1. Introduction

Sparse aperture imaging systems are capable of producing high resolution images while maintaining an overall light collection area that is small with respect to a fully-filled aperture achieving the same resolution. Such systems have advantages over resolution-equivalent monolithic apertures due to their lower size, weight and volume [1], and cost [2].

The modulation transfer function (MTF) of sparse aperture imaging systems will, in general, have a main lobe whose midband spatial frequencies contain lower energy than that of a fully-filled aperture with the same spatial frequency cutoff. If the midband MTF level becomes too low, or drops to zero, then all information at the corresponding image frequencies will be lost. A sparse array could in theory be tailored on an object-dependant basis such that the pass-band of the system is in the region of interest [3]. However, this is unlikely to be the case for any practical implementation of an imaging system designed for unknown targets.

In general, a quality sparse aperture imaging system will be designed such that the autocorrelation of the aperture function (i.e. the imaging system’s pupil) produces an MTF that covers most of the spatial frequencies that a filled aperture yielding the same image resolution would capture [4]. For example, the Golay family of arrays are of particular interest due to their compact, non-redundant autocorrelations which can nearly uniformly fill the spatial frequency space with minimal or no zeros present in the MTF [5].

Although the Golay family of arrays provide nearly uniform coverage in Fourier space, they are still hindered by reduced midband contrast, which can in turn give rise to signal-to-noise ratio (SNR) difficulties and poorer image quality overall. Solutions to this problem, including the use of long integration times and sophisticated post processing algorithms have been examined [6,7]. However, our work has been centered on increasing the midband contrast through the systematic introduction of redundancy in the aperture arrays themselves. In particular, we have focused on increasing the midband contrast of sparse aperture imaging systems based upon the Golay-9 array, as we have previously shown that a uniform sub-aperture diameter Golay-9 array, with an expansion factor of 1.4, can optimally maximize resolution and minimize overall pupil plane collection area while also maintaining acceptable MTF side-lobe levels [8].

2. Golay-9 review

It is well known that the MTF describes an incoherent imaging system’s ability to transfer object contrast to an image as a function of spatial frequency [9]. The MTF is preferable for quantifying image quality because it contains more information regarding the overall imaging process than the Rayleigh, Strehl or other single-number measures [10]. This is due to the fact that the MTF is direction dependant in the spatial frequency plane and contains information specific to resolution and image contrast.

The spatial frequency response of a diffraction-limited incoherent imaging system can be found by examining its optical transfer function (OTF), given by

In an aberration-free system it can be shown that H(f_x, f_y) = P(λz_if_x, λz_if_x) where H(f_x,f_y) is the two-dimensional spatial Fourier transform of the amplitude impulse response h and P is the imaging system’s pupil function. It then follows from Eq. (1), after a change of variables and application of Parseval’s Theorem, that the OTF can also be written as

For example, a uniform sub-aperture diameter Golay-9 array is shown in Fig. 1 . Individual sub-apertures each have diameter D and the center-to-center spacing of the most closely spaced sub-apertures is S∙D, where S is the expansion factor. Note that in Fig. 1, the expansion factor is set to the optimum value of S = 1.4 [8]. The corresponding MTF for this array, computed using Eq. (2) when D = 1mm, is then provided in Fig. 2 .

 

Fig. 1 A uniform sub-aperture diameter Golay-9 array with an expansion factor of S = 1.4.

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Fig. 2 MTF of a Golay-9 array with D = 1mm and S = 1.4. The spatial frequency cutoff ρ_min, determined by the conservative maximum inscribed circle approach, is shown by the white circle.

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Recall that the goal of sparse aperture imaging is to achieve resolution comparable to that of a fully-filled monolithic aperture while maintaining a sparse fill factor. A sparse array will then correspond to an effective fully-filled aperture with diameter D_eff, related to the spatial frequency cutoff ρ_min of the MTF according to

Previously we have taken the most conservative approach in defining the spatial frequency cutoff ρ_min by taking the diameter of the largest circle which could be inscribed within the contiguous portion of the MTF, as shown by the white circle in Fig. 2.

It should be noted that the effective diameter D_eff can be calculated from Eq. (3) if the spatial frequency cutoff ρ_min is found numerically.

Another important quantitative performance metric in sparse aperture imaging systems is the fill factor α, which we define to be the ratio of the light collection area normalized with respect to the area of a fully-filled aperture capable of achieving the same resolution. As seen in Fig. 1 the inner-most set of sub-apertures lie on a unique radius R₁ from the center of the array. Likewise, the intermediate set of sub-apertures lie on a radius R₂ and the outer-most set of sub-apertures lie on a radius R₃. The fill factor for a Golay-9 array is then given by

3. Increasing midband contrast

In the Golay-9 array there are three sets of three sub-apertures that lie at unique radii from the center of the array. To better understand how to increase mid-frequency contrast we closely examined the MTF as we sequentially attenuated the different radial sets of sub-apertures. From this we were able to determine which sets of sub-apertures were responsible for sampling particular spatial frequency regions, as demonstrated in Fig. 3 .

 

Fig. 3 Two-dimensional MTF for a uniform sub-aperture diameter (D = 1mm) Golay-9 array with an expansion factor of 1.4 for: (a) the R₁ set of sub-apertures attenuated by 75%; (b) the R₃ set of sub-apertures attenuated by 75%; and, (c) the R₂ set of sub-apertures attenuated by 75%.

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As seen in Fig. 3(a), the R₁ set of sub-apertures is mainly responsible for the low spatial frequency region as well as some midband frequency content, but are not at all responsible for the outlying high spatial frequency content. Therefore, we did not further explore perturbing the R₁ set of sub-apertures. Also, note that the introduction of autocorrelation redundancy by increasing the diameters of the R₁ set of sub-apertures is unrealistic since their diameters may only be increased up to the expansion factor before they become tangent.

Next, as shown in Fig. 3(b) the R₃ set of sub-apertures is responsible for only high spatial frequency content, including all of the non-contiguous outlying high spatial frequency region. Increasing the diameter of the R₃ set of sub-apertures would therefore not generally be beneficial to increasing the midband contrast. As a result, perturbing this set of sub-apertures was not considered further.

Finally, from Fig. 3(c) we see that the R₂ set of sub-apertures is responsible for over 50% of the mid-frequency autocorrelation and exactly 50% of the outlying high spatial frequency content. Furthermore, the R₂ set of sub-apertures are positioned in such a way that there is plenty of room to increase their diameters before causing overlap with any other sub-aperture. For these reasons we have chosen to systematically increase the diameters of the R₂ sub-apertures in order to increase autocorrelation redundancy, thereby boosting the MTF at the desired midband frequencies.

4. Metrics and theoretical expectations

Figure 4(a) depicts the MTF generated from the same Golay-9 array corresponding to the MTF in Fig. 2, except that the R₂ sub-aperture diameters have been increased by a factor of 2.8. From this figure it is apparent that our efforts to increase mid-frequency contrast by increasing the diameter of the R₂ sub-apertures can make the contiguous portion of the MTF start to overlap with what was previously outlying high spatial frequency content. This effect gives rise to reduced gaps in the MTF and causes the spatial frequency cutoff to be poorly described using our previous methods. We have therefore developed a new algorithm, consistent with our conventional approach, which fairly incorporates some of the new energy introduced into the MTF as D_R2 increases relative to D_R1 and D_R3.

 

Fig. 4 (a) The MTF of a Golay-9 pupil plane array with S = 1.4, D_R1 = D_R2 = 1mm and D_R2 = 2.6mm; and, (b) the binary version of (a) demonstrating the nominal frequency cutoffs as determined by the conventional conservative approach (ρ_min-old) and by use of the new spatial frequency availability algorithm (ρ_min-new).

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4.1 Determining spatial frequency cutoff, ρ_min

Our new spatial frequency cutoff algorithm is based on an imaging system’s ability to successfully image radial spatial frequencies. We have come to call it the “spatial frequency availability” algorithm. This algorithm is implemented by first setting a threshold T₁ for acceptable optical power in the midband spatial frequencies and then creating a binary version of the MTF, as shown for example in Fig. 4(b) for T₁ = 0. For practical imaging systems T₁ would be set according to the minimum acceptable SNR level within the pass-band of the MTF.

The fractional open areas of a series of thin concentric annuli (each about 0.01 cyc/mm wide when D = 1mm) are then calculated to determine the availability of particular radial spatial frequencies, after which a second threshold T₂ is chosen. This second threshold specifies the minimum fraction of any given radial spatial frequency that must be available in the final image. Setting a high threshold (T₂ ≈1) will yield our previously used conservative approach to defining spatial frequency cutoff, and would reduce to choosing the radius of the largest circle inscribed within the contiguous portion of the MTF, as shown in Fig. 4(b).

4.2 Measuring midband average relative contrast, ARC

A mid-frequency optical power metric was then defined to compare the average midband contrast of various Golay-9 arrays to that of a fully filled aperture with the same effective diameter. In order to consider only midband contrast improvement we elected to exclude the optical power present in the center lobe of the MTF (with uniform sub-aperture diameters) as well as all optical power outside ρ_min, as determined according to our new algorithm. Specifically, the midband average relative contrast (ARC) is defined as the integrated midband optical power of a sparse array, normalized with respect to the integrated midband optical power of a resolution equivalent fully-filled aperture according to the following relationship:

4.3 Numerical analysis

Numerical calculations of spatial frequency cutoff, fill factor and ARC were performed for several R₂ sub-aperture diameters and various T₁ values_. The spatial frequency cutoff was found using the methods described in section 4.1 while the fill factor and ARC were then calculated from Eqs. (4) and (5), respectively. In all cases the positions of the sub-apertures were fixed to that of the uniform sub-aperture diameter Golay-9 array with an expansion factor of 1.4. The R₁ and R₃ sets of sub-apertures were set to 1mm and the distance from the pupil plane to the image plane was taken to be one focal length. The diameters of the R₂ set of sub-apertures were then increased. Plots for T₁ = 0 and T₁ = 0.01, with T₂ = 0.98, are provided in Fig. 5 . Setting T₁ = 0 corresponds to the perfect, noiseless case, while the T₁ = 0.01 threshold was chosen by examining noise levels in our experimental setup. In particular, T₁ was chosen to be very small while also allowing as many detector noise induced artifacts as possible to be removed from the MTFs created from measured data. Note that T₁ = 0.01 was chosen to be approximately three times the peak noise values in otherwise dark pixels of our MTFs. The choice of T₂ = 0.98 was, however, made somewhat arbitrarily, though a large value close to unity will be preferable in many practical applications. Corresponding to the experimental conditions described in the following section, a focal length of 750mm was assumed and the source wavelength was set to 632.8nm.

 

Fig. 5 Numerical simulations of spatial frequency cutoff ρ_min, fill factor α and midband average relative contrast (ARC) for varying D_R2 and initial thresholds of (a) T₁ = 0; and, (b) T₁ = 0.01. The dots in (a) denote the values associated with the MTF shown in Fig. 4.

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As shown in Fig. 5, for small increases in the R₂ sub-aperture diameters a slight increase in spatial frequency cutoff is observed. An appreciable increase in the average relative mid-band contrast is also seen while maintaining a relatively sparse fill-factor. Notice in Fig. 5(a) that there is a sudden increase in ρ_min when the relative diameter of the R₂ set of sub-apertures increases beyond ~2.6. We believe this is an anomaly of the noiseless case and arises from the significant overlap of the contiguous portion of the MTF with what was previously outlying high spatial frequency content. As the SNR threshold (T₁) increases a more realistic sampling of the MTF occurs and this effect is much less severe, as seen in Fig. 5(b). Note also that the dots in Fig. 5(a) denote the values associated with the MTF shown in Fig. 4.

5. Experimental verification

Figure 6 shows a diagram of our experimental setup. Multiple Golay-9 masks with varying R₂ sub-aperture diameters were placed before an imaging lens L₂. The masks were then illuminated with a spatially filtered and collimated laser source having a wavelength of λ = 632.8nm. Both lenses L₁ and L₂ were well corrected three inch diameter doublets having focal lengths of 750mm. In order to magnify the PSF and allow better spatial sampling, a microscope objective was placed one focal length after the imaging lens. The magnified intensity PSF was then recorded by the CCD array. Finally, after scaling the measured PSF dimensions by the system magnification, for each value of D_R2 the MTF was found by taking the modulus of the OTF calculated by use of Eq. (1).

 

Fig. 6 Diagram of our experimental setup.

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The magnification factor for our system was measured by comparing the pixel distance between three sets of distinct outer lobes in the theoretical, unmagnified PSF, and the experimental, magnified PSF. See, for example, the white arrowed lines of Fig. 7 . Using the average experimental center-to-center pixel spacing between the three sets of distinct lobes, taken over three experimental PSF realizations, the magnification factor was found to be 3.63.

 

Fig. 7 An example of one set of distinct lobes for which the center-to-center pixel spacing was recorded in (a) an experimental, magnified PSF; and (b) the theoretical, unmagnified PSF.

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The MTF and associated metrics were calculated (assuming T₁ = 0.01 and T₂ = 0.98) for a uniform diameter (D = 2.54mm) Golay-9 mask, as well as for four other Golay-9 masks whose R₂ sub-aperture diameters were increased by factors of 1.7, 2.0, 2.4 and 2.8. Our theoretical expectations are plotted in Fig. 8 along with the values determined from experimental data. Note that the theoretical curves of Fig. 8 describing both fill factor and ARC are identical to the corresponding curves of Fig. 5(b), as they are each independent of D.

 

Fig. 8 Theoretical (solid) and experimental (dashed) calculations of spatial frequency cutoff ρ_min, fill factor α and midband average relative contrast (ARC) using an initial threshold of T₁ = 0.01 and a spatial frequency availability threshold of T₂ = 0.98.

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We believe our experimental results match our theoretical expectations very well. Notice, though, that the experimentally determined spatial frequency cutoffs are consistently a bit smaller than theory predicts. This intuitively produces higher fill factors and lower ARC values according to Eqs. (4) and (5). Though not shown here, the same set of data was also analyzed for several other spatial frequency availability thresholds T₂. In all cases we found that the data followed the theory in the same manner shown in Fig. 8.

6. Summary and conclusions

Sparse aperture imaging is a useful technique for obtaining high resolution images while maintaining a small light collection area with respect to a large, resolution-equivalent monolithic aperture. However, conventional sparse aperture systems suffer from reduced midband contrast.

A new method for determining spatial frequency cutoffs for sparse aperture arrays having semi-redundant autocorrelations was developed based upon the notion of radial spatial frequency availability. The quantitative average relative contrast metric was then developed in order to describe the recovery of mid-frequency contrast as sparse array autocorrelation redundancy is increased. From these metrics we were able to compare perturbed versions of the nominal Golay-9 array to their resolution equivalent monolithic apertures.

In particular, numerical simulations were performed to examine the midband contrast recovery caused by increasing the diameter of the intermediate radius set of sub-apertures in a Golay-9 array. We found that an increase in average relative contrast of over 55% can be achieved, but at the expense of an increased fill factor. These numerical expectations were then experimentally verified with very little disagreement observed between theory and data.

Moreover, our spatial frequency cutoff algorithm and the average relative contrast metric can be extended to describe the performance of any sparse aperture imaging system. As the results will scale with wavelength, sub-aperture diameter and system magnification we believe that our techniques will be useful to other researchers examining a wide variety of other sparse aperture imaging systems.