this term rapidly oscillate about unity within the usable bandwidth, this contribution to the PTF is a phase value that oscillates about π/4 radians except at u = 0, where it becomes zero. It is apparent from Eq. (11) that the PTF exhibits a strong dependence on defocus. Variations in the PTF as a function of defocus thus cause disparities in spatial frequency shifts for objects at different distances in a scene. This disparity in turn results in image artifacts when the captured image is restored with a single restoration filter. These effects have been previously studied in , wherein an approximate expression for the PTF that does not rely on Fresnel integrals has been given. In section 4.4, the experimentally measured PTF is shown to closely resemble the theoretical prediction given by Eq. (11). It is further shown in section 4.5 that the PTF is more reliable than the MTF in estimating the strength α of the cubic phase mask.
In view of the aforementioned characteristics of CPM wavefront coding imaging systems, the goal of this paper is to present experimental validation of the mathematical predictions derived from the exact OTF given by Eq. (5) and previously presented in .
3. Experimental setup
A simple optical design that nominally consisted of an imaging lens, a cubic phase mask and a digital sensor was used in this experiment. The spatial frequency response was measured for various defocus magnitudes across a wide range of working distances by incrementally moving the sensor with a motorized actuator and measuring the system OTF at each location. Figure 1 shows the experimental setup used in the data collection process.
The optical subsystem consisted of a half-inch, achromatic element (Thorlabs AC127-075-A) as the sole imaging lens in the system. The lens was coated to operate in the visible spectrum (400nm to 700nm) and had an effective focal length of 75mm. Given that the original principle behind the CPM imager involved a rectangularly separable system, a 7.5mm square aperture enclosure was designed to house the imaging lens, thus yielding an F/10 imager.
The cubic phase mask used in this system was a model CPM127-R60 element provided by OmniVision Technologies Inc. This mask was designed for optimal operation at 550nm and sported an overall lateral phase profile variation of 60 waves across the diagonal of a square whose side equaled the diameter (12.7mm) of its working surface. Since this overall variation included both transverse directions (along x and y axes), the end-to-end variation along one dimension was one-half of this value. It is recalled that the extent of the 1D variation across the open aperture is used in calculating the strength of the phase mask. The maximum surface sag ξ′ = ξ/λ of the phase mask along 1D across its 12.7mm face was therefore 30 waves for λ = 550nm in free space. Since the aperture width was 7.5mm and the surface profile of the phase mask varied as a cubic function, the effective maximum sag along one dimension in multiples of wavelengths across the face of this aperture was ξ/λ = 30 × (7.5/12.7)3 = 6.179 waves. Multiplying this quantity with 2π as per Eq. (2) then yielded the effective strength of the phase mask as α = 38.822 for this experiment.
A Sumix SMX-M95M monochrome CMOS detector with 2.2μm square pixels and unity fill factor was used as the imaging sensor. The optical cutoff and Nyquist spatial frequencies were calculated to be ηo = 180cyc/mm and ηN = 227.27cyc/mm respectively, thus ensuring an absence of aliasing. To keep out stray light as the sensor was incrementally moved forward, a telescoping baffle was used between the imaging lens and the sensor, which was designed to fold as the working distance was reduced.
An on-axis point source was placed at a distance of 7.5m from the imaging system to yield a magnification M of approximately –0.01. The observed image of this point source yielded the PSF, which was then directly used to measure the OTF. This point source was realized by back-illuminating a 100μm pinhole by a white-light LED (Thorlabs model# MCWHL2). Given that theoretical analyses of the spatial frequency response assumed a single wavelength, a color filter (Thorlabs model# FB550-10) with center wavelength and full-width-half maximum passband of 550nm and ±10nm respectively was included in the imaging chain. This filter helped emulate monochromatic illumination while assuring near-diffraction-limited imaging performance by the lone imaging lens.
The working distance za was then incrementally reduced by moving the sensor towards the lens in steps of 10μm across a defocus range far exceeding the imaging system’s operating limits. The range of working distance variation was 27.285mm which, when expressed relative to the in-focus working distance zi was –15.7mm < zi – za < 11.6mm. In terms of defocus, this range of distances from best focus corresponded to 4.685α > ψ > –4.937α. A linear motorized actuator (Zaber actuator model# T-LA28A) with a 28mm maximum travel length was employed to move the sensor.
For purposes of noise mitigation, several PSF images were captured at each working distance and then averaged. In order to ease computational burden, each averaged image was then cropped to a smaller region of interest (ROI). This ROI consisted of an odd number of pixels along both the horizontal and vertical directions. An important note is that the ROI window was centered about the peak intensity pixel of the in-focus PSF and the ROI coordinates were kept fixed throughout the experiment. The latter step is crucial since the goal of the experiment was to collect information about the complex OTF, that is, to measure both the MTF and the PTF of the system. As the PTF reveals information on the extent of spatial frequency shifts including a global shift of the PSF, a fixed set of ROI coordinates becomes necessary to ensure a reliable reference system against which such global shifts may be compared. The reason for maintaining odd numbers of pixels along each dimension of the ROI window was to minimize the accumulation of a linear phase error when measuring the PTF via a Fourier transform operation.
For each working distance, a 1D PSF was then computed by either extracting a vertical slice of the cropped PSF image or averaging pixel intensities across all columns within the ROI. In the former method, a 3-pixel-wide slice was selected such that the central column of this slice matched the image column containing the peak intensity pixel within the ROI. Mean intensities across the three slices were then computed to yield the 1D PSF. The latter method of averaging intensities across all columns helps to further reduce noise; however, slice extraction is preferred when hard-to-eliminate spurious specular reflections within the image could potentially cause artifacts in the measured PSF. A slice of the overall PSF could then avoid such artifacts, provided that these spurious locations do not lie within the columns being extracted. On the other hand, when the defocus is so severe that noise dominates over the signal, the peak intensity location within the image ROI no longer reliably marks the PSF peak and one would then need to resort to column averaging to obtain the 1D PSF.
In this experiment, the former method of slice extraction was utilized when the working distance satisfied the condition –3α < ψ < 3α and the latter method of all-column averaging was used for all other locations outside this range. Performing a Fourier transform on the 1D PSF then yields the 1D OTF of the imaging system. It is recalled that a qualification for a 1D slice to be adequate for analysis is that the PSF be rectangularly separable. This was indeed the case for this experiment, as the inset within Fig. 1 attests. Figure 2 shows various MTF and PTF data at the in-focus plane (ψ = 0) against the spatial frequency axis region of 0 ≤ u ≤ 1.
The theoretical MTFs Ma(u) = |Ha(u)| and Mt(u) = |Ht(u)| in Fig. 2 exclusively pertain to the imaging system’s optics, whereas the measured MTF Mm(u) encompasses the modulation transfer function of the imaging system as a whole. Therefore, a more accurate comparison would be of the latter with the theoretical MTF lowered by factors such as the pixel MTF, effective optical SNR and other sources of noise that contribute to the overall system response. While an extended treatise on the effective optical SNR and other noise factors is beyond the scope of this paper, the pixel MTF Mp(u) is relatively straightforward to evaluate and is hence included while comparing theoretical versus observed data in subsequent discussions. For a square pixel of pitch and size p, the pixel MTF is given by Fig. 2 shows Mp(u) for this experiment, as well as the product Mt(u) × Mp(u). From Fig. 2(a), it is seen that Mm(u) is slightly lower than Mt(u) × Mp(u), as naturally expected of the aggregate system MTF.
On the PTF plots of Fig. 2(b), it is seen that the approximate PTF Θa(u) and the exact theoretical PTF Θt(u) are nearly identical, whereas the measured PTF Θm(u) deviates from the first two curves. This deviation is primarily due to the fact that the ROI window was centered about the peak intensity pixel of the in-focus PSF. However, it has been shown that for an on-axis point object, the peak intensity of the CPM imaging system’s PSF is offset from the optical axis even when the defocus is zero . Centering the ROI about the peak intensity pixel is then akin to globally shifting the PSF from its true position, which in turn manifests as a linear phase error in the frequency domain . Since the ROI window was kept fixed at all working distances relative to the in-focus plane, this error remains fixed across all defocus values. The measured PTF at any working distance can thus be corrected via a linear compensation function. The dashed curve in Fig. 2(b) shows the result of such a linear phase correction Θc(u) = muηo applied to the measured PTF, where m is the slope of this correction term. In this experiment, this slope was estimated to be m = 0.04 radian-mm/cyc or in the spatial domain, a global PSF shift of 6.366μm or 2.89 pixels based on the analysis in .
The single measurement for the in-focus plane seen in Fig. 2 attests that the experimental OTF closely follows the theoretically predicted exact OTF given by Eq. (5) for ψ = 0. In the following section, it is shown that this prediction holds across all defocus values of interest. The measured available spatial frequency bandwidth as a function of defocus and the design range of the imaging system are also compared with theoretical calculations.
4. Experimental results and discussion
In this section, experimental results of a CPM wavefront coding imager are presented showing the MTF and cutoff frequency uc as a function of working distance (i.e., defocus). Experimental findings supporting the validity of the theoretical design range defocus limit |ψm| are also included. Additionally, theoretical versus measured PTFs are compared across the design range and an imaging system characterization scenario is presented wherein the value of α is estimated from the MTF as well as the PTF.
4.1. The modulation transfer function
The validity of the exact expression for the MTF of a CPM imaging system as given by Eq. (7) was tested by generating three-dimensional (3D) plots of the MTF as a function of normalized spatial frequency u and distance from best focus zi – za. It is herein recalled that negative values of the defocus parameter ψ correspond to positive values of the distance from best focus, that is, the value of ψ decreases as the sensor is moved towards the lens.
Figure 3 shows a comparison of the theoretical approximate, theoretical exact and measured MTFs of the CPM wavefront coding imager used in this experiment. As mentioned earlier, the approximate and exact theoretical MTFs Ma(u,ψ) and Mt(u,ψ) respectively, are lowered by the pixel MTF Mp(u,ψ) = Mp(u) before comparing them to the measured MTF Mm(u,ψ). The plots in Fig. 3 are shown for a defocus range of 3α > ψ > –3α, corresponding to a distance from best focus of roughly –9.34mm < zi – za < 7.5mm. This set of working distances encompassed the entire design range of this imaging system.
It is clear from Fig. 3(a) that as expected, the approximate MTF Ma derived from Eq. (4) does not reveal the effects of defocus on the spatial frequency response of a CPM imaging system. On the other hand, the exact MTF Mt given by Eq. (7) and represented by Fig. 3(b) predicts a reduction in spatial frequency bandwidth when the magnitude of defocus is increased. Inspecting Fig. 3(c), it is seen that apart from noise and a slight overall contrast reduction, the measured MTF Mm accurately follows the theoretically predicted frequency response embodied by Fig. 3(b). As noted in the previous section, the contrast reduction seen in Mm as relative to Mt × Mp is to be expected, given that the quantity being measured was the overall system MTF. The experimental evidence presented in Fig. 3 hence corroborates the theoretical prediction of the modulation transfer function of a cubic phase mask wavefront coding imaging system as given by Eq. (7) and previously presented in .
4.2. Available spatial frequency bandwidth
Having experimentally proved that Eq. (7) accurately describes the effects of defocus on the MTF of a CPM imaging system, the next step is to examine the bandwidth reduction imposed by defocus upon this MTF. Figure 4 compares the theoretical expression for the cutoff spatial frequency uc as given by Eq. (8) versus its experimentally estimated counterpart. In order to better understand this comparison, the approach used to estimate the value of uc from the experimental data is explained herein.
For a given value of defocus ψ, the cutoff frequency uc is defined as the location on the spatial frequency axis where the exact theoretical MTF Mt drops to one half the approximate MTF Ma . Now consider the definition of the diffraction limit, which states that the MTF drops to zero at the optical cutoff frequency. The approximate MTF Ma however, does not accurately reflect this fact as seen in Fig. 2(a) and Fig. 3(a). On the other hand, at ψ = 0, the entire diffraction-limited bandwidth is in principle available to the imaging system, and uc as given by Eq. (8) is unity. Nonetheless, the exact MTF Mt drops to zero at the optical cutoff as expected, rather than reduce to one half the height of Ma. Hence for very low defocus magnitudes, a more accurate approach would be to identify the height of Mt × Mp at the theoretical value of uc and then locate the spatial frequency at which Mm drops to this threshold value. This approach continues to apply as |ψ| is increased and hence may be employed across the entire design range. The measured values of uc shown in Fig. 4 were thus obtained using this method.
From Fig. 4, it is seen that the theoretical and estimated values of uc for a given distance from best focus exhibit very similar trends, thereby strongly supporting the validity of Eq. (8). However, inspecting the left-hand side of Fig. 4 reveals that the computed values of uc from measured data are somewhat lower than its theoretical counterpart. This is because for a given aperture, the effective optical SNR is lower for longer working distances and therefore Mm is observed to be slightly lower than Mt × Mp. The value of Mm therefore drops to the threshold value of Mt × Mp evaluated at the theoretical prediction of uc, at a spatial frequency that was fractionally lower than that predicted by theory. When |zi – za| and hence |ψ| approaches zero, both Mt × Mp and Mm converge to the optical cutoff frequency and therefore the theoretical and estimated values of uc converge to unity at zi – za = 0, as seen in Fig. 4. As zi – za increases, the peak PSF intensity and hence the optical SNR increases, thereby causing Mm to closely follow Mt × Mp especially at lower spatial frequency values of uc, as seen on the right-hand side of Fig. 4. In the next subsection, this observation is illustrated via the magnitude of the ambiguity function as obtained from experimental data.
4.3. The design range
From Eq. (8), it is seen that the bandwidth cutoff reduces to zero as the magnitude of defocus |ψ| approaches 3α. However, as seen in Figs. 3(b) and (c), the MTF does not summarily drop to zero at |ψ| = 3α for non-zero values of spatial frequency. Yet, the frequency response under such severe defocus conditions is so poor that it is effectively useless for most practical imaging scenarios. It may be observed via the AF that as |ψ| approaches 3α, the MTF rapidly deteriorates and therefore, defining a limiting defocus magnitude |ψm| as given in Eq. (10) would be helpful in determining the design range, i.e., the system’s useful operating limits. Depending on the imaging application at hand, it may be desirable to limit bandwidth reduction to an acceptable lower bound of uc. This choice of uc in turns specifies the value of |ψm| and by extension, the threshold parameter t in Eq. (10). For purposes of the following illustration, the value of t was chosen as 0.25, yielding |ψm| = 2.575α.
In the AF magnitude plot of a CPM imaging system, radial lines with slopes of ±2|ψm|/π mark the extent of the operating region of such a system. Figure 5 demonstrates this concept in the form of AF magnitude plots derived from Mt × Mp as well as from Mm.
It is seen in Fig. 5(a) that nearly all the power contained in the theoretical AF magnitude plots is bounded by the radial lines corresponding to ±|ψm|. The measured AF shows a nearly identical result. To quantify the estimate of the design range, the percentage of power contained within the region bounded by the design range of –|ψm| ≤ ψ ≤ + |ψm| with respect to the overall plot was calculated for the theoretical prediction as well as the experimental data. To ensure a fair comparison, only those regions were included in the theoretical plot for which experimental data was collected. The resulting calculations showed that for the theoretical data, 95.38% of power was contained within the radial lines marking the defocus values of ±|ψm|. This ratio was 93.00% in the experimental data, resulting in a difference of 2.38% or an error of 2.49%. As a benchmark, the percentage of power within ±|ψm| in the theoretical case with respect to the entire plot was 93.37%. The experimental results thus validate the expression for the defocus design range as previously presented in  and shown in Eq. (10).
Another observation that may be made in Fig. 5 is regarding the symmetry of power distribution within the AF magnitude plots. Since the MTF is symmetric about u = 0 on account of the Hermitian symmetry of the OTF, it follows that the AF magnitude plot is symmetric about u = 0 along a radial line of arbitrary slope 2ψ/π. Furthermore, in systems where the MTF as a function of u is symmetric about ψ = 0, the AF magnitude plot is also symmetric about the horizontal axis, that is, about the line 2uψ/π = 0 or ψ = 0. In theory, the CPM imaging system exhibits both the above traits as may be seen in Fig. 5(a). In practice however, factors such as variations in the peak intensity of the PSF as a function of working distance cause the MTFs at ψ < 0 to not exactly match their counterparts for ψ > 0. In such a situation, the resulting AF magnitude plot exhibits an asymmetry about the horizontal axis, i.e., about ψ = 0, while maintaining symmetry about u = 0 along a radial line. This effect is observed in Fig. 5(b). Regions where ψ < 0 (second and fourth quadrant) correspond to working distances where zi – za > 0, whereas those areas where ψ > 0 (first and third quadrant) mark working distances where the sensor lay between the lens and the in-focus plane.
Inspecting the first and the fourth quadrant of Fig. 5(b) reveals that the extents of power distribution about ψ = 0 along the vertical direction are not identical as in the theoretical case. Specifically, power extends vertically outward to a greater degree in the fourth quadrant than in the first quadrant. This is because the former corresponds to sensor locations nearer to the lens and thus yield higher optical SNR than the latter. This effect results in lower estimates of the cutoff frequency uc from the first quadrant as revealed by the left-hand side of Fig. 4.
4.4. The phase transfer function
Unlike in many traditional imagers, the PTF may not be summarily ignored in computational imaging systems whose complex OTF contains a substantial phase component . This fact is particularly true of wavefront coding architectures where PTF signatures vary as a function of object distance. Employing a single restoration filter in such cases could then result in undesired artifacts in the final images . In view of this insight, the PTF of the CPM imaging system used in this experiment was also measured. The resulting data is presented in Fig. 6 and compared against the theoretical expression given by Eq. (11). Figure 6(a) shows the theoretical PTF obtained from Eq. (11) as a function of u and zi – za, while Fig. 6(b) presents the corresponding experimentally measured PTF. In order to offset the effects of centering the ROI window to the peak intensity location of the PSF, a linear PTF function has been added to the measured data, whose slope is m = 0.04 radian-mm/cyc as described in section 3. This compensation function is independent of defocus and hence constant across all working distances.
In Fig. 6, it is noted that reliable PTF information is available only within the usable spatial frequency bandwidth, i.e., in those regions where the MTF remains above the noise floor. Deviations of the measured data from theoretical prediction may be attributed to factors such as (1) variations in sampling phase as a changing ψ induces the PSF to shift across the sensor ; (2) a lateral motion of the sensor as the working distance is varied; and (3) noise.
Examining the two plots in Fig. 6 shows that the characteristics of the measured PTF show a strong correlation to that of its theoretical counterpart. Closer inspection however reveals that the measured PTF is “warped” about zi – za = 0 and along u. In other words, the measured PTF at working distances where zi – za is positive has an overall trend that is higher than its theoretical counterpart; whereas the opposite is the case for locations corresponding to zi – za < 0. One reason for this effect may be due to the aforesaid lateral drift of the sensor as the working distance is varied. That is, when misalignment causes the longitudinal travel of the sensor to occur along a path that is not perfectly parallel to the optical axis, a linear phase error accumulates due to the resulting PSF shift within the fixed ROI window. The slope of this phase error monotonically increases as zi – za increases. It is expected that experimental setups with enhanced motion precision capabilities could help mitigate this error.
4.5. Estimation of α
A useful application of the PTF is in characterizing CPM imaging systems where the effective strength α of the cubic phase mask is to be estimated. In general, the value of α would be a known quantity since it is a design parameter of the phase mask. However, scenarios could exist where the effective value of α differs from that given by specifications. For instance, when the cubic phase mask is placed in an imaging system with a variable aperture, the effective strength of the mask would vary depending on the size of the aperture. Another example is when the mask is placed at a location other than the traditionally prescribed pupil planes within the imaging chain. In such a situation, the functional strength of the phase mask would disagree with the design specification of α. Estimating the true value of α would then be required in such scenarios.
In principle, α may be estimated from either the MTF as per Eq. (7) or the PTF as described by Eq. (11). However, it is herein demonstrated that estimating α via the PTF is a more reliable approach than by using the MTF. In the following exercise, a polynomial curve fitting approach is used to determine the value of α. For the sake of simplicity, only those MTF and PTF values for which u > 0 are considered.
The right-hand side of the MTF equation in Eq. (7) consists of two components. The first component indicates a polynomial relationship between the MTF M and the spatial frequency u. The second term is a scaled function of Fresnel integrals which oscillates rapidly about unity as a function of u. For a low-order polynomial curve fitting approach, this term may hence be ignored. As a result, only the stationary phase component of the overall MTF as given by the modulus of Eq. (4) is utilized in this application. The MTF in terms of this component is then M = [π/(24αu)]½, which may be rewritten as
Upon curve-fitting the right-hand side of the above equation to a first-order polynomial with respect to u, namely a straight line, the resulting coefficient of the first degree exponent yields the slope of the fit, which is also the value of α. It is apparent from Eq. (13) that the above polynomial in u is highly sensitive to the MTF. It is further noted that larger values of α serve to lower the MTF. If the MTF used in the estimation is then lower than that predicted by Eq. (7) due to a host of system or environmental factors, this reduced MTF would result in a higher but erroneous estimate of α. Given that the system MTF is almost always lower that its theoretical optical-only counterpart, using the measured MTF to estimate this parameter would virtually assure incorrect results.
In the case of the PTF as given by Eq. (11), the first term, namely 2αu3, is a cubic exponent with respect to u and only contains the parameter α within its coefficient. The second term 2ψ2u/(3α) is a linear function of u and a quadratic function of ψ. It has been shown that the quadratic dependence on ψ may be exploited to locate the in-focus plane of a CPM imaging system . The third term contributes to a rapid oscillation of the PTF about π/4 radians within the available spatial frequency bandwidth of the system. This term may therefore be ignored in a low-order polynomial estimation problem. The equation used for curve-fitting would then be
The advantage of using the PTF to estimate α as opposed to the MTF is that most measurement errors tend to introduce a linear phase error as explained earlier, which would affect the second term on the left-hand side of Eq. (14). On the other hand, the first term containing the cubic exponent in u is independent of defocus or other linear effects. Thus, in the absence of optical aberrations specifically contributing to the cubic exponent in the above equation, the coefficient of this exponent directly yields an estimate of α while remaining largely unaffected by noise or other measurement errors. Practical utility of this approach is herein demonstrated via theoretical and experimental data.
Estimation of α was performed using three theoretical data sources Mt, Mt × Mp, and Θt, and three experimental sources, namely Mm, Mm/Mp, and Θm. It is noted that no linear or other phase correction was employed on the experimentally measured PTF Θm prior to its use in the estimation exercise. The value of α was estimated for each data source corresponding to a given defocus value within the range –20 ≤ ψ ≤ 20 and then averaged. The curve-fitting approach used normalized spatial frequencies uest within the range 0 ≤ uest ≤ ucc as the base variables whose coefficients were to be estimated. To ensure consistency across all ψ, the value of ucc was kept constant throughout this range. The value of ucc was determined by inspecting the measured MTF at ψ = 20 and identifying the actual cutoff frequency uca at this defocus value. The upper limit ucc was then chosen to be 0.95uca|ψ = 20. Figure 7 presents the results of the above exercise. Each colored curve represents the estimated value of α, namely αest, from one of the abovementioned data sources and across the specified range of defocus values. Estimates from theoretical data are shown in Fig. 7(a) while those from experimental data are illustrated in Fig. 7(b).
Table 1 shows the resulting mean estimates of α from each of the data sources. The numbers shown in green represent reasonably accurate estimates, while those in red indicate unreliable results.
Figure 7(a) shows that both Mt and Θt yield reasonably close estimates of α as expected, whereas the result obtained by evaluating Mt × Mp shows a significant error amounting to a deviation of roughly 33% from the design value. The fact that visual inspection of Fig. 2(a) reveals only small differences between Mt and Mt × Mp underscores the high sensitivity of the estimation method to variations in MTF height. This sensitivity is further highlighted upon inspecting Fig. 2(a) while examining αest in Fig. 7(b) obtained from the measured MTF Mm. It is seen that the estimate of α from Mm produced an error of nearly 120%. Attempting to raise the measured MTF by the pixel MTF does little to help as seen in Fig. 7(b), since Mm/Mp would remain below Mt considering that Mt × Mp is higher than Mm, especially at higher frequencies as attested by Fig. 2(a).
Inspecting the results obtained from the PTFs, it is seen that the measured PTF Θm yields a value of αest that is very close to the design value, with an error of less than 2%. This is in spite of the fact that Θm is lower than Θt as seen in Fig. 2(b). Given that the correction factor Θc used in Fig. 2(b) and Fig. 6(b) is a strictly linear function, it does little to influence the estimation of α via the cubic exponent αu3. It is therefore seen that the PTF is a far better property than the MTF for estimating the strength α of a cubic phase mask imaging system, at least in cases where a polynomial curve-fitting method is used. Another insight that may be gleaned from Fig. 7 is that oscillations due to the Fresnel integrals are more pronounced in the MTF than in the PTF. The values of αest as a function of ψ thus exhibit greater deviations from their respective mean values when the MTFs are used as a data source.
The evidence presented in this paper serves an essential requirement of validating previous mathematical analyses of the spatial frequency response of cubic phase mask wavefront coding imaging systems. It was shown that experimental data strongly substantiated theoretical predictions of the MTF as a function of defocus. The mathematical expression for the available spatial frequency bandwidth at a specified value of defocus was demonstrated to reliably predict the observed cutoff frequency at any given working distance within the design range. It was further proved that the analytical equation describing the design range accurately reflects the extent of working distances for which the CPM imaging system can usefully operate. The experimentally measured PTF was also shown to closely agree with its theoretical counterpart. In a problem of interest to imaging system characterization tasks, it was illustrated through measured data that the PTF is more reliable than the MTF in estimating the strength of the cubic phase mask via a polynomial curve-fitting approach. In summary, the experimental data collected in the course of this work has clearly authenticated mathematical analyses describing the spatial frequency response of this type of imaging system.
The results presented in this paper are intended to help bridge the gap between theoretical understanding and observed behavior regarding the frequency response of cubic phase mask wavefront coding systems. Given the increasing interest shown by the computational imaging community in cubic phase mask imagers, it is expected that these results could assist scientists and engineers in the research, design and characterization of such systems.
The authors would like to acknowledge OmniVision Technologies Inc., Sunnyvale, CA for providing the cubic phase masks used in the wavefront coding experiments. The authors wish to recognize Paulo E. X. Silveira and Chris Linnen for facilitating the procurement of the cubic phase masks and for providing useful suggestions during the experimental process. Expressions of thanks also go to Prasanna Rangarajan and Indranil Sinharoy for their insightful discussions and thoughtful recommendations; and Andrew Scott and Necdet Yildirimer for their assistance with component fabrication. In addition, the authors thank the reviewers of this paper for their valuable feedback. Research was sponsored by the Army Research Laboratory and was accomplished under Cooperative Agreement Number W911NF-06-2-0035. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Laboratory or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation herein.
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