1. Introduction

In many incoherent imaging applications, the quality of imagery is degraded by atmospheric turbulence. In a previous paper [1], we used wave-optics simulations to explore the potential for post-detection restoration of image objects in the presence of deep-turbulence, notably when the isoplanatic angle is smaller than the image diffraction-limited resolution. For the objective scenario studied (300 km slant path viewing the ground from a 20 km altitude), extreme anisoplanatisim was encountered yielding 355 isoplanatic angles (θ₀) across a single resolution angle (ρ). Considering this implies there are on the order of 10⁵ independent turbulence PSFs combining within a single resolution patch, it seems surprising that fine detail would be retained for any extended object, even in a short-exposure. To test this, we conducted wave-optics image simulations using what we termed a minimally extended object (MEO), a square object matching a single critical sample (Q = 2) for a diffraction-limited optical system. We found that 50% of simulated short-exposure MEO image realizations had central lobes with radii less than half the long-exposure average. We applied the term “spatial stabilization” to the behavior of the turbulence PSFs enabling this preservation of fine resolution when combined over the extent of an MEO.

This paper presents initial experimental support for this result. Our experiment addresses a severe case of deep-turbulence anisoplanatism (θ₀ /ρ < 0.1) as a stepping stone to the more extreme objective scenario. Section 2 describes the experiment design, including an Isoplanatic-Angle Sensor (IAS) providing direct measurements of turbulence PSF spatial correlation. Section 3 provides a direct comparison of measured and simulated central lobe statistics for point-like and MEO images. IAS results are compared to scintillometer-based calculations of the Fried isoplanatic angle [2] using a correlation threshold suggested in our original paper [1]. Conclusions are drawn in Section 4.

2. Experiment description

Our experiment consisted of collecting short-exposure image sequences of point-like and MEO targets viewed over a 5.1 km slant path looking northeast (Heading 037) from the Building 620 Tower (camera location) to a ground-level site (target location) located at Wright-Patterson Air Force Base, Dayton, Ohio. The path altitude profile is shown in Fig. 1. The experiment was supported by scintillometer measurements and a special laser target designed to directly measure a correlation-based isoplanatic angle. The later measurement directly verifies the collected imagery is subject to significant PSF decorrelation at scales much finer than the camera resolution. Table 1 compares the anticipated test scenario conditions (computed for representative C_n² profiles and the path geometry) with those assumed for previous object scenario simulations [1]. The very shallow slant path of the test scenario causes the expected spherical-wave coherence diameter (r₀) [3] to be relatively small, leading to modest telescope aperture diameters (D) for the desired level of blur (D/r₀ ∼2-3). To accommodate the expected variability of actual test conditions, we prepared several circular apertures measuring 38.1 to 76.2 mm. The expected test-scenario isoplanatic angles (θ₀) are >100x larger than for the objective scenario, but still provide ρ > 10θ₀ for available resolution angles ρ = λ/D (9.8 to 19.7 µrad) at wavelength λ (750 nm).

 

Fig. 1. Altitude profile for the camera-to-target imaging path.

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Table 1. Parameters characterizing imaging performance for two scenarios

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2.1 Instrumentation and targets

The primary test sensor was a PCO Edge 5.5 scientific CMOS camera (6.5 µm pitch) mated with a 950 mm focal length Williams Optics FLT-132 f/7.0 telescope. The telescope was operated at f/12.1 to f/24.3 when fitted with apertures scaled for prevailing test conditions. The camera was selected for low read noise (2.5e-) while supporting windowed fast framing (51 × 1920 at 1493 Hz) in global shutter readout mode. The assembly supported spectral filters for imaging the sunlit MEO target (745 ± 47 nm) and to minimize background for laser-based isoplanatic angle measurements (787 ± 10 nm). The sampling Q is 2 (Nyquist sampling) for our nominal aperture size (57.2 mm) at the laser point source simulator wavelength (785 nm). The sampling Q varies between 1.4 and 2.9 for all aperture and spectral filter combinations.

The IAS is a specialized laser source operated in combination with the above camera and telescope. The source (Fig. 2) consists of two pulsed lasers (Thorlabs LD785-SE400) combined with a polarizing beam splitter. Each laser aperture is 57 µm making it an excellent point-source emulator. One laser is translated with respect to the other (up to 40 mm separation) so that its displacement as seen by the receiver defines an angle over which isoplanatism is measured (maximum of 7.8 µrad for 5.1 km range). The lasers are sequentially pulsed so that the receiver camera images each laser in successive frames. Using fast pulse rates, the temporal decorrelation is minimized so that the decorrelation between measured PSFs is dominated by anisoplanatism.

 

Fig. 2. Isoplanatic angle measurement source and receiver configuration.

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The target site setup is shown in Fig. 3. The center (mostly black) panels measure 8 × 8 feet and are designed to provide a uniform background to facilitate isolating image signal from IAS (left) and MEO targets (right). At 5.1 km range, the panel size supports a 40 × 40 pixel target-imaging area, a 4-pixel wide band for estimating the background level, and a 10-pixel wide band (5ρ for the nominal 57.2 mm aperture) isolating the background estimate from the surrounding scene. The IAS laser source projects through a 4“x6” slot in the center-left panel. White square MEO objects attach to the middle of the right-center panel. The MEOs are paired with telescope apertures so that they meet their definition of a diffraction-limited critical sample at the maximum wavelength of interest (sized for λ=787 nm). For example, with the nominal 57.2 mm aperture, the corresponding MEO measures 35.1 × 35.1 mm. Edge and tribar targets are included to make the dataset useful for testing future image-restoration algorithms. The source for the Scintec BLS 900 scintillometer, providing path-average C_n² measurements (weighted by an instrument function peaking at the path midpoint), is seen to the left of the IAS panel. All targets are located on a concrete pad.

 

Fig. 3. Target site configuration. From left to right: tilted MTF edge target, scintillometer source, IAS panel, MEO panel, and tribar resolution target.

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2.2 Data set

The experiment was performed on 22 September 2015. Scintillometer data were logged at 1-minute intervals and used to inform adjustments to planned aperture selections. In total, 13 IAS and 11 MEO data sets were collected. For IAS collections, the receiver and source were set to capture a 3000-frame (∼2-second) rapid sequence of pulse pairs at each of 41 laser positions (0 to 40 mm separation in 1 mm intervals). A complete IAS measurement sequence took about 7 minutes. The receiver camera (configured with the 20 nm bandwidth background filter) was operated at a frame rate of 1493 Hz, with the exposure set to the maximum possible 360 µs, resulting in a duty cycle of 54%. The lasers were adjusted to emit 10-µs pulses with 30-mW peak power, and a beam divergence of 1°. Signal generators triggered the lasers to send out a pulse from the first laser, followed by a pulse from the second laser and then a blank pulse at an effective pulse frequency of 1471 Hz. The laser pulse frequency is offset from the camera frame rate to ensure ample pulse capture within the integration time period of the camera duty cycle, without requiring complex transmitter-receiver synchronization circuitry. The blank pulse was used for differential-mode processing, where the blank pulse frame is subtracted from the previous and post pulse frames to eliminate background. Using this setup, for each 2-second data acquisition, nominally 500 pulse pairs were available for correlation processing. Figure 4 shows a sequence of ROIs extracted from a moderate turbulence case with a 6-mm source separation. The pulse image pairs appear correlated but also exhibit clear differences.

 

Fig. 4. Example IAS image sequence for 6 mm laser separation in moderate turbulence.

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A complete MEO data set consisted of 1000-frame sequences collected at 30 Hz with exposures of 0.36, 1, 2, 5, and 10 ms. The 30 Hz frame rate was selected to promote independence of image realizations within a reasonably short collection time. All MEO data sets were collected with the nominal 100 nm bandwidth filter allowing simultaneous imaging of the IAS lasers serving as a point-source emulator (separation set to 0.0 mm), used to estimate r₀. The 0.36 ms exposure guarantees at most a single 10 µs laser pulse per frame to obtain very-short-exposure PSFs, though with poor duty cycle. Longer exposures accumulate multiple pulses at an average rate of ∼1 pulse per ms. The 5 and 10 ms exposures are primarily needed to obtain useful signal levels in blurred (sunlit) MEO images. These signal levels are constrained by aperture choices needed to meet D / r₀ and sampling-Q test objectives. The 5 ms exposure provides the shortest-exposure simultaneous imaging of the point and MEO targets.

3. Analysis

The complete data set is summarized in Fig. 5 using spherical-wave r₀ and θ₀ computed from scintillometer data, compared with r₀ measured directly from point source emulator image sequences (5 ms exposures) and θ₀’ measured using the IAS. For completeness, we provide the uniform path expressions for computing spherical-wave r₀ and θ₀ from the scintillometer path-averaged C_n² values, evaluated at the laser wavelength 785 nm and propagation path length L = 5.1 km.

 

Fig. 5. Comparison of r₀ and θ₀ computed from scintillometer data and extracted from point-source emulator imagery and IAS correlation measurements, respectively.

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The coherence diameter was estimated from point-source emulator data by fitting a sensor MTF model to the sum of short-exposure sequences (1000 frames, each with 5 ms exposure). Prior to summing, frames with saturated data were excluded and the background was estimated (mean estimate obtained from a 4-pixel wide band surrounding a 39 × 39 pixel target window) and subtracted (any negative-value pixels set to zero). The normalized spectrum was computed (DFT of the summed image zero-padded to size 128 × 128) and fit to the following expression using r₀ to minimize the sum-square error (SSE).

The IAS θ₀’ values were determined by measuring intensity correlation as a function of source separation. This begins by manually specifying an image ROI centered on the IAS transmitter. At each source separation, frames containing pulses are identified when the maximum value within the ROI exceeds a user-specified threshold. A second threshold is applied to eliminate frames which are near saturation. Pulse pairs are found by stepping through the frames and recording the location whenever two sequential non-saturated pulse-containing frames are found and are bracketed by two frames without pulses. Background is removed from each pulse image by subtracting the nearest pulse-free image. The image containing the pulse from the translated source is then shifted horizontally to align it to the image from the stationary source. The correlation coefficient is then computed for each pair of pulse images. The median correlation value among all pulse pairs found in a given image sequence is taken as the representative value for that spatial separation. The correlation-based isoplanatic angle θ₀’ is assigned by finding the separation where the correlation coefficient falls to 0.86 (Fig. 6).

 

Fig. 6. PSF decorrelation with spatial separation measured using the IAS. The correlation-based isoplanatic angle is determined by the separation at which the correlation coefficient falls to 0.86 and the measurement range.

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Care must be taken in comparing the scintillometer-derived and direct measurements of r₀ and θ₀ since the scintillometer assumes a constant C_n² profile whereas the direct measurements are impacted by the actual profile. The scintillometer C_n² value is weighted toward the path midpoint, while r₀ and θ₀ are weighted by turbulence at the sensor and target, respectively. We note that during the morning and early afternoon hours, the image-derived coherence diameters tend to be smaller. This would be consistent with stronger turbulence near the camera location; the face of the tower was sunlit during the morning. The agreement is better in the late afternoon when the tower has been shaded and allowed to cool. All of the imaging experiment r₀’s fall within the planning range of 1.1-2.7 cm. The IAS measurements (θ₀’) are in good agreement with scintillometer-derived values (θ₀) for most of the day, adding to confidence in the 0.86 correlation-coefficient threshold. Most intriguing is the result measured at 17:04EDT (see also Fig. 6). During the 6.5-minute period required to sample 40 PSF separations, the correlation never fell below 0.92 suggesting the isoplanatic angle was greater than the instrument limit of 7.8 µrad. In examining the data, we found no reason to discount the measurement.

Figure 5 highlights two imaging data sets selected for more detailed analysis: Experiment 7 (Exp. 7) at 14:22 EDT and Experiment 13 (Exp. 13) at 16:43 EDT. These were selected to span the strongest and weakest turbulence encountered during the afternoon period when favorable solar illumination angles provided the best signal levels for the MEO target. The test conditions, aperture, and MEO target selections for these cases are listed below in Table 2. The experiment r₀ values are derived from the laser images whereas θ₀ and σ_χ² are computed from scintillometer data. Two IAS measurements are listed corresponding to the closest available times bounding the MEO collections.

Table 2. Parameters and test conditions for selected experiments

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One of our goals for this work was to assess the validity of the Atmospheric Compensation Simulation (ACS) wave-optics code [4] results that formed the basis of our previous paper [1]. To accomplish this, we applied the same simulation approach to modeling these experiments, notably modeling the MEO image formation process by incoherently summing the PSFs produced by an array of point sources distributed over the MEO. We used 25 × 25 and 11 × 11 grids for Exp. 7 and 13 respectively, achieving finer than ½ θ₀ sampling in both cases. We used the propagation path profile of Fig. 1, modeling C_n² as a function of path height using the Hufnagel-Valley vertical profile [5]. ACS has an internal algorithm to position phase screens (ten in this case) along the slant path with realizations initially defined by a modified von Karman spectrum and the Hufnagel-Valley model. The screens were then multiplicatively scaled so the simulated r₀ matches the test condition. The average modeled crosswind (1.1 m/s) was estimated from experiment truth. The resulting θ₀ and σ_χ² are shown in the table.

The experiment and simulated imagery were both analyzed using central-lobe half-intensity radii statistics as described in [1]. The experiment data are corrupted by noise, accommodated as follows. Data were converted to photoelectron units using the factory-supplied conversion gain (0.46 e-/count). Background was estimated and removed as described above for r₀ estimation. The peak value (P) was found and a conservative 50% threshold (T) set at $T = {\raise0.7ex\hbox{$P$} \!\mathord{\left/ {\vphantom {P 2}} \right.}\!\lower0.7ex\hbox{$2$}} - 1.6{\sigma _n}$ where $\sigma _n^2\; $ is the estimated noise variance at threshold T given by: $\sigma _n^2 = T + 9$, and the read noise variance is estimated at 9 e-². Images were up-sampled by a factor of 4 prior to computing a contour at the threshold level T and measuring radii between the peak pixel and every contour pixel.

Whereas most images contained a well-defined central lobe that could be measured, this was not the case for all images owing to strong scintillation, signal fades and noise. Table 3 summarizes the numbers of images used for results presented below. The simulation data sets include a total of 500 frames; all experiment data sets contained 1000 frames.

Table 3. Frame counts contributing to central lobe radii statistics

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Figure 7 shows our key experimental result using the presentation format of reference 1. The results confirm that even moderately-short-exposures of MEO objects (10 ms in this example) retain significant fine-resolution information below the long-exposure limit. In this example, 15% of all measured radii are less than half the long-exposure radius. Interestingly, 10% of all radii are indicative of super-resolution (finer resolution than the vacuum diffraction limit).

 

Fig. 7. Histogram showing the relative occurrence of main-lobe extent in Exp. 7 MEO images collected with a 10 ms exposure time. Separate histograms are displayed for the mean radius of each main lobe (red) and all radii from all images (blue). The solid lines indicate the cumulative probabilities for the 898 selected images.

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To provide a wider set of comparisons we have selected the more compact presentation of Figs. 8–10 which show samples from the cumulative probability distributions. We display short-exposure radii normalized by the radius of the corresponding long-exposure image (produced by summing the short-exposure frames). This normalization removes possible differences in turbulence conditions when comparing different data sets collected at different times, and the wavelength differences for point emulator and MEO data. Figure 8 compares point and MEO image results from Exp. 7 for selected exposures. Recalling that point exposures sum laser pulses at an average rate of ∼1/ms, the blue bars representing simultaneous 5 ms exposures provide our best point of comparison. Clearly the point object images preserve smaller central lobes than the MEO images, but the MEO images retain significant benefit compared to long-exposures. Comparison of the shortest point exposures (1-2 pulses per image) to the 5 ms point exposures indicates there is temporal decorrelation at 5 ms exposure. There is very little change in the MEO central lobe moving from 5 to 10 ms exposures. Note that a significant fraction of the 1 ms point object images exhibit evidence of super-resolution, notably at the shortest exposures, even using the mean radius metric. The MEO images do not show evidence of super-resolution at the 5% probability level.

 

Fig. 8. Normalized short-exposure per-image mean radii at selected cumulative probabilities comparing Exp. 7 point and MEO results for three exposures. The diffraction limit reference (0.36) is indicated for the 1 ms point data set, the only one demonstrating super-resolution for the metrics shown. The diffraction limit references (normalized radii units) are 0.38, 0.42, and 0.45 for the 5 ms point, 5 ms MEO, and 10 ms MEO data sets, respectively.

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Figure 9 summarizes the 5 ms exposure results for both experiments, further illustrating the value retained in the short-exposure MEO images. Considering all radii individually adds evidence of super-resolution detail retained in MEO images, albeit this detail will be limited to specific orientations in individual images. Such detail can be accumulated in many orientations by using multi-frame restoration. Further, in the Exp. 7 MEO case, more than 20% of all radii measured less than half their long-exposure counterpart. There is a weak trend showing less short-exposure (5 ms) benefit for both point and MEO images in Exp. 13, the more moderate of the two turbulence conditions. This is not surprising given the smaller differential between the long-exposure blur and the diffraction limit.

 

Fig. 9. Normalized short-exposure radii (mean, left and all radii, right) for selected cumulative probabilities comparing simulated point and MEO results for Exp. 7 and Exp. 13, all for 5 ms exposure. The diffraction-limit references are shown for the MEO data sets. The diffraction-limit references are 0.38 and 0.42 for the Exp. 7 point and MEO data, respectively, and 0.47 and 0.49 for the Exp. 13 point and MEO data, respectively.

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Figure 10 provides simulated results in direct comparison to the data shown in Fig. 9. The trends seen in the experimental results are duplicated in the simulations – MEO images suffer some degradation of short-exposure central lobe radius compared to a PSF, but retain significantly more detail than a long-exposure image. However, the simulations are clearly optimistic, underestimating the short-exposure central lobe radii for both object types, and indicating the small central lobes are preserved with higher probability. This optimism is most evident in Fig. 11 which compares full cumulative probability distributions for simulations and measurements of the mean radius metric. The PSF results agree best for Exp. 7 1-ms data, the closest match to the instantaneous-exposure simulations. The simulation and experiment agree that for the more severe turbulence case (Exp. 7), the differential between PSF and MEO image central lobes is greater than in the more moderate case (Exp. 13).

 

Fig. 10. Normalized instantaneous-exposure mean radii for selected cumulative probabilities comparing simulated point and MEO image results representing Exp. 7 and Exp. 13. All simulations were monochromatic (750 nm); the diffraction-limit references shown apply to MEO simulations (0.45 and 0.50 for Exp. 7 and Exp. 13, respectively). The corresponding point object references are 0.43 and 0.50, respectively.

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Fig. 11. Normalized short-exposure radii (per-image mean) cumulative probability distributions comparing simulated and measured, point object and MEO results for Experiments 7 and 13.

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Given that the simulation long-exposure blur was forced to agree with the experiment (the simulation C_n² profile was scaled to match the experiment r₀), and that the normalized presentation eliminates any minor difference in long-exposure blur present in the specific simulated frame sequence, it is clear that the simulation is predicting more image distortion (e.g., tip-tilt wavefront error) and less short-exposure blur than is evident in the experiment data. The explanation for this is not known, but two possibilities we have considered are the mismatch in exposure time, and a mismatch in the path C_n² distribution. The simulations represent instantaneous exposures while signal-to-noise considerations led us to 5 ms minimum MEO exposures which certainly suffer some temporal decorrelation. Second, the side of the tower hosting the camera (facing east) is sunlit during the morning and early afternoon, and could be introducing significant turbulence close to the telescope aperture. As mentioned previously, this is consistent with the smaller image-based r₀ estimates compared to those computed from scintillometer C_n² measurements observed during the 10am to 3pm EDT period (see Fig. 5). The scintillation-based estimate is weighted by mid-path turbulence while the imaging r₀ measurements are most sensitive to turbulence near the camera. In addition, the calculations use the classical Fried definition for r₀ which does not take into account inner- and outer-scale features found in real turbulence [6]. This could also play into the observed differences between measurements and simulations.

4. Conclusions

We have performed experiments that explore short-exposure imaging through deep-turbulence causing severe anisoplanatism and resulting sub-resolution space variance of the turbulence-degraded PSF. In these experiments, the strongest turbulence provided θ₀/ρ = 0.066, or 15 isoplanatic angles per diffraction-limited resolution angle ρ. At this level, the data confirms short-exposures of minimally extended objects retain detail, measured by the radius of the image central lobe, intermediate to short- and long-exposure PSFs. This is a very promising result supporting the potential for post-detection image restoration from short-exposure images, in the presence of deep-turbulence.

A new isoplanatic angle measurement technique was demonstrated, based on prior simulations of short-exposure PSF decorrelation as a function of point-source separation [1]. These measurements provide direct verification of the deep-turbulence test conditions; comparisons of source separations yielding 0.86 correlation coefficients with Fried isoplanatic angles computed from scintillometer data were in good agreement, supporting the threshold derived from simulation.

Simulations of specific experiments generally resulted in optimistic point-source and MEO image central lobes as compared to the experiment, even though the simulation C_n² profile was scaled to match image-based coherence diameter (r₀) observations. While temporal decorrelation during MEO exposures and C_n² profile differences were discussed as plausible explanations, we have not attempted simulation trade studies to explore these possibilities.