Amber L. Iler, KBR; Bobby R. Hunt, KBR, INC; David G. Sheppard, KBR
Keywords: Image resolution, atmospheric turbulence, compressive sensing
Abstract:
Atmospheric turbulence is a major constraint on the ability of uplooking optical systems to gather information (including imagery) of objects orbiting the Earth. Better observations, for space domain and space situational awareness, depend on mitigating the effects of this turbulence.
An emerging, and fruitful, method in signal processing, Compressive Sensing (CS) through sparse and redundant representations, has been applied in recent years to atmospheric turbulence in optical uplooking telescopes [1, 2]. These new methods adopt alternative concepts for the representation of data in terms of a set of basis functions. Conventional tools of signal and image processing focus on Fourier series and transforms to generate representations of data. Conversely, CS creates basis sets of functions that do not possess the usual properties of orthonormality and minimality. CS compiles these basis sets from actual data and contains them in a collection of functions called a dictionary [3]. These dictionary methods have led to improvements in the mitigation of turbulence by the methods of blind deconvolution [2, 4]. However, the success of these dictionary methods relies on knowledge and access to dictionaries that encompass the turbulence present in observations.
Additionally, a continuing problem in atmospheric turbulence is anisoplanatism, where the point-spread-function (PSF) of the optical system varies with position across the image focal plane. In this situation, mitigation deblurring actions for one position must be adapted to different actions in a different focal plane position. Further, because a PSF determines resolution in an image region, anisoplanatism alters inherent resolution in different image positions. This was graphically pointed out by Fried in his classic paper on the probability of a diffraction-limited Lucky Image observed through turbulence [5]:
It is appropriate to note that the probability we have calculated applies independently to separate isoplanatic patches on the image. This means that in any one image, rather than its being entirely good or entirely poor resolution, there will be distributed over the image field- of-view a set of rather small regions, isoplanatic patches, in which the resolution is good. The rest of the image area will have much poorer resolution.
Frieds comments prompt an important question: What do the statistics of the resolutions, present in observations through turbulence, predict? The answer to this question is critical to planning and having available the necessary variety of dictionaries to be used in advanced blind deconvolution algorithms [4].
The analysis of Fried, deriving the probability of diffraction-limited imaging in turbulence, has a logical complement mandated by the laws of probability, i.e., the probability of Unlucky Imaging. This is the probability that, in each anisoplanatic patch as described by Fried, there will be less than diffraction-limited resolution. From this viewpoint we show that:
The Unlucky Image probability can be computed in simple numerical fashion from the calculated form of Lucky Image probability in Frieds analysis.
The definitions of probability mean the Unlucky Image statistics describe a Cumulative Distribution Function (CDF).
The CDF of the Unlucky Image can be converted into the corresponding Probability Density Function (PDF) of different resolutions in turbulent images.
A PDF, so derived, gives the distribution of local resolution variability for Frieds descriptions of the Lucky Image behavior (quoted above), and the resolutions are directly related to the turbulence, parameterized by the Fried parameter, r0, and the pupil diameter used in image formation.
We conclude by presenting a PDF of resolutions, computed from real PSFs experimentally collected in atmospheric turbulence, displaying the same shape and behavior as predicted by our Unlucky Image analysis. We disclose, further, another set of numerical observations, reported in the literature, that have the same behavior for the case of uplooking observations, as well as similar results for additional uplooking calculations beyond those referenced from the literature. Thus, we verify, for planning resources when imaging in turbulence, it is possible to estimate, from optical system properties and turbulence strength, the resolution variations expected in space-variant blind deconvolution of anisoplanatic behavior. This has immediate and direct application to the planning of resources and systems for collection of data for Space Domain Awareness (SDA). From these estimates, it is then possible to determine the range of dictionary resolution behaviors, which must be provided for dictionary-based turbulence mitigation to achieve adaptive blind deconvolution of observations for SDA.
References
[1] B. R. Hunt., K. Knox, Sparse and Redundant Dictionaries in Representation of Atmospheric Turbulence Point Spread Functions, OSA Classical Optics Congress, Big Island, Hawaii, June 2014.
[2] B. R. Hunt, K. Knox, Optimal Dictionaries for Sparse Solutions of Multi-frame Blind Deconvolution. Proceedings, AMOS Conference, Maui, Hawaii, September 2014.
[3] M. Elad, Sparse and Redundant Representations: From theory to applications in signal and image processing, Springer, New York, 2010.
[4] D. G. Sheppard, A. L. Iler, B. R. Hunt, Block-based streaming blind deconvolution for space-variant turbulence mitigation, Proc. SPIE. Vol. 11836, San Diego, August 2021.
[5] D. L. Fried, Probability of getting a lucky short-exposure image through turbulence, J. Opt. Soc. Amer., 68 (22), 1651-1658, 1978.
Date of Conference: September 27-20, 2022
Track: Atmospherics/Space Weather