Real Time Deconvolution of Adaptive Optics Ground Based Telescope Imagery

Abstract

This article presents new developments in a multiframe blind deconvolution algorithm for imaging low earth orbiting objects during flyover. The foundational aspects of the algorithm rely on the constrained maximum likelihood (ML) formulation in the presence of Poisson noise, previously developed in Schulz et al. (2018). The new algorithm achieves real time evaluation capability at over two frames per second, which is allowed by two novel aspects. First, the prototype algorithms are transferred to highly parallelized computations on graphical processing units (GPUs) with CUDA implementation that reduce the computational time of a single iteration by a factor of up to 100. Second, new numerical optimization strategies are developed and demonstrated to accelerate the convergence of the algorithm by a factor of 5 to 10. Several other new capabilities are also demonstrated in this article. We derive and implement a modified variation of the algorithm that achieves subpixel resolution, which is shown effective on real and simulated data. Finally, a new post-processing visual enhancement technique is proposed with several examples, which in part helps deal with the dynamic range degradation due to glint.

Appendix A: PSF Iterations

Recall from “Deconvolution Model” the blurry image data model given in Eq. 1. To solve the objective function (3) for the object u and the PSFs simultaneously, an alternating maximization is used: the set of estimated PSFs is fixed and a new estimate of u is iterated, and then u is fixed and the PSFs are iterated. For a fixed set of PSFs, the MLE iteration on the image was given in Eq. 4, and its corresponding accelerated version in Eq. 7. For a fixed image variable u, the iteration for the PSFs was given in [22], and we provide the main ideas again here.

The approach uses a constrained model for the PSFs based on the optics and aperture of the telescope. The aperture provides a model for the PSFs, with a free parameter to optimize for being the optically and turbulence-induced phase aberration. The PSF iterations are determined similarly by differentiating the objective function in Eq. 3 with respect to the phase aberration.

Let |A(ξ)| denote the magnitude of the aperture function and 𝜃_k(ξ) denote the optically and turbulence-induced phase aberration of the k-th image frame. Then the corresponding coherent and incoherent PSFs are given by (see [4] for details)

$$ g(x;\theta_{k}) = \mathcal{F}^{-1} \left\{ A(\xi) e^{i\theta_{k}(\xi) } \right\} , $$

(15)

and

$$ h_{k}(x) = |g(x; \theta_{k})|^{2}. $$

(16)

Differentiating the ML objective function in Eq. 3 with respect to 𝜃_k, the fixed point phase iteration step is determined to be

$$ \theta^{n+1} = \text{Phase} {\left [ \mathcal{F} \left\{ \left( \frac{d_{k}} { {i_{k}^{n}} } * \tilde u^{n} (x) \right) g(x;{\theta^{n}_{k}}) \right\} \right ] }. $$

(17)