# Post-AO high-resolution imaging using the Kraken multi-frame blind
deconvolution algorithm

Douglas A. Hope  Georgia State University, 25 Park Place, Atlanta, GA 30303,
USA  Stuart M. Jefferies  Georgia State University, 25 Park Place, Atlanta, GA
30303, USA  University of Hawaii, 34 Ohia Ku Street, Pukalani, HI 96768, USA
Gianluca Li Causi  INAF-IAPS, Via del Fosso del Cavaliere, 100, 00133 Roma,
Italy  INAF-OAR, Via Frascati, 33, 00040 Monte Porzio Catone (RM), Italy
INAF-ADONI, Adaptive Optics National Laboratory, Italy  Marco Landoni  INAF -
Osservatorio Astronomico di Cagliari. Via della Scienza - Selargius (CA) -
Italy  INAF - Osservatorio Astronomico di Brera, via E Bianchi 46 Merate (LC)
- Italy  Marco Stangalini  ASI Italian Space Agency,Via del Politecnico snc,
00133, Rome, Italy  INAF-OAR, Via Frascati, 33, 00040 Monte Porzio Catone
(RM), Italy  Fernando Pedichini  INAF-OAR, Via Frascati, 33, 00040 Monte
Porzio Catone (RM), Italy  INAF-ADONI, Adaptive Optics National Laboratory,
Italy  Simone Antoniucci  INAF-OAR, Via Frascati, 33, 00040 Monte Porzio
Catone (RM), Italy  INAF-ADONI, Adaptive Optics National Laboratory, Italy

###### Abstract

In the context of extreme adaptive optics (ExAO) for large telescopes, we
present the Kraken multi-frame blind deconvolution (MFBD) algorithm for
processing high-cadence acquisitions, capable to provide a diffraction-limited
estimation of the source brightness distribution. This is achieved by a data
modeling of each frame in the sequence driven by the estimation of the
instantaneous wavefront at the entrance pupil. Under suitable physical
contraints, numerical convergence is guaranteed by an iteration scheme
starting from a Compact MFBD (CMFBD) which provides a very robust initial
guess which only employs a few frames. We describe the mathematics behind the
process and report the high-resolution reconstruction of the spectroscopic
binary $\alpha~{}And$ ($16.3~{}mas$ separation) acquired with the precursor of
SHARK-VIS, the upcoming high-contrast camera in the visible for the Large
Binocular Telescope.

High angular resolution, Deconvolution, Ground-based astronomy, Astronomical
seeing, Computational astronomy

††journal: ApJ

##  1 Introduction

Turbulence in the Earth’s atmosphere unavoidably degrades the sharpness of
astronomical images obtained with ground-based telescopes. The turbulence
distorts the planar wavefront coming from the astronomical point sources and
prevents the formation of the ideal Airy diffraction pattern at the telescope
focal plane. Fortunately, by using Adaptive Optics (AO), an advanced
instrumentation technique developed over the last few decades, we can
partially correct these distortions in real-time. The Large Binocular
Telescope (LBT) currently hosts one of the most advanced extreme AO systems
(ExAO) built to date, called FLAO for First Light Adaptive Optics (Esposito et
al., 2010; Pinna et al., 2015). The LBT’s ExAO corrects the first 550 modes of
the wavefront thanks to its Pyramid Wavefront Sensor (Ragazzoni, 1996) at a
cadence of 1kHz with 3ms of lag time that, in the visible bands (550-850nm),
provides a sharp point spread function (PSF) with a core width very close to
diffraction limit (rms wavefront error of about 90-100nm on stars brighter
than mag 6).

Performance of ExAO correction is commonly measured by the “Strehl ratio”
(Roggemann & Welsh, 1996). For the best frames we have acquired at the LBT in
the visible band the Strehl ratio is around 0.3. This highlights the fact
that, unfortunately, even ExAO correction is never perfect, and the resultant
post-AO imagery does not reach the diffraction limit: a significant fraction
of the starlight remains in many “speckles” outside the sharp PSF core, as
depicted in the first panel of figure 1. Moreover, the morphology of these
speckles rapidly evolves with a typical timescale of a few ms (see Stangalini
et al. (2016)).

In practice, such varying speckle patterns prevent us from reaching the
theoretical angular resolution of the telescope, thus also reducing the
contrast. They also block us from resolving closely spaced sources, especially
when dealing with faint sources close to bright objects. Digital post-
processing of the images acquired after the adaptive optics can effectively
help to recover the full spatial resolution of the telescope, so long as the
speckles’ evolution is measured with a sufficiently fast temporal cadence in
order to freeze the PSF in each frame.

As a part of the SHARK-VIS project (Pedichini et al., 2016; Mattioli et al.,
2018), the high constrast visible imager at the Large Binocular Telescope
(LBT), we are investigating new post-processing techniques, e.g. Speckle-Free
Imaging and Speckle-Free Angular Differential Imaging (SFI and SFADI and, Li
Causi et al. (2017); Mattioli et al. (2019)), with the aim of both achieving
diffraction-limited resolution performance and noise-limited high contrast
imaging at sub-arcsecond separation.
In this context, we present here the first astronomical results obtained with
a novel implementation of the Multi-Frame Blind Deconvolution (MFBD)
algorithm, called Kraken MFBD (Hope et al., 2016). The Kraken MFBD algorithm
was initially developed for high-resolution, high-contrast imaging of Earth-
orbiting artificial satellites, which, like astronomical sources, suffers from
significant image degradation due to atmospheric turbulence.

Figure 1: Data modeling and object estimation in MFBD. Panels from left to
right: one of the selected frames (i) and its Kraken model (ii); SFI
reconstruction from Mattioli et al. (2019) (iii); CMFBD initial estimate based
on the best 2 frames (iv); full Kraken MFBD reconstruction (v).

##  2 The Kraken MFBD algorithm

MFBD is a known mathematical technique (Jefferies & Christou, 1993; Schulz,
1993) able to estimate both the object and the PSF evolution directly from the
recorded images, under the only requirement that the object is stationary
during the observations and the changing PSF demonstrates significant
diversity. Although MFBD proved to surpass the performance of other speckle
imaging methods (Matson, 2008), and is nowadays routinely used in solar
observations (van Noort et al., 2005), its use has not gained traction in the
nigthtime astronomy community. This lack of traction is due to the high
computational cost of the MFBD algorithm, difficulties associated with
processing many images, and uncertainty in the reconstructions’ photometric
quality. Research to overcome these obstacles has been limited, but our recent
studies have provided several algorithmic improvements, that allow the current
Kraken software to address these problems and provide state-of-the-art
reconstructions.

We have discovered that restoring images acquired through strong atmospheric
turbulence is often best achieved using a variety of restoration techniques in
combination. Our software is built on this premise and provides access to
several restoration approaches and the combinations of these approaches, which
include: Basic MFBD, Compact MFBD Hope & Jefferies (2011); Hope et al. (2019),
Myopic MFBD with WFS data Conan et al. (2000), Aperture diversity Hope et al.
(2016), Wavelength diversity Ingleby & McGaughey (2005), and Phase diversity
Paxman et al. (1992).

Probably the largest shortcoming of MFBD algorithms is their sensitivity to
entrapment in local minima; especially, when using many frames in the
restoration process. CMFBD provides a means to produce good initial estimates
for the object and PSFs whilst using a greatly reduced number of variables,
reducing the susceptibility to entrapment in local minima. We have found that
a multi-step restoration process in which the estimates for the object and
PSFs are refined before committing to a full MFBD restoration provides the
best performance. We note that myopic MFBD and the diversity imaging
techniques supported in Kraken, provide a constraint on the atmospheric
wavefront, and thus the PSFs for the images. These constraints thus provide
strong leverage for separating the object from the PSFs by MFBD.

One development is Compact MFBD (CMFBD: Hope & Jefferies (2011); Hope et al.
(2019)), a means to acquire high-quality initial estimates of the object and
PSFs for the MFBD, by using a relatively small number of variables and prior
information on the nature of the PSFs. Another development is modeling the PSF
by estimating the real and imaginary parts of a band-limited function whose
cut-off frequency is commensurate with the telescope’s cut-off frequency
(Miura, 2003), a model able to accommodate any wavefront amplitude variation
faster than the atmospheric coherence time, as well as the errors incurred
from using a Fourier Optics model of the PSF for long exposure times.

In practice, the basic Kraken MFBD algorithm estimates the blur-free object
and the PSFs for a set of data frames by minimizing the error metric

$\epsilon=\Sigma_{l}\Sigma_{x,y}\left[(g_{l}(x,y)-f(x,y)\odot
h_{l}(x,y))/\sigma_{l}(x,y)\right]^{2}$ (1)

Here $g(x,y)$ is an observed image, $f(x,y)$ is the object, $h(x,y)$ is the
point spread function, $\sigma(x,y)$ is a weighting term that depends on the
noise in the observed image, $\odot$ denotes convolution, and $l$ is the
number of frames.

The object is modeled via the reparameterization $f(x,y)=\phi^{2}(x,y)$ and
the PSF is modeled via the wavefront amplitudes, $A(u,v)$, and phases,
$\psi(u,v)$. That is, $h(x,y)=FT^{-1}\left[P(u,v)\star P(u,v)\right]$ where
$P(u,v)=A(u,v)\exp{(-i\psi(u,v))}$ and $\star$ denotes correlation.

The quality of an MFBD restoration is expected to improve as the number of
frames ($l$) used in the restoration increases (because there is more
information available). However, as the number of frames increases, so does
the number of variables required to model all the data. Eventually, the
estimated variables’ quality deteriorates due to the large dimensionality of
the parameter hyperspace that leads to inevitable entrapment in local minima
during the optimization. These local minima can be far from the global
minimum. To reduce the probability of entrapment in a local minimum, we use
CMFBD to provide high-quality estimates for the object and PSFs for all the
data frames. These estimates are then used in a traditional MFBD restoration.
That is, we use CMFBD to provide estimates of $f(x)$ and $h_{l}(x)$ which are
then iteratively updated using equation (1) and a conjugate gradients-based
optimization algorithm.

The CMFBD algorithm is a two-step process. In the following we use the Fourier
transform pair identities $[g(x,y);G(u,v)],[f(x,y);F(u,v)]$ and
$[h(x,y);H(u,v)]$.

###  2.1 CMFBD: Step 1

We control the number of variables that need to be estimated by restricting
the restoration to a subset of the data that contains the selected best
frames. Here the selection is done to provide the highest signal-to-noise
signal across a large fraction of the object’s Fourier spectrum, using the
smallest number of frames. The downside to this approach, of course, is that
it only uses a sub-set of the data, and therefore we are not capitalizing on
all of the information that is available to us. However, we can use the non-
selected data frames to provide additional constraints on the PSF estimates
for the selected data frames without increasing the number of model
parameters. That is, the parameters that describe the object are determined by
the information available in the selected data frames. The parameters that
describe the PSFs for the selected frames, on the other hand, are determined
by the information in all of the images in the data set. We also invoke prior
knowledge we have about the PSFs for the non-selected frames via two penalty
functions. The first enforces the prior knowledge that PSFs are positive
functions. The second enforces the prior knowledge that the PSFs should not
depend on how they are estimated. We then minimize the error metric

$\displaystyle\epsilon$ $\displaystyle=$
$\displaystyle\Sigma_{k}\Sigma_{x,y}\left[(g_{k}(x,y)-f(x,y)\odot
h_{k}(x,y))/\sigma_{k}(x,y)\right]^{2}$ (2) $\displaystyle+$
$\displaystyle\alpha\Sigma_{k}\Sigma_{j\neq
k}\Sigma_{x,y}m_{j}(x,y)h_{j,k}^{2}(x,y)$ $\displaystyle+$
$\displaystyle\gamma\Sigma_{u,v}\Sigma_{k}\Sigma_{k^{\prime}\neq
k}~{}M_{k,k^{\prime}}(u,v)\Sigma_{j}M_{j}^{H}(u,v)$ $\displaystyle|$
$\displaystyle\beta_{j,k}(u,v)H_{k}(u,v)-\beta_{j,k^{\prime}}(u,v)H_{k^{\prime}}(u,v)|^{2}$

Here the index $k$ is over the frame-selected images, the index $j$ is over
the remaining images (the non-selected frames), and $k+j=l$ (the total number
of frames). The penalty scalar factors $\alpha$ and $\gamma$ are set to 0.1
(Hope et al., 2019). The second term in equation 2 is the non-control frame
PSF non-negativity penalty function. Here the estimate of the PSF for the
$j^{th}$ non-control frame, estimated via the optical transfer function (OTF)
for the $k^{th}$ control frame is given by

$h_{j,k}(x,y)=|FT^{-1}\left[H_{k}(u,v)\beta_{j,k}(u,v)\right]|^{2}~{},$ (3)

where

$\beta_{j,k}(u,v)=G_{j}^{obs}(u,v)/G_{k}^{obs}(u,v)$ (4)

and $m_{j}(x,y)=1$ if $h_{j,k}(x,y)<0$, and $m_{j}(x,y)=0$ otherwise. The
third term in equation (2) is a PSF consistency metric which enforces the
prior knowledge that the PSF estimate for a non-control frame should not
depend on which control frame is used to estimate it. In this term
$M_{k,k^{\prime}}(u,v)$ is a binary mask that is unity at spatial frequencies
where the image spectra, $G^{obs}_{k}(u,v)$ and $G^{obs}_{k^{\prime}}(u,v)$,
have an SNR above some threshold (we used SNR $>$ 5), and zero elsewhere.
Similarly, $M^{H}_{j}(u,v)$ is a binary mask that is zero at spatial
frequencies where the Fourier spectrum $G^{obs}_{j}(u,v)$ is zero, and is
unity elsewhere.

During the minimization we further control the number of variables by assuming
that the amplitudes $A(u,v)$ are constant, and only solve for the variables
$\phi(x,y)$ and $\psi(u,v)$. We maintain this assumption through to the full
MFBD step (see section 2.3).

###  2.2 CMFBD: Step 2

Once the object and PSFs for the control frames have been estimated, we can
use the recovered object in a forward deconvolution problem (i.e., hold
$F(u,v)$ fixed) to provide good initial PSF estimates for the non-control
frames, $h_{j}(x,y)$. Similar to step 1, we enforce the prior knowledge that
the PSFs should not depend on how they are estimated. We then minimize the
error metric

$\displaystyle\epsilon$ $\displaystyle=$
$\displaystyle\Sigma_{j}\Sigma_{x,y}\left[(g_{j}(x,y)-f(x,y)\odot
h_{j}(x,y))/\sigma_{k}(x,y)\right]^{2}$ (5) $\displaystyle+$
$\displaystyle\gamma\Sigma_{j}\Sigma_{u,v}M_{k,j}(u,v)$ $\displaystyle|$
$\displaystyle P_{j}(u,v)\star
P_{j}(u,v)-\Sigma_{k}H_{k}(u,v)\beta_{j,k}(u,v)/K|^{2}$

###  2.3 MFBD

At the end of step 2, we have reasonable initial estimates for the object and
all the PSFs. We then run a traditional MFBD with all the frames by minimizing
equation (1). This final step capitalizes on the information on the object
that is available in the non-control frames. Because the MFBD algorithm is
provided with high-quality estimates for the object and PSFs, the optimization
algorithm has a better chance of not getting trapped in a local minimum far
from the global minimum. We run the MFBD until we reach convergence, defined
by a change in error metric of less than $10^{-5}$. At this point, we restart
the MFBD and allow $P(u,v)$ to vary without restricting the amplitudes. This
is important for achieving the best restoration (Jefferies et al., 2002).
Practically, we move from estimating the variables $\psi(u,v)$ to new
variables $P_{R}(u,v)$ and $P_{I}(u,v)$, the real and imaginary parts of
$P(u,v)$ (Miura, 2003).

We note that high-fidelity modeling of the speckle pattern requires that the
restoration is performed on a pixel grid that samples the wavefront phase with
at least 4-5 pixels sampling the atmospheric coherence length ($r_{0}$) for
the observing conditions. It is also essential to accurately model the
telescope aperture’s details, including the structure associated with the
spider supporting the secondary mirror.

##  3 The Cloud Computational Architecture

The workload of Kraken in terms of CPUs availability and type of access, which
is mainly burst, is best suited in the context of Cloud Computing (Williams et
al., 2018). For this reason, following a similar approach as described in
Landoni et al. (2018, 2019), we prepared a Platform as Service architecture on
the Amazon Web Services (AWS)111https://aws.amazon.com/ cloud. In particular,
we deployed a custom machine image with a fully working MATLAB license aiming
to exploit, throughout the Parallel Computing Toolbox offered by the language,
up to 96 vCPU cores available on the computer node on AWS. For the storage, we
made use of both Elastic Block Storage (EBS) for the local storage of the node
and Amazon Simple Storage Service (S3) for long-term archiving.

##  4 Observations and results on $\alpha$ And binary

For our first application of the MFBD to astronomical targets we chose the
close binary star $\alpha$ And, which our team directly resolved through
imaging for the first time in Mattioli et al. (2019) by means of the Speckle
Free Imaging technique (SFI) that we developed for processing SHARK-VIS image
sequences (Li Causi et al., 2017). This binary is composed by a very bright
$mag_{R}=2$ primary star, $\alpha$ And A, and a $1.3\cdot 10^{-1}$ contrast
companion, $\alpha$ And B, at a separation of $16.3mas$ at the moment of the
observations, comparable to the LBT theoretical diffraction limit of
$\lambda/D=16.4mas$ at the observation wavelength of $656nm$.

Observations were performed at LBT on 2018 December 23. A sequence of fast-
cadence (1 kHz) short-exposure (1 ms) images were acquired continuously for 1
minute with the SHARK-VIS Forerunner instrument (Pedichini et al., 2017),
observing at $656nm$ with a $10nm$ bandwidth filter. A data set of 60,000
frames of 200x200 pixels size were aquired, corresponding to a field of view
of $\sim 1~{}arcsec^{2}$, while the FLAO AO system (Esposito et al., 2010) was
correcting 300 modes in a closed loop at 400 Hz. The seeing was unstable and
above 1 arcsec, according to the DIMM telemetry.

Due to the poor seeing, a best-frames selection was necessary. To do this, we
made a best-frame list, consisting of 3000 frames, by maximizing peak value
and sharpness, and run the Kraken reconstruction of the first 219 of them, in
order to be comparable with our previous result in Mattioli et al. (2019). We
also did not perform PSF amplitude estimation during the restoration process.

Thanks to the good quality of the acquired frames, the initial CMFBD steps
only needed to use the best two frames out of these to provide excellent
estimation of the object and the PSFs (see figure 1, panel (iv)). This allowed
the complete Kraken MFBD to return the very clean image of the binary star
shown in figure 1, panel (v), in which the two binary components are fully
disentangled. This result clearly surpasses our first imaging detection in
Mattioli et al. (2019) (shown in panel (iii) of the same figure for
comparison), achieving an effective spatial resolution of $6.7mas$ full width
at half maximum, as measured from a two-gaussian fit of the reconstructed
image. It’s also noticeable that no prior information on the nature of the
source has been given to the algorithm, in which each pixel in the final image
is estimated independently on the others.

Figure 2: Spectro-interferometric orbit of $\alpha$ And B overlaid on the
Kraken MFBD reconstructed image, shown in log color scale. Black crosses
indicate the gaussian-fitted positions for the two stars, while the green
triangle is the expected location of the companion at the observation date.
Sky orientation is shown by North and East arrows, while the circle in bottom
left corner indicates the achieved FWHM resolution.

By fitting a double gaussian on this resulting image we derive a separation of
$16.97\pm 1.17mas$ and a position angle $123\pm 4.3\deg$. These numbers are
compatible with the nominal expected position, derived from the orbital
parameters given by Branham (2016) (position angle $126.7\deg$, separation
$16.3mas$), which were obtained by combining previous spectroscopic and
interferometric observations (e.g. Pan et al. (1992); Pourbaix, D. et al.
(2004)), as shown in Figure 2. Also, the flux ratio that we measured between
the components, of $6.29\pm 0.83$, is compatible with the expected value of
$7.9\pm 3.6$ (at $656nm$) considering the spectral types (B8IV and A3V) and
ratio of radii ($1.64\pm-0.23$) given in the literature (Ryabchikova et al.,
1999).

Errors in our photometric and astrometric result include both instrument
uncertainties (namely a $\pm 1\deg$ in the orientation of the Forerunner
instrument and a $\pm 0.1mas/pix$ in the estimated image scale of
$5.08mas/pix$), plus the variance of the MFBD image reconstruction, estimated
by repeating the full Kraken restoration process on the next few subsets of
219 frames in our best-frames list.

##  5 Conclusions

We presented the first application of the Kraken algorithm to the case of post
adaptive optics high-resolution imaging in the visible on a large telescope,
describing its mathematical basis and our implementation solutions.

Kraken is a high performace multi-frame blind deconvolution algorithm, working
on fast-cadence frame sequences at KHz rate, which freeze the speckles
evolution typical of the AO residual PSF. It works by first computing a very
precise initial guess by means of the Compact MFBD estimator, then following a
complete MFBD procedure which performs a physical modeling of the
instantaneous wavefront for each frame, ending up with an high-resolution
reconstruction of the source brightness distribution.

We shown the result obtained on a fast-cadence observation of the
spectroscopic binary $\alpha~{}And$ ($16.3mas$ separation) acquired at the
Large Binocular Telescope with the SHARK-VIS Forerunner instrument.
Noticeably, the Kraken result clearly reconstructed two disentangled
components without any prior information about the nature of the source,
definitely surpassing the first direct detection that we obtained with the
speckle-free imaging method.

We demonstrated that photometry and astrometry performed on the Kraken
reconstruction are compatible with those expected from the companion’s orbit,
already known by means of spectroscopy and interferometry. Also, the
estimation of these important astronomical observables are very stable with
respect to frames replacements. These facts together guarantee both the
accuracy and the robustness of the Kraken output and its adequacy to be
applied to high-resolution imaging in astronomy.

The Air Force Office of Scientific Research funded DAH and SMJ’s contributions
to this work through contract FA9550-14-1-0178. Cloud computing resources have
been sustained by ICT-INAF (see https://www.ict.inaf.it/computing/amazon-web-
services-on-demand-access/) .

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