We present a numerical wave propagation method for simulating imaging of an extended scene under anisoplanatic conditions. While isoplanatic simulation is relatively common, few tools are specifically designed for simulating the imaging of extended scenes under anisoplanatic conditions. We provide a complete description of the proposed simulation tool, including the wave propagation method used. Our approach computes an array of point spread functions (PSFs) for a two-dimensional grid on the object plane. The PSFs are then used in a spatially varying weighted sum operation, with an ideal image, to produce a simulated image with realistic optical turbulence degradation. The degradation includes spatially varying warping and blurring. To produce the PSF array, we generate a series of extended phase screens. Simulated point sources are numerically propagated from an array of positions on the object plane, through the phase screens, and ultimately to the focal plane of the simulated camera. Note that the optical path for each PSF will be different, and thus, pass through a different portion of the extended phase screens. These different paths give rise to a spatially varying PSF to produce anisoplanatic effects. We use a method for defining the individual phase screen statistics that we have not seen used in previous anisoplanatic simulations. We also present a validation analysis. In particular, we compare simulated outputs with the theoretical anisoplanatic tilt correlation and a derived differential tilt variance statistic. This is in addition to comparing the long- and short-exposure PSFs and isoplanatic angle. We believe this analysis represents the most thorough validation of an anisoplanatic simulation to date. The current work is also unique that we simulate and validate both constant and varying Cn2(z) profiles. Furthermore, we simulate sequences with both temporally independent and temporally correlated turbulence effects. Temporal correlation is introduced by generating even larger extended phase screens and translating this block of screens in front of the propagation area. Our validation analysis shows an excellent match between the simulation statistics and the theoretical predictions. Thus, we think this tool can be used effectively to study optical anisoplanatic turbulence and to aid in the development of image restoration methods.

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Simulation of anisoplanatic imaging through optical turbulence using numerical wave propagation with new validation analysis

A time-lapse imaging experiment was conducted to monitor the effects of the atmosphere over some period of time. A tripod-mounted digital camera captured images of a distant building every minute. Correlation techniques were used to calculate the position shifts between the images. Two factors causing shifts between the images are: atmospheric turbulence, causing the images to move randomly and quickly, plus changes in the average refractive index gradient along the path which cause the images to move vertically, more slowly and perhaps in noticeable correlation with solar heating and other weather conditions. A technique for estimating the path-averaged C 2n from the random component of the image motion is presented here. The technique uses a derived set of weighting functions that depend on the size of the imaging aperture and the patch size in the image whose motion is being tracked. Since this technique is phase based, it can be applied to strong turbulence paths where traditional irradiance based techniques suffer from saturation effects.

Estimation of the path-averaged atmospheric refractive index structure constant from time-lapse imagery

Measurements of an object viewed at low-light levels through weak atmospheric turbulence are modeled mathematically as a time-space doubly stochastic Poisson process in which the intensity function moves randomly in time but is otherwise undistorted. This model is used with the maximum-likelihood method of statistics to derive a new method for forming an image of the object. A simple computer simulation of a moving one-dimensional binary star suggests that improved images may be produced by this method in comparison with others that have been suggested in the literature, but it remains to be demonstrated that this improvement is realized in practice with real imagery data.

High-resolution imaging at low-light levels through weak turbulence

Accurate characterization of atmospheric turbulence is useful for performance assessment of optical systems operating in real environments and for designing systems to mitigate turbulence effects. Irradiance-based techniques such as scintillometry, suffer from saturation, and hence commercial scintillometers have limited operational ranges. A method to estimate turbulence parameters, such as path weighted Cn2 and Fried’s coherence diameter r0 from turbulence-induced random, differential motion of extended features in the time-lapse imagery of a distant target is presented. Since the method is phase-based, it can be applied to longer paths. It has an added advantage of remotely sensing turbulence without the need for deployment of sensors at the target location. The approach uses a derived set of path weighting functions that drop to zero at both ends of the imaging path, the peak location depending on the size of the imaging aperture, and the relative sizes and separations of the features whose motions are being tracked. Using different sized features separated by different amounts, a rich set of weighting functions can be obtained. These weighting functions can be linearly combined to approximate a desired weighting function such as that of a scintillometer or that of r0 in inhomogeneous turbulence. The time-lapse measurements can thus mimic the measurements of a scintillometer or any other instrument. The method is applied to images captured along two different paths, and the estimates are compared to co-located scintillometer measurements.

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Estimation of atmospheric turbulence using differential motion of extended features in time-lapse imagery

Atmospheric turbulence parameters are estimated for an imaging path based on time-lapse imaging results. Atmospheric turbulence causes frame-to-frame shifts of the entire image as well as parts of the image. The statistics of these shifts encode information about the turbulence strength (as characterized by Cn2, the refractive index structure function constant) along the optical path. The shift variance observed is simply proportional to the variance of the tilt of the optical field averaged over the area being tracked and averaged over the camera aperture. By presuming this turbulence follows the Kolmogorov spectrum, weighting functions, which relate the turbulence strength along the path to the shifts measured, are derived. These weighting functions peak at the camera and fall to zero at the object. The larger the area observed, the more quickly the weighting function decays. One parameter we would like to estimate is r0 (the Fried parameter or atmospheric coherence diameter.) The weighting functions derived for pixel sized or larger parts of the image all fall faster than the weighting function appropriate for estimating the spherical wave r0. If we were to presume that Cn2 is constant along the path, then an estimate for r0 could be obtained for each area tracked, but since the weighting function for r0 differs substantially from that for every realizable tracked area, it can be expected that this approach would yield a poor estimate. Instead, the weighting functions for a number of different patch sizes can be combined through the Moore–Penrose pseudoinverse to create a weighting function that yields the least-squares optimal linear combination of measurements for the estimation of r0. This approach is carried out for one example and is shown to give noisy results. A modified version of this approach that creates larger patches by averaging several smaller patches together solves this noise issue. This approach can also work to estimate other atmospheric parameters.

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Estimation of turbulence from time-lapse imagery

The significance of tilt anisoplanatism is established, and a measurement theory, based on aperture-averaging intensity scintillation, is developed. The theory is a direct extention of the technique currently used to determine the isoplanatic angle as defined by Fried
	 [ J. Opt. Soc. am.72, 52 ( 1982)]. By using this theory, a physically realizable binary aperture-weighting function is derived for a particular case of interest. It is noted that direct quantitative measurements of tilt anisoplanatism can also be made, under specific circumstances, by tracking the relative centroid motion of a binary star pair. Thus independent verification of the remote-sensing theory for tilt anisoplanatism, based on aperture-averaging scintillation measurements, should be possible.

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D. vL. Marquis

Papers

Stellar scintillation technique for the measurement of tilt anisoplanatism