A theoretical model ACIS BI CCD was modeled using the CCD simulator described in Chapter 7. We placed 4000 1-keV photons on a simulated ACIS BI 1024 X1024 array, with the subpixel position fixed but the depth and array position random. This was done 121 times for 121 subpixel positions, mapping out a pixel in 0.1-pixel increments.
Each output image was run through an event-finding algorithm to generate
a list of event energies and grades, and a histogram of the distribution
of grades. Events are detected by considering a 3 X 3 pixel subarray centered
on a bright pixel. For clarity, the pixels in this subarray are assigned
numbers as given in the array below. Here, pixel number 4 is the brightest
pixel in the subarray. (Note: this pixel numbering is not the same
as used in ACIS flight or calibration event neighborhood coding.)
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The grades used are defined in Table 6.19,
which refers to the pixels by the numbers given above. The cryptic grades
S+, P+, and Other are equivalent to the ASCA grades of the same name and
refer to the few unusual events that contain diagonal pixels or don't fit
into any of the other shape catagories.
Table 6.19: Grade definitions and subpixel positions of probability maxima
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Dividing the grade distribution histogram by the total number of detected events then gives a fractional grade distribution. One of these was generated for each subpixel position. Of the 16 possible grades, all but S+, P+, and Other are useful in determining event positions. These three grades were not used because there were too few events in each grade to yield a meaningful event position probability distribution (see below). Trimming these left a 13-element vector of fractional grade distribution. The normalization occurred before these grades were removed. The event-finding algorithm used an event detection threshold of 50 electrons and a split-event threshold of 20 electrons - these thresholds do affect the splitting.
We can assemble all the 13-element vectors for each subpixel position
into a 13-plane-deep 3-D array, each plane representing the fractional
grade distribution of a given grade across the pixel. Examples of these
planes are given in Figures 6.24
through
6.27. The peaks of the distributions
are marked in the figures and given in Table 6.19
for each plane. We can treat these planes as probability distributions
for each grade, showing the likelihood that a photon impinging at a certain
subpixel position will yield an event of the shape (grade) being considered.
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Note the bimodality in these distributions - the upwards single-split
distribution shown in Figure 6.25,
for example. Some events that occurred near the lower edge of the
pixel were detected as up splits - this is because enough of the
charge clouds from these events were detected in the adjacent (lower) pixel
that they were detected as upward splits. This illustrates the fact that
we cannot know in which pixel the event really occurred, we can
only assume that it occurred in the brightest pixel, and this assumption
is not always right. Note also that, by replicating these single-pixel
distributions for adjacent pixels, a consistent probability distribution
appears, shaped similarly to the single-pixel distribution but centered
on different subpixel coordinates ((0.3,-0.3) for the L-shaped events shown
in Figure 6.26, instead of (0,0)
for the single-pixel events) and having somewhat different widths. This
implies that, if we are given an event's grade, we have a distribution
showing the likelihood that the event came from a certain subpixel position.
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Returning to the simulations, we deposited 4000 photons at the same subpixel position and came up with a distribution of grades. We want to use that distribution of grades to estimate the position of this ensemble of events. We have probability distributions of positions for each event, but how do we combine these to yield the best-estimate position for the ensemble?
As an initial step, we chose the simplest conceivable mapping. We assumed that a given grade came from a photon which interacted at the most likely subpixel position for that grade, i.e. wherever the peak is in the plots mentioned above. The subpixel coordinates of the peak are given as the third column of Table 6.19. Then we just averaged these positions to get the most likely subpixel position for the ensemble.
We computed ensemble position estimates as above for each subpixel position.
Then to test the accuracy of the method, we generated a ``distortion map''
by subtracting the true position from the estimated position (separately
for x and y), then computing a radial distortion. Figure 6.28
shows this distortion map. This map is intended to illustrate the degree
to which this simple algorithm is able to recover positions. Note that
there are regions on the pixel where the algorithm works well, and regions
where it does not. This leads to ambiguities in a photon's true subpixel
position and indicates that this algorithm provides accurate subpixel positioning
to about 1/16 pixel.
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A more relevant test of the algorithm is the degree to which it can
recover an accurate subpixel position of a point source smeared by a PSF.
To simulate this, we generated photons with a two-dimensional Gaussian
distribution about some pre-determined subpixel position. We deposited
these photons on simulated CCD frames one at a time, then simulated frame
readout and event detection and grading. This rate of one photon per source
per frame is consistent with the readout rate expected for modest sources
with AXAF. Using each event's grade as above, and ignoring events with
grades S+, P+, and Other, we assigned to the event a subpixel position
(the most likely position for that grade). Once we had accumulated an ensemble
of events, we computed the simple average and standard deviation for the
x and y positions separately and compared these estimates of the source's
position and the PSF widths to the input values. We also made these estimates
using only integer pixel positions for each photon and using the true subpixel
positions for each photon, for comparison. Since the positions and PSF
widths obtained from the true subpixel positions of the photons are the
best estimates we can make given the finite sample size, the fairest comparison
is between these results and those for the two algorithms in question,
not between the input values and the results for the two algorithms. The
results are summarized in Table 6.20. Note that
some source positions were deliberately located at subpixel positions that
suffered large distortions in the earlier tests (see Figure 6.28).
These results confirm that the subpixel position mapping algorithm described above does a better job of recovering the source position than integer pixel positioning, when the PSF is small compared to the pixel size. Not surprisingly, the algorithms converge when the PSF is comparable to the pixel size (as we approach critical sampling). The results of the grade-based algorithm are affected by energy since the splitting is affected by energy - as fewer events are split, the subpixel position mapping algorithm collapses to integer pixel positioning. Subpixel position mapping appears to perform marginally better than integer pixel positioning for small numbers of photons.
