Blind deconvolution of the SXT PSF core part
Szymon Gburek
Janusz Sylwester
Space Research Centre, Polish Academy of Sciences
Wrocław,Poland
Petrus Martens
Montana State University
Bozeman MT
Multi-Wavelength Observations of Coronal Structure and
Dynamics -- Yohkoh 10th Anniversary Meeting
January 21-24, 2002, Kailua-Kona, HI
ABSTRACT
We show the first results of blind iterative deconvolution (BID) of the core part of
SXT point spread function from flare images. As the performance and speed of BID
algorithms depend on good initial guess for PSF function shape and quality of data
used, we took special care of the data selection and processing. From the analysis of
several compact flare kernels we came to conclusion that a good guess for PSF can
be provided directly from images of X-ray compact structures observed by SXT.
Next, we conducted extensive mission-long searches for compact structures through
entire database of SXT full resolution frames. The searches returned plenty compact
structures which may be used for BID restoration purposes. The period of data
selection has been then constrained to the year 2000. A period relatively short in
comparison with Yohkoh mission duration but long enough to ensure good SXT
CCD data coverage. Also optical transfer functions for the all compacts of 2000
found have been tested for their signal value distribution. Using observation for this
selected set of structures we construct constraints for Al12 PSF shrouds and
compare them with ground calibration data.
Introduction
The main purpose of our contribution is to explore blind iterative deconvolution
for determination of the core part of the SXT PSF.
We decided to deconvolve the SXT partial frame images (PFI) taken in Al12 filter.
PFI images comes from telemetry at highest possible SXT resolution and therefore are
particularly useful for finding fine sources which can then yield appropriate PSF core
approximation. From the searches that we performed on the entire PFI images archive
we recognised that the thick aluminium filter was one of the most frequently used.
Hence, the choice of Al12 data allows for obtaining a representative image sample
from the whole PFI database. On the other hand majority of the SXT ground
calibrations in soft X-rays were performed in aluminium K spectral line which lies
close to the maximum of effective area for Al12 filter. This coincidence makes
possible the most thorough cross-comparison between flight recorded and ground
based data. Because one can not exclude that PSF would change in time, it seems to
be important to look on PFI data in relatively short time interval, say one year, in
comparison with Yohkoh mission duration. That is why we focused our research on
period covering the year 2000.
Blind Deconvolution Algorithm
We have adapted the blind deconvolution algorithm (Ayers, Dainty, 1988) for restoration of SXT
data and PSF. The algorithm starts from an initial approximation of the PSF which is constructed
directly from SXT images of compact sources. The initial PSF is then Fourier transformed and a
restoration is performed by division of the Fourier transform of noisy and blurred SXT data by the
transformed PSF. Next, by inverse Fourier transform an estimate of image is obtained. Then a
positivity and total count conservation constraint is imposed on the image to yield the first
deconvolved image approximation. These three steps are then repeated for the deconvolved image
in order to find a new approximation of the PSF, what closes first iteration of blind deconvolution
loop. The iterations are repeated until a minimum of chi-square parameter is found. In addition,
apodization of high frequencies is introduced into the loop after Fourier transform division. The
two, available at each iteration estimates of image Fourier transforms are averaged to improve
transform smoothens. The algorithm is shown pictorially in the figure below with the following
designations:
i1 , i2 – image (true brightness distribution estimates),
p1 , p2 – point spread function estimates,
d – SXT data (blurred & noisy),
I1 , I2 , P1 , P2 , D – Fourier transforms of i1 , i2 , p1 , p2 , d respectively.
Fourier domain constraint
averaging I1 and I2
Initial
approximation for
PSF
THE
ALGORITHM
I2
Apodization
Find new estimate
of I2 from D and P2
P2
Inverse Fourier Transform
Fourier Transform
i2
p2
Impose image constraints
Impose PSF constraints
• positivity
• positivity
• conservation of total counts
• normalization
p1
i1
Inverse Fourier Transform
Fourier Transform
I1
Apodization
Find new estimate
of P1 from D and I1
P1
Blind Deconvolution Test on Synthetic Data
We have tested the algorithm extensively on synthetic models of SXT images of compact X-ray
sources. The test images were blurred with a model of a PSF function. To the blurred image a noise
were also added to obtain model of SXT data. The picture below shows an example of synthetic
image, PSF, and data which is going to be disused also in the sequel.
Model of SXT image
Model of a PSF
Model of SXT data
(image and PSF convolution
with addition of a noise)
Noise Reduction and Signal Amplification Handling
A significant enhancement in quality of deconvolved image and determinations of PSF
shape can be obtained when a reduction of noise and removal of the method artefacts is
performed at each iteration loop. Tests show that even in case of very low noise, restored
data and PSF approximation get speckled by small spurious sources. Noise causes also
deformations of structures in low signal regimes to appear in image or PSF
deconvolution output estimates. Another problem here is that for high frequencies large
regions of low signal value are often present in the processed spectra. The division of
Fourier transforms used twice at each iteration loop can cause then extreme amplitude
amplification for certain frequencies, what again spoils deconvolution results.
The left picture above shows effects of the noise on synthetic data. Background area is speckled
by spurious sources and peaks base degradation is also seen in the image. Similar defects
develop for PSF during deconvolution. To the right, power spectrum for deconvolved synthetic
image (low frequencies were shifted to the corners for the presentation purpose). Large signal
amplification is clearly seen for higher frequency regions in a form of large spikes.
Both of the mentioned problems are mainly related to higher frequencies in analysed spectra.
A research on power spectra for SXT images (Martens, Gomez 1992) showed that signal rapidly
decreases to low values with increasing frequency. It has been found also there, that noise
contribution dominates signal for high frequency image spectrum content. In source paper (Ayers
and Dainty, 1998) for the algorithm we present here, a solution to the problems in high frequency
regions was suggested via multiplying Fourier transform of data by appropriate apodization
function at each algorithm iteration. Such a method cuts off certain amount of high frequency area
completely in order to smooth the deconvolution output.
In our approach we are dealing with SXT images of very compact peak sources. In this case a
complete suppression of high frequencies is rather undesirable as they build the main spiky parts
in our data. Therefore, we process also the high frequencies image content and, at each iteration,
perform their inspection in order to recycle information about sharp structures that may be present
there. These structures are then identified and subtracted. A residual, left after the subtraction,
image consist mainly of noise. It may, however, still contain some remains of the genuine signal.
Hence, it has to be taken into consideration. An estimate of deconvolved image is formed as a sum
of structures build from lowest frequencies and identified sharp structures in high frequency
image content. Then residuals with smoothed out noise are added to the estimate. Eventually, in
the next step of the algorithm, output image is formed by imposition of positively and count
conservation constraint. To shed more light on the above let consider the following example on
synthetic data.
a)
f)
b)
e)
c)
d)
Clockwise, in the picture above: a, b) the image content formed from low and high frequencies of its
spectrum respectively (sum of these images yields what comes stright from division of data Fourier
transform by PSF transform), c) shows the noisy residuals from b) left after subtraction of structure
seen in the middle, d) residuals with smoothed out noise contribution, e) deconvolved image- the
sum of a) , d) and structures seen in f), final algorithm output with positivity and normalisation
constraint for the deconvolved image at each iteration.
Blind Deconvolution Test Results
Deconvolution of test images reveals that the use of the algorithm presented here, with the
described above high frequency treatment, is capable of producing good image and PSF
restorations. The following features characterises the restoration results:
1)
2)
3)
small spurious sources are no longer seen in deconvolved image and PSF
sharp peak structures are well resolved
deconvolved image and PSF have similar signal range like the original one
In the picture below a final result of blind deconvolution of chosen synthetic data are
shown.
test image
deconvolved
test image
test PSF
deconvolved
test PSF
SXT Data Selection
The first important question that arises here is in which part of the CCD one can determine
approximation for SXT PSF from selected Al12 PFI data. The answer comes from analysis of
situation of PFI frames within CCD detector for the year 2000. It turns out that they are not
evenly spread all over the CCD surface. Vast majority of them comprise an about 1000x650
pixels belt only, wee bit shifted from CCD centre to its upper left corner.
To the left, a coverage map of the CCD detector surface by full resolution SXT frames. Gray intensity
says how many times a given pixel was captured within a full resolution frame during year 2000. To the
right a shaded surface for the coverage map.
We found that the compact sources which
we identified in SXT data for the year 2000
pretty well cover the belt of PFI data
occurence shown above. All of the sources
are well localised structures with signal
rapidly vanishing with distance from the
peak and good signal to noise ratio. Special
care has been taken, during searches for
these sources, to not mistake them with
ordinary spikes and to filter out fuzzy Xray structures observed by SXT. Search
criteria and result are discussed in detail in
our paper (Gburek S., Sylwester J., in print)
In total we detected above 104 SXT
compact source images in flare and quiet
instrument operation mode for the year
2000. Their location on CCD detector is
shown to the right. Eventually we have
selected about 2500 of the best images of
compact sources for blind deconvolution
purposes.
Map of all compact sources
found in full resolution SXT
frames during year 2000 (in CCD
co-ordinates)
All of selected images satisfy the following criteria:
1)
The temperature of CCD during the image exposure was lower than -20 C,
2)
The image is taken outside the South Atlantic Anomaly SAA passages,
3)
The image possess only a single pixel with maximum value
(i.e. global maximum was present),
4)
The pixel with maximum value is situated at least seven pixels away from the
image boundaries,
5)
The maximum value is above 1000 DN,
6)
The image is not saturated (i.e. maximum value below 3313 DN),
7)
8)
The maximum dark current level in the image is below 50 DN
(50 DN is an average dark current in reference frames for the year 2000),
The Fourier transform of the image is broad and possess more uniform distribution
of signal value.
We have also inspected SXT calibration data in order to select material for comparison of
deconvolved PSFs from in-flight collected data with pre-flight calibrations. Extensively
ground calibration tests of SXT telescope were performed at High Energy Laser System Test
Facility at White Sands Missile Range (WSMR) in New Mexico Four months before launch
of Yohkoh satellite. (Martens, P. et al 1995). The X-ray tests objectives were to check the
SXT mirror parameters, fix focus position and collect material for determining telescope blur.
During the WSMR X-ray calibration tests SXT CCD detector was lit up by a microfocus
source beam in different energy ranges and positions relative to telescope axis. The Al.-K line
(1.49 keV) was used in the majority of calibration test exposures in soft X-rays. For
comparison of ground calibration data with in flight recorded AL12 images we have chosen
bx01_apr23 series because:
a) it was taken in AL.-K line which lies near maximum of the SXT effective area
curve for Al12 filter,
b) the SXT focus was set in the flight position during bx01_apr23 frames acquisition,
c) peak position of x-ray beam in images of bx01_apr23 series equidistantly
covers the entire CCD area.
The above mentioned properties of bx01_apr23 allows for most thorough comparison of
deconvolution results with point source phantom observed on ground in test data.
The image above shows 49 images of bx01_apr23 ground calibration series. Enlarged contour
plots of their peak portions in the image bellow.
Initial Approximation of PSF for Blind Deconvolution
Steepest Descent Method
Inspection of collected SXT Al12 data for
year 2000 shows many very well localised Xray sources. It could be rather hardly expected
that a single image can yield a good
estimation of the SXT PSF however. We have
checked that to achieve improvement in PSF
shroud determination it is better to work with
several to several tens of images from
approximately the same CCD region
simultaneously. In the approach presented
here, we took square sub-arrays of all images
centred at the maximum, normalised them to
[0, 1] signal range and stack them onto a data
cube. Then we construct a surface of the same
size as the sub-arrays taken but with signal
value at each pixel equal to minimal value we
found at relative pixel position along the data
cube of normalised images. Such a surface is
considered as a final PSF approximation in or
method.
We
constructed
our
PSF
approximations in the neighbourhoods of peak
observed in calibration data from bx02_apr23
series. Peak positions for in-flight data which
we selected and calibration data are shown on
the left.
Compact flare images and WSMR
peak positions on CCD surface.
Let us illustrate a construction of PSF shroud by steepest descent method in what follows.
First, a sequence of compact source images placed
nearly at the same location on CCD needs to be
found. Three images from such a sequence of 144
images are shown below. To the left, the situation
of sequence image peaks in CDD coordinates
• • •
Next, signal in images is normalised in certain sub-arrays centred at the peak. (here, 15x15
pixel square sub-arrays)
sub-array sequence
normalized
• • •
Finally, a PSF approximation is constructed by taking at each
pixel the minimal signal value possible to find in the whole
normalized sequence at the respective pixel position.
Example of Blind Deconvolution of SXT Data
Let us focus now on blind deconvolution of real observations from SXT telescope. As an example
we deconvolved the image and PSF from SXT flare frame taken on 8-MAY-00 at 10:50:21 in
Al12 filter. It is one of the frames from the sequence of 144 images discussed earlier.
Hence, we used initial approximation of the PSF which was constructed by Steepest
Descent Method in the previous section. Blind deconvolution output for the frame
selected is shown in the pictures below.
Discussion of the Result Obtained
Similarly as it was observed for test on synthetic data, deconvolution of true SXT data
reveals expected, desirable features:
- In deconvolved image increase in signal range and separation of proximate sources
occur,
- The structures seen in data become much sharper after deconvolution,
- Sharpening is also seen in the restored PSF shape.
No significant deformations of the image that comes from noise or method artefacts are
observed either in restored image or PSF. The deconvolved PSF is still more fuzzy that
the calibration beam profile in Al.-K line (see the comparison of cross-sections below). It
is difficult to find explanation to that. It could be caused however by the Al12 filter and its
support. On the other hand, the solar radiation is not monochromatic and one would
expect broadening of the true PSF in comparison with the calibration one observed in
Al-K spectral line. The deconvolved PSF profile sharpening is also an expected
result.The initial PSF approximation that comes from SXT data by steepest descent
method should be rather an upper limit (envelope) of the true PSF.
SXT flare frame
Deconvolved image
8-MAY-00 at 10:50:21
SXT flare frame (Log10 scale)
Deconvolved image (Log10 scale)
PSF approximation from
steepest descent method
Calibration beam seen
in Al.-K line
Deconvolved PSF
PSF cross sections (x-cross section in the left panel, to the right y-cross section.
Thin solid line – cross sections of the PSF found by steepest descents method
Thick solid line – cross sections of the calibration beam
Diamonds – deconvolved PSF cross-sections
Conclusions
The material for blind deconvolution of the SXT core part from
flare images and has been collected.
First result obtained, shows that blind deconvolution can produce
PSF shape close to the calibration data.
References,
Ayers G. R., Dainty J. C., Iterative blind deconvolution method and its applications,
Optic Letters, vol .13, no. 7, 1988
Martens P., Acton L., Lemen J., The Point Spread Function of the Soft X-ray Telescope Aboard Yohkoh,
Sol. Phys. 157, 141, 1995.
Martens P. and Gomez D., Spatial Power-Spectra from Yohkoh Soft X-Ray Images,
1992, Publ. Astron. Soc. Japan 44,
L187-L191.
Gburek S., Sylwester J., Search for Compact X-ray Sources in SXT Observations,
sent to Solar Physics, 2001