Blind deconvolution of the SXT PSF core part

Szymon GburekJanusz Sylwester

Space Research Centre, Polish Academy of SciencesWrocław,Poland

Multi-Wavelength Observations of Coronal Structure and Dynamics -- Yohkoh 10th Anniversary Meeting

January 21-24, 2002, Kailua-Kona, HI

Petrus Martens

Montana State UniversityBozeman MT

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

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.

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:

Find new estimate

of I2 from D and P2

Find new estimate

of P1 from D and I1

Impose image constraints

• positivity

• conservation of total counts

Fourier Transform

Inverse Fourier Transform

Impose PSF constraints

• positivity

• normalization

Fourier Transform

Inverse Fourier Transform

I2

i2

i1

I1

P1

p1

p2

P2

Fourier domain constraint

averaging I1 and I2

Initial approximation for

PSF

THE

ALGORITHM

Apodization

Apodization

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) b) c)

d)e)f)

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) small spurious sources are no longer seen in deconvolved image and PSF2) sharp peak structures are well resolved3) 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 imagedeconvolved

test image

test PSFdeconvolved

test PSF

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.

SXT Data Selection

Map of all compact sources found in full resolution SXT frames during year 2000 (in CCD co-ordinates)

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 X-ray 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.

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) 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),

8) The Fourier transform of the image is broad and possess more uniform distribution

of signal value. 

All of selected images satisfy the following criteria:

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 MethodInspection of collected SXT Al12 data for year 2000 shows many very well localised X-ray 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 sequencenormalized

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

8-MAY-00 at 10:50:21

Deconvolved image

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