## Abstract

We demonstrate the use of image support constraints in a noise-reduction algorithm. Previous work has revealed serious limits to the use of support if image noise is wide-sense stationary in the frequency domain; we use simulation and numerical calculations to show these limits are removed for nonstationary noise generated by inverse-filtering adaptive optics image spectra. To quantify the noise reduction, we plot fractional noise removed by the proposed algorithm over a range of support sizes. We repeat this calculation for other noise sources with varying degrees of stationarity.

©1997 Optical Society of America

## 1. Introduction

A bound around the area of an image where the object intensity is nonzero is referred
to as a *support constraint*. All measurements outside the support
are assumed to be optical system or detector noise and set to zero. Knowledge of
image support is information in addition to the measured data, and can be used in a
variety of image reconstruction algorithms [1] to estimate an object map in the presence of detector noise
and time-varying optical aberrations. In previous work, Matson [2] showed that the fundamental mechanism behind image noise
reduction is noise correlation in the Fourier spectrum of the image; that is,
application of support in the image domain imposes a convolution of the image
Fourier spectrum with the Fourier transform of the support. If, as a result of the
convolution, noise power increases relative to the measured data in any band, the
original (unconstrained) Fourier data can be substituted for or used to bound the
constrained components in that band. Since all image pixels *outside*
the support have been set to zero, the noise reduction occurring as a result of this
substitution must occur *inside* the support. This algorithm can be
continued iteratively, since the substitution of unconstrained Fourier components
causes the reappearance of image-domain noise components outside the support [2].

An obvious and critical question is: under what circumstances will application of
support cause noise power to increase in some regions of the image spectrum? Matson
showed that if the noise is wide-sense stationary (WSS) in the frequency domain, the
noise power in the real and imaginary parts of the image spectrum must have a
*minimum ratio*. The term “wide-sense
stationary” designates a random process for which the ensemble first and
second moments are independent of coordinate (spatial frequency in this case). In
this paper, we will use “WSS” and
“stationary” synonymously. Since the ensemble second moment of
a Fourier-domain noise distribution is the mean noise power spectrum, WSS noise has
by definition equal average power at all frequencies in the relevant image spectrum.
This means the only way noise may be reduced inside the support is if the
application of support causes noise power to couple from the real part of the image
spectrum to the imaginary part or vice versa. Further, the only way this coupling
can occur is if the support is to some extent *asymmetric* with
respect to the image origin. Noise reduction will occur only in the asymmetric part
of the support.

Any pixel-independent noise source is WSS over frequency [3]; examples include CCD read noise and Poisson (shot) noise.
Many real-world noise sources, though, are nonstationary. Two examples are
encountered when imaging astronomical objects through the atmosphere: adaptive
optics noise (noise features resulting from uncompensated atmospheric turbulence)
and speckle imaging noise (features resulting from speckle processing with
inadequate data SNR) [4]. Nonstationarity is also imposed by deconvolution (inverse
filtering) [5]. We have extended by hypothesis the minimum real-imaginary
ratio for stationary noise to a more general requirement for sufficient
“structure” in the noise power spectrum. We propose that with
sufficiently large variations in (rather than between) the real and imaginary noise
power spectra, noise may be coupled from band to band within the real and imaginary
spectra by a *symmetric* support constraint. Noise reduction would
occur in both the symmetric and asymmetric components of the processed image.

To analyze this hypothesis further, consider that if the support-constrained image
*i*_{c}
(**x**) is given by
*i*(**x**)*w*(**x**), where
*w*(**x**) is the support weighting function, then the
corresponding image spectrum is given by

where *I*(**u**) can be decomposed into signal and noise
components *S*(**u**) and
*N*(**u**). Substitution of the above equation into
expressions for the spectral variance distribution (the noise power spectral density
or PSD) of support-constrained data yields complicated expressions involving, for
nonstationary noise, convolutions of covariances which do not lend themselves well
to analysis except in limiting cases [6]. However, we know that convolution or correlation operations
tend to “smooth” and enlarge the extent of the function
involved. For example, just as the bandwidth limitation of telescope data, imposed
by diffraction, results in corresponding imagery which is in principle infinite in
extent (consider the Airy disk), constraining an image to finite support requires
the associated spectrum to be infinite in bandwidth. The application of support to a
bandlimited image, then, causes the associated spectrum to spread beyond its
original bandwidth. The practical extent of the bandwidth increase (in principle
infinite) depends on the bandwidth of *W*(**u**) [7] *and the degree to which the spectrum was already
“smooth.”* As an example, consider the case of a
very-wide-band, nearly constant spectrum. The corresponding image
*i*(**x**) would be very nearly a point, so while the
application of some finite-diameter support would in principle extend the bandwidth
of the image spectrum, the spectrum *W*(**u**) would be so
narrowband as to be nearly a δ-function relative to the width of
*I*(**u**). Conversely, if
*I*(**u**) is confined to a very small region about zero
frequency, application of even a large-diameter support constraint will extend the
bandwidth by smoothing the “spike” near DC. These notions are
at the heart of some superresolution concepts. Rather than study extension of the
*S*(**u**) bandwidth (superresolution), we study the
redistribution of *N*(**u**) induced by the application of
support. As a consequence, we are concerned less with the structure of the image
spectrum than we are with the those of the associated noise PSDs. Preliminary
theoretical treatment of the support-induced “noise transport”
phenomenon shows the redistribution of noise power at a point in the spectrum
depends in part on the local gradient of the PSD, as expected.

Matson and Roggemann have previously analyzed noise reduction with AO imagery using asymmetric support [6]; these results were analyzed in terms of the real-imaginary variance ratio. In this paper, we demonstrate noise reduction using symmetric support and qualitatively link the amount of noise reduction with the structure, or nonstationarity, of the noise PSD. We study our “sufficient structure” hypothesis using computer simulations of both CCD read noise and adaptive optics noise.

## 2. Calculations and results

We generated the results shown here by calculating sample statistics over ensembles
of 200 independent realizations of noisy imagery. Each data ensemble was corrupted
with a different noise source. Although each source is common in telescope imagery,
the associated noise PSDs are different in the extent to which they are stationary.
First, we created an ensemble of realizations with the center pixel of each array
having an amplitude of 10^{6}. Zero-mean, independent Gaussian noise with
*σ* = 10 was then added to every pixel in each array.
This data is diffraction-free point-source imagery with WSS noise from a CCD
amplifier as the only noise. Next, we used a computer simulation to model data from
an AO system with 45 deformable mirror actuators. The simulated atmospheric
turbulence was characterized by a Fried parameter *r*_{o}
of
10 cm. On a 1.6-m pupil, 45 actuators leads to about 22 cm spacing between
actuators, larger than *r*_{o}
. This scenario mimics the
not-uncommon case where turbulence is severe enough that the AO only partially
compensates turbulence effects, resulting in AO noise, which is nonstationary. Two
data sets of AO noise combined with different levels of CCD noise were then created
by adding white Gaussian noise with *σ* = 150 and 250 to
these images. To create a fourth data set, we used the simulation to produce data
modelling AO correction of *r*_{o}
= 10 cm turbulence with
only 21 actuators. The separation between actuators for this case is 29 cm.
Typically, a deconvolution algorithm is applied to boost the spectral amplitudes of
partially-compensated AO data [5], and we have done that here. This is the only data set where
deconvolution is applied.

We estimate the OTF of the model “telescope” and AO system by
averaging 10,000 frames of simulated data. This OTF served as our deconvolution
inverse filter. We attenuated some of the noise amplification caused by
deconvolution by applying a regularizing, or smoothing, filter
*F*(**u**, *u*_{c}
) to the
deconvolved spectrum:

where *u*_{c}
is the cutoff frequency, determined by the data
ensemble SNR. However, sharply localized noise “spikes” remain
near the diffraction cutoff of the data. These structures can be seen in Fig. 1, where we show a surface plot of
${\mathit{\text{PSD}}}_{R}^{u}$
(**u**), the
unconstrained PSD of the noise in the real part of the data spectra. These
structures indicate that deconvolution-amplified adaptive optics noise is far from
stationary.

The quantities of interest here are noise PSDs, estimated separately for the real and imaginary parts of the image spectra from the sample moments

and

where *I*(**u**) is the Fourier transform of the image
distribution *i*(**x**) and
*Ī*(**u**) is the sample mean of the Fourier
transform. The support constraints used are circular, with diameters varying from
182 pixels (just large enough that a 128 × 128-pixel image array can be
entirely contained by the support) to 8 pixels. For each constraint diameter,
support is applied to each of the 200 noisy images and the PSDs above are estimated.
Then, using the PSDs calculated with constrained and unconstrained data, we
calculate the fractional noise increase *F* in the constrained data
using

where PSD(**u**${)}_{R\left(I\right)}^{c}$ and PSD(*u*${)}_{R\left(I\right)}^{u}$ are the constrained and
unconstrained PSDs for the real (imaginary) part of the data spectra, respectively.
The integration in the numerator includes only those values of the integrand greater
than zero, so the numerator gives the total amount of noise that can be removed from
the support interior by substituting spectral components from the original,
unconstrained data. Normalizing by the total noise in the unconstrained data gives
*F* for the real (imaginary) part of the noise spectrum.

In Fig. 2, we show our main result:
*F*_{R}
for one step of the noise reduction algorithm is
plotted vs. support diameter *d* for both simulated data cases. For
deconvolution-amplified AO (DAAO) noise, over 10 percent of the total noise can be
removed with a single application of support. This shows a symmetric support
constraint can be used to increase image SNR when the data is corrupted with
nonstationary noise from a common source. Note the support diameter
*d*_{o}
yielding the largest noise reduction is
conveniently large (112 pixels). The maximum *F*_{R}
for DAAO
noise is reasonably large due to the large amplitude of the band-to-band variations
in ${\mathit{\text{PSD}}}_{R}^{u}$
(**u**). While we can argue
heuristically that *d*_{o}
should occur where the bandwidth
of *W*(**u**) is similar to the bandwidth of the largest and
most localized features in
${\mathit{\text{PSD}}}_{R}^{u}$
(**u**), a quantitative
understanding of this and other features of our present results are the focus of
ongoing work.

For CCD noise, the maximum value of *F*_{R}
is nearly 100
times less than for DAAO noise. According to theory, there should be no noise
transport at all for what seems to be WSS noise. However, the finite size of the
data ensemble used means very small variations in
${\mathit{\text{PSD}}}_{R}^{u}$
(**u**) remain; that is,
the noise is not quite stationary (see the
${\mathit{\text{PSD}}}_{R}^{u}$
(**u**) distribution for
CCD noise in Fig. 3). As *d* decreases, however,
*W*(**u**) rapidly becomes large enough that the CCD
noise power integrated over a support constraint bandwidth is constant in
**u**. The finite-ensemble band-to-band variations are small, so the peak
value of *F*_{R}
is small relative to the peak
*F*_{R}
for DAAO noise. An implication of the
non-zero *F*_{R}
for CCD noise, though, is that
*some* nominally WSS noise can be removed in practice using
*symmetric* support for finite ensemble sizes. In future work, we
will study the exploitation of this phenomenon.

For the two combinations of AO and CCD noise, we observe an immediate rise and
subsequent rolloff in *F*_{R}
as *d* is
decreased below 182 pixels for both *σ* = 250 and
*σ* = 150. This feature is similar in behavior to the
large-*d* behavior of *F*_{R}
with CCD
noise. The difference in the slopes of the CCD and combination curves for
*d* < 160 can be attributed to the contribution of AO
noise to ${\mathit{\text{PSD}}}_{R}^{u}$
(**u**), causing the
noise to be nonstationary in character. As can be seen in Fig. 4, AO noise (observed inside the diffraction cutoff
frequency) causes significant variation in
${\mathit{\text{PSD}}}_{R}^{u}$
(**u**), which can no
longer be considered uniform.

We also note *F*_{R}
increases for both combination-noise
curves for *d* less than about 35. These increases are again
artifacts of AO noise, which tends to be correlated to a much higher degree than CCD
noise. Image realizations from the CCD and combination
(*σ* = 150) ensembles are shown in Figs. 5 and 6, respectively; from these, it is apparent that even for
smaller values of *d*, there is little variation in the intensity
subtended by the support for pure CCD noise. Conversely, the correlated structure of
the AO noise contribution in the combination-noise realization implies that for
smaller values of *d*, there can be significant frame-to-frame
variation in the intensity subtended by the support. This variation is observed as
an increase in the zero frequency value of the noise PSD. Finally, we note that for
*d* less than the diameter of the long-exposure point-spread
function, application of support corresponds to a form of “blind
deconvolution.” Without additional knowledge, such as that obtained by
estimating the system OTF, an implicit assumption is being made concerning the
angular extent of the object. In Fig. 7, we show a cross-section of the PSF for
*r*_{o}
= 10 cm and turbulence compensation by a
45-actuator AO system. The core of this PSF, defined as the diameter containing 90
percent of the energy, is 16 pixels.

## 3. Summary

We have shown that large, simple, and symmetric support constraints can be used to remove noise inside the constraint boundary if the noise is nonstationary in the frequency domain. Using data corrupted by adaptive optics noise, we showed that deconvolution amplified noise provides enough variance structure, or nonstationarity, to allow the removal of over 10 percent of the total image noise in a single step of a noise-removal algorithm using support.

## Acknowledgement

The authors gratefully acknowledge grants from the Air Force Office of Scientific Research, Bolling AFB, MD. and the USAF/Maui High Performance Computing Center Research and Development Consortium.

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