## Abstract

Multiframe blind deconvolution - the process of restoring resolution to blurred imagery when the precise form of the blurs is unknown - is discussed as an estimation-theoretic method for improving the resolving power of ground-based telescopes used for space surveillance. The imaging problem is posed in an estimation-theoretic framework whereby the object’s incoherent scattering function is estimated through the simultaneous identification and correction of the distorting effects of atmospheric turbulence. An iterative method derived via the expectation-maximization (EM) procedure is reviewed, and results obtained from telescope imagery of the Hubble Space Telescope are presented.

© Optical Society of America

## 1. Introduction

The distorting effects of atmospheric turbulence on ground-based imagery of
astronomical and space objects are well-documented, and numerous methods have been
proposed, implemented, and tested for their correction^{1}. These methods can be broadly classified into two categories: post-processing
through signal processing whereby the image distortions are identified and
compensated by computer processing of the recorded imagery; and pre-processing
through adaptive optics whereby the distortions are identified and compensated with
wavefront sensors and deformable mirrors so that undistorted imagery is, in
principle, recorded directly. With current technology the cost of adaptive
compensation can be prohibitive, particularly if full compensation is desired.
Because of this, many systems in use today benefit only from partial compensation or
none at all, and post-processing methods such as the one discussed in this paper
provide an economical means for restoring a telescope’s lost resolving
power.

It is widely accepted that only modest improvements in image quality can be attained
through the measurement and processing of long exposure (on the order of minutes to
hours) telescope imagery. This difficulty is often quantified through
Fried’s seeing parameter *r*
_{0}, which provides a
measure of the largest spatial frequency that is present in the long-exposure
imagery. Whereas a diffraction-limited telescope records spatial frequencies as high
as the ratio *D*/*λd*_{o}
, where
*D* is the telescope diameter, *λ* is
the nominal wavelength being sensed, and *d*_{o}
is the
distance from the telescope to the object, long-exposure imagery acquired through
atmospheric turbulence will only contain frequencies to the turbulence-limited
cut-off frequency
*r*
_{0}/*λd*
_{0}, where
*r*
_{0} can take values under 10cm. For a 2m telescope
with *r*
_{0} = 10cm, then, recovery of diffraction-limited
resolution requires one to super-resolve the long-exposure imagery by a factor of
20. Such improvements are rarely, if ever, accomplished with real data.

Exposure times on the order of a few milliseconds, however, provide imagery with
spatial frequency information out to the diffraction limit. This fact was first
exploited by Labeyrie^{2}, and has since spawned a number of signal-processing methods broadly
classified as *speckle imaging techniques*. A review of these is
found in Ref. 1 and the many references within. Many of these, such as
Labeyrie’s speckle interferometry, the Knox-Thompson method^{3}, and the triple-correlation method^{4}, rely on the existence of a data transformation that is invariant to the
effects of atmospheric turbulence. After performing this transformation along with
any required calibration, a non-trivial image-recovery problem must be solved to
obtain an estimate of the underlying object. As an alternative to these methods, the
data can be processed to directly estimate and correct for the effects of
atmospheric turbulence. Methods based on this approach are broadly classified as
*multiframe blind deconvolution*
^{5} techniques. The application of a multiframe blind deconvolution method to
real telescope data is the focus of this paper.

## 2. Mathematical model

For vertical path, ground-to-space imaging, the turbulence-induced distortions are commonly modeled as a time-varying phase screen that induces a point-spread function given by:

where: *u* and *y* are two-dimensional spatial
variables in the telescope-pupil and image planes, respectively;
*A*(·) is the telescope pupil function;
*θ*_{t}
(·) is the time-varying
phase distortion caused by turbulence-induced changes in the refractive index of the
atmosphere; *λ* is the nominal wavelength; and
*d*_{o}
is the distance from the object to the
telescope. This is the isoplanatic approximation and results in an imaging model
that is linear and spatially invariant. In the absence of other effects, then, the
time-varying intensity recorded by the telescope’s camera is:

$$\phantom{\rule{3.5em}{0ex}}=h(y;{\theta}_{t})*o\left(y\right),$$

where *o*(·) is the object’s incoherent
scattering function, after magnification and inversion^{6}. The notation *i*(·;
*θ*_{t}
, *o*) shows the image
intensity’s explicit dependence on the unknown object
*o*(·) and turbulence-induced phase aberrations
*θ*_{t}
(·).

Many telescope imaging systems in use today use charge-coupled-device (CCD) cameras
to record the image intensity. As discussed by Snyder et al. in Ref. 7, however, the data acquired by these cameras include other
effects such as internal and external background counts, nonuniform camera response,
electronic read-out noise, and other sources of camera bias. Accordingly, a
reasonable model for the data acquired by the *n*th detector element
in a CCD camera array during the *k*th exposure is:

where *N*_{k}
[*n*] is a Poisson distributed
random variable that models the number of object-induced photocounts recorded by the
detector, *M*_{k}
[*n*] is a Poisson
distributed random variable that models the number of background-induced (both
internal and external) photocounts recorded by the detector,
*g*_{k}
[*n*] is a Gaussian distributed
random variable that models the CCD read-out noise, and
*b*[*n*] is a deterministic bias induced by the
electronic circuitry used to acquire and record the data. All of these variables are
usually independent from each other and across detector elements.

The expected number of object-induced photocounts is modeled as:

$$\phantom{\rule{13.2em}{0ex}}=\gamma \left[n\right]{\int}_{{y}_{n}}{\int}_{{t}_{k}}^{{t}_{k}+T}i(y;{\theta}_{t},o)\mathit{dydt},$$

where *t*_{k}
is the time and *T* the duration
of the *k*th exposure, *y*_{n}
is the spatial
region over which the *n*th detector element integrates the image
intensity, and *γ*[*n*] is a
spatially-varying scale factor that accounts for detector efficiency. Pixels that
record meaningless data are accommodated with this model by setting
*γ*[*n*] = 0. The phase aberrations are
usually constant over a short exposure, and the detector regions are often small in
size relative to the fluctuations in the image intensity so that the integrating
operation can be modeled by the sampling operation:

where *y*_{n}
is the location of the *n*th
detector element and |*y*| is its integration
area. Furthermore, the object and point-spread functions are often approximated on a
discrete grid so that:

and

where Δ_{y}, Δ_{x}, and Δ_{u} are the image-, object-, and pupil-plane grid sizes, respectively. When the
pupil-plane array size is *N* × *N* and Δ_{u}Δ_{y}/*λd*_{o}
= 1/*N*, the
computations required to create the point-spread function can be accomplished
efficiently by using a fast Fourier transform (FFT) algorithm.

The expected number of background-induced photocounts
*M*_{k}
[*n*] is denoted by
*I*_{b}
[*n*] and assumed to be the same
for each exposure. The CCD read-out noise
*g*_{k}
[*n*] are typically modeled as
zero-mean, independent random variables with variance
*σ*
^{2}[*n*]. Appropriate
values for the gain function *γ*[·], background
mean *I*_{b}
[·], read-noise variance
*σ*
^{2}[·], and bias
*b*[·] are usually determined through a carefully
controlled study of the data acquisition system. This typically involves the
analysis of data acquired both without illumination (dark frames) and with uniform
illumination (flat fields).

## 3. Maximum-likelihood estimation

Consistent with the noise models developed in the previous section, the data recorded
by each detector element in a CCD camera are a mixture of independent Poisson and
Gaussian random variables. Accordingly, the probability of receiving
*N* object- and background-induced photocounts in the
*n*th detector element during the *k*th exposure is:

where

$$=\gamma \left[n\right]\sum _{m}h({y}_{n}-{x}_{m};{\theta}_{{t}_{k}})o\left[m\right].\phantom{\rule{1.0em}{0ex}}$$

Here, *o*[*m*] =
*T*|*y*|${\mathrm{\Delta}}_{x}^{2}$
*o*(*x*_{m}
) contains the dependence on
the unknown object, and *h*(·;
*θ _{tk}*) contains the
dependence on the unknown phase aberrations. Furthermore, the probability density
for the read-out noise is

and the density for the measured data at each detector element is:

For a given set of data {*d*_{k}
[·]}, the
maximum-likelihood estimate of the unknown object and phase aberrations is selected
to maximize the likelihood:

or, as is commonly done, its logarithm (the log-likelihood):

$$\phantom{\rule{11.0em}{0ex}}=\sum _{k=1}^{K}\sum _{n}\mathrm{ln}\phantom{\rule{.2em}{0ex}}{p}_{{d}_{k}\left[n\right]}({d}_{k}\left[n\right];{\theta}_{{t}_{k}},o).$$

The complicated form for the measurement density (Eq. 11) makes the maximization of 𝐿(*o,
θ*) an overly complicated problem. When the read-out noise
variance is large (greater than 50 or so), however, the data can be modified
according to:

$$\phantom{\rule{7.5em}{0ex}}={N}_{k}\left[n\right]+{M}_{k}\left[n\right]+{g}_{k}\left[n\right]+{\sigma}^{2}\left[n\right],$$

and these modified data are then similar (in distribution) to the sum of three
Poisson-distributed random variables^{8}. Accordingly, the modified data are approximately Poisson distributed with
the mean value *I*_{k}
[*n*] +
*I*_{b}
[*n*] +
*σ*
^{2}[*n*]. The probability
mass function for these data is then modeled as:

so that the log-likelihood is:

where terms not dependent on the object or phase aberrations have been omitted. The
maximum-likelihood phase aberration and object estimates are then those that
maximize 𝐿(*o, θ*) subject to any physical
constraints that are imposed on these parameters. For all problems we have
considered, the object intensity *o*[·] is constrained to
be a nonnegative function. Known object support is another constraint that is
commonly used; however, implementation of this constraint in an optimization
procedure usually results in a trivial reduction of the number of unknown
parameters.

#### 3.1 Comments

In recent years, many methods for the blind deconvolution of astronomical and
space-object imagery have been proposed. These can be broadly classified into
two categories: i) the Ayers-Dainty^{9} class of iterative algorithms; and ii) constrained numerical optimization
algorithms. The methodology described here is one of constrained numerical
optimization, and is a straightforward extension of the method proposed by Schulz^{5} with important modifications to accommodate real sensor artifacts such as
read-noise, internal and external background, and non-uniform gain. Other
methods of constrained optimization have been proposed^{10,11,12}, and some have relied on essentially the same
methodology as the one proposed in Ref. 5 with the differences being the method used for numerical optimization^{12}, and the utilization (or lack thereof) of realistic sensor models. Key
points regarding the methodology presented in this paper are:

- A practical and faithful model is used for the data collection process. By using the maximum-likelihood method for estimation, this model is in turn used to induce a cost function for a constrained optimization problem.
- The importance of the point-spread function model (Eq. 7) should not be understated. This point was emphasized by Cornwell in his summary of the 1994 workshop on the Restoration of HST Images and Spectra II as he commented on the subject of blind deconvolution
^{13}:“One always gains by adding more information in the form of known imaging physics. For example, one can ask whether the closure relations are enforced? Alternatively, is the PSF forced to be the Fourier transform of a cross-correlation of a pupil plane with phase errors? If not, then it should be.”

#### 3.2 Numerical optimization

Many methods are available for performing the optimization required to solve for
the maximum-likelihood aberration and object parameters. Gradient-based methods
and the expectation-maximization (EM) method proposed in Ref. 5 are two popular choices. However, one should not confuse
the method used for solving the optimization problem with the methodology used
to pose the problem. Provided the problem is successfully solved, the
optimization method will only influence *algorithm* performance
(computation time). The methodology used to define the problem, which is greatly
dependent upon the use of sound mathematical and physical models, will determine
*estimator* performance (bias and variance).

With straightforward modifications to accommodate sensor effects such as non-uniform gain and backgrounds, the EM method of Ref. 5 can be applied to this problem. Additionally, the derivatives of the log-likelihood with respect to the object and phase-aberration parameters are easily derived and can be used in a gradient-based optimization method. Both methods have performed well for this problem, and the determination of an optimal method of optimization is still an open problem.

## 4. Results with real data

Telescope data were acquired at the Air Force Maui Optical Station (AMOS) on March 9, 1995 and subsequently processed to determine maximum-likelihood object estimates. The telescope and imaging-system parameters are listed in Table 1. The object under observation was the Hubble Space Telescope in its 600 km orbit, and 656 short exposure images were acquired, each with an 8 ms exposure time. Four of these images are shown in Fig. 1. The 656 images were partitioned into 41 sets of 16, and each set in the partition was used to obtain one object restoration. The optimization method used was a straightforward extension of the expectation-maximization method described in Ref. 5. Four of these restorations are shown in Fig. 2. Each restoration required approximately 15 minutes of processing time on 9 nodes of an IBM SP2 at the Maui High Performance Computing Center. Whereas very little detail is available in the raw data, the restored images allow one to clearly identify the satellite’s solar panels along with the cover (lower left) for the space telescope’s 2.4 m primary mirror. An mpeg video showing all data and restorations is available at the URL listed as Ref. 14.

## 5. Summary

For ground-based surveillance of astronomical and space objects, post-detection
processing through multiframe blind deconvolution is a viable method for restoring
image resolution. To support this claim, a maximum-likelihood estimation method has
been implemented and tested on real telescope data, and preliminary results of this
study are presented in this paper. Much work still needs to be done, however, and
important topics for consideration include: the use of prior models for the object
and phase aberration parameters; the optimal use of measurement diversity as
supplied by a wavefront sensor or phase-diversity system^{15}; clever methods for generating initial parameter estimates; and the
development and implementation of improved methods of optimization.

## 6. Acknowledgments

This work was supported by the Air Force Maui Optical Station, the Maui High Performance Computing Center, and in part by the Air Force Office of Scientific Research through grant F49620-97-1-0053.

## References and links

**1. **M. C. Roggemann and B. Welsh, *Imaging Through Turbulence*, CRC Press, Inc. (1996).

**2. **A. Labeyrie, “Attainment of diffraction limited resolution in large telescopes by Fourier analyzing speckle patterns in star images,” Astron. Astrophys. , **6**, 85 (1970).

**3. **K. T. Knox and B. J. Thompson, “Recovery of images from atmospherically degraded short exposure images,” Astrophys. J. , **193**, L45–L48 (1974). [CrossRef]

**4. **A. W. Lohmann, G. Weigelt, and B. Wirnitzer, “Speckle masking in astronomy: triple correlation theory and applications,” Appl. Opt. , **22**, 4028–4037 (1983). [CrossRef] [PubMed]

**5. **T. J. Schulz, “Multi-frame blind deconvolution of astronomical images”, J. Opt. Soc. Am. A , **10**, 1064–1073 (1993). [CrossRef]

**6. ** Strictly speaking, because of image inversion and magnification an imaging system is never spatially invariant; however, if one views the input signal as the inverted and magnified object, many imaging systems are then well-modeled as spatially invariant.

**7. **D. L. Snyder, A. M. Hammoud, and R. L. White, “Image recovery from data acquired with a charge-coupled-device camera,” J. Opt. Soc. Am. A , **10**, 1014–1023 (1993). [CrossRef] [PubMed]

**8. **D. L. Snyder, C. W. Helstrom, A. D. Lanterman, M. Faisal, and R. L. White, “Compensation for readout noise in CCD images,” J. Opt. Soc. Am. A , **12**, 272–283 (1985). [CrossRef]

**9. **G. R. Ayers and J. C. Dainty, “Iterative blind deconvolution method and its application”, Opt. Lett. , **13**, 547–549 (1988). [CrossRef] [PubMed]

**10. **R. G. Lane, “Blind deconvolution of speckle images,” J. Opt. Soc. Am. A , **9**, 1508–1514 (1992). [CrossRef]

**11. **S. M. Jefferies and J. C. Christou, “Restoration of astronomical images by iterative blind deconvolution”, Astrophys. J. , **63**, 862–874 (1993). [CrossRef]

**12. **E. Thiebaut and J.-M. Conan, “Strict *a priori* constraints for maximum-likelihood blind deconvolution,” J. Opt. Soc. Am. A , **12**, 485–492 (1995). [CrossRef]

**13. **T. J. Cornwell, “Where have we been, where are we now, where are we going?”, in *The Restoration of HST Images and Spectra II*, B. Hanisch and R. L. White, editors, (Space Telescope Science Institute, Baltimore, MD, 1993) pp. 369–372.

**14. **T. J. Schulz, “Movie of processed Hubble Space Telescope imagery,” http://www.ee.mtu.edu/faculty/schulz/mpeg/hst.mpeg

**15. **R. G. Paxman, T. J. Schulz, and J. R. Fienup, “Joint estimation of object and aberrations using phase diversity,” J. Opt. Soc. Am. A , **9**, 1072–1085 (1992). [CrossRef]