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Sub-pixel measurements

The HWHM of the PSF for AXAF's High Resolution Mirror Assembly is expected to be roughly 0.5 arcsec, the same size as a pixel in the AXAF CCD Imaging Spectrometer (ACIS). In order to reconstruct the PSF and obtain source positions accurate to less than 0.5 arcsec, we would like to locate individual photon interaction sites on a subpixel scale. We have explored this goal by using the intrinsically larger event splitting tendencies of back-illuminated (BI) CCDs at moderate (1 keV) X-ray energies.

An ACIS BI CCD was modeled using the frame simulator described above. We placed 4000 1-keV photons on a simulated ACIS BI 1024$\times$1024 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$\times$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.

Table 6.16: Pixel numbering for grade subarrays
6 7 8
3 4 5
0 1 2

The grades used are defined in Table 6.17, 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 6.17: Grade definitions and subpixel positions of probability maxima  
event type constituent pixels grade probability maximum
single 4 0 (0,0)
S+ 4 + others 1 --
up vertical 4,7 2 (0,0.4)
down vertical 1,4 3 (0,-0.4)
left horizontal 3,4 4 (-0.4,0)
right horizontal 4,5 5 (0.4,0)
P+ 4 + others 6 --
down left L 1,3,4 7 (-0.3,-0.3)
down right L 1,4,5 8 (0.3,-0.3)
up left L 3,4,7 9 (-0.3,0.3)
up right L 4,5,7 10 (0.3,0.3)
down left quad 0,1,3,4 11 (-0.4,-0.4)
down right quad 1,2,4,5 12 (0.4,-0.4)
up left quad 3,4,6,7 13 (-0.4,0.4)
up right quad 4,5,7,8 14 (0.4,0.4)
Other 4 + others 15 --

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 6.12 through 6.15. The peaks of the distributions are marked in the figures and given in Table 6.17 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.

Figure 6.12: Single-pixel event position probability distribution  
\psfig {file=subpixelres/g0prob.eps}

Note the bimodality in these distributions - the upwards single-split distribution shown in Figure 6.13, 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 6.14, 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.

Figure 6.13: Two-pixel event position probability distribution  
\psfig {file=subpixelres/g2prob.eps}

Figure 6.14: Three-pixel (L-shaped) event position probability distribution  
\psfig {file=subpixelres/g8prob.eps}

Figure 6.15: Four-pixel (square-shaped) event position probability distribution  
\psfig {file=subpixelres/g13prob.eps}

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 6.17. 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 6.16 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.

Figure 6.16: Radial distortion map for one pixel, showing the difference between the true and reconstructed positions for 121 sample positions in a pixel. Each arrow starts at the true subpixel position and points to the position estimated by the algorithm.  
\psfig {file=subpixelres/newdistor.eps}

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 6.18. Note that some source positions were deliberately located at subpixel positions that suffered large distortions in the earlier tests (see Figure 6.16).

Table 6.18: Results of subpixel position testing using a PSF-convolved point source  
energy (eV) number of
true PSF estimate using
integer pixel positions
estimate using
mapped subpixel positions
estimate using
real photon positions
x y $\sigma_{x}$ $\sigma_{y}$ x y $\sigma_{x}$ $\sigma_{y}$ x y $\sigma_{x}$ $\sigma_{y}$ x y $\sigma_{x}$ $\sigma_{y}$
1000 3000 5.30 5.00 0.85 0.85 5.30 5.00 0.90 0.89 5.30 5.00 0.85 0.86 5.30 5.00 0.85 0.85
1000 3000 5.38 5.35 0.85 0.85 5.37 5.34 0.90 0.89 5.37 5.34 0.85 0.85 5.37 5.34 0.84 0.85
1000 3000 5.20 5.50 0.85 0.85 5.18 5.48 0.90 0.91 5.18 5.49 0.85 0.87 5.18 5.49 0.84 0.86
1000 500 5.20 5.50 0.85 0.85 5.21 5.51 0.89 0.90 5.20 5.52 0.84 0.87 5.20 5.52 0.83 0.86
1000 100 5.20 5.50 0.85 0.85 5.31 5.45 0.94 0.83 5.25 5.51 0.86 0.80 5.25 5.50 0.88 0.80
1000 1000 5.05 5.35 0.85 0.85 5.03 5.30 0.88 0.90 5.07 5.30 0.84 0.86 5.06 5.30 0.84 0.86
500 1000 5.05 5.35 0.85 0.85 4.96 5.31 1.02 1.03 4.97 5.30 1.00 1.01 5.02 5.37 0.85 0.85
5000 1000 5.05 5.35 0.85 0.85 5.08 5.33 0.91 0.89 5.10 5.34 0.88 0.86 5.09 5.34 0.86 0.85
1000 1000 5.03 5.35 0.10 0.10 5.18 5.50 0.47 0.00 5.05 5.38 0.13 0.16 5.05 5.38 0.10 0.10
5000 1000 5.03 5.35 0.10 0.10 5.18 5.50 0.47 0.00 5.04 5.30 0.12 0.18 5.05 5.35 0.10 0.10
1000 1000 5.05 5.35 0.40 0.40 5.07 5.36 0.51 0.49 5.07 5.35 0.40 0.42 5.07 5.35 0.40 0.41
1000 20 5.05 5.35 0.40 0.40 5.00 5.45 0.51 0.51 5.02 5.48 0.42 0.40 5.08 5.43 0.37 0.40
1000 5 5.05 5.35 0.40 0.40 5.30 5.50 0.45 0.71 5.30 5.42 0.35 0.48 5.27 5.35 0.30 0.47

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.

next up previous contents
Next: Spatial linearity Up: ACIS/HRMA Performance Prediction Previous: Ray projection on ACIS

Mark Bautz