In order to show the extent to which event coincidence affects centroiding resolution, the author has carried out a series of com puter resolution simulations. These simulations model the response of the centroiding electronics by showing how it would centroid two events which are initially separated by 5 CCD pixels and then brought together in steps of 0.1 CCD pixels. The com puter program simulates tw o asym m etrical gaussian profiles (the shown in F ig 5.1) w ith a given separation (one rem aining stationary while the other is moved 0.1 pixels closer each tim e) and places them in to CCD pixel bins. The energy in each CCD pixel is integrated and the resulting event profiles passed to a software version of the appropriate centroiding algorithm . The program calculates the distance from the tru e event centre to the centre of the subpixel bin into which each event is centroided. The results are presented by showing the centroiding error (in subpixels) against the true event separation.
Three sets of simulations were carried out. They each show the effect of
• the type of centroiding algorithm (3 or 5 pixel) used,
• the subpixel size and event w idth,
• the coincidence correction circuit
on the centroiding resolution. Simulations were firstly carried out for events having a FW HM of 1.2 CCD pixels, being centroided to ^ of a pixel. Both events were assumed to have the same to ta l energy which was 100 ADU (each). The two events were then bought closer together in steps of 0.1 CCD pixels. Comparisons were made between events which were centroided
• w ith a 5-pixel centroiding algorithm F ig 5.4
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Centroided to 1 /4 of e pixel. E ren t FWHU- 1.2 pixels. Event 1 - Stetionary. Event 2 - Moving.
s o
0 1 2 3 4 5
Event Separation (Pixels)
Fig 5.4. C entroiding erro rs w hen using a 5-pixel cen tro id in g alg o rith m
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Event 1 - Stationary. Event FWHU 1.2 CCD PlxeU.
C entroiding to 1 /4 Of a Pixel. Event 2 - Moving. 'Ô) c c CO g L, O 00 c o
g
u 0 1 2 3 4 5Event Separation (Pixels)
The results of this set of simulations show firstly th a t th e separation a t which two events s ta rt to affect the centroid position of the other is larger when using th e 5-pixel algorithm instead of the 3-pixel algorithm . A t separations above 3.3 CCD pixels (for the 5-pixel algorithm ) and 2.2 pixels (for the 3-pixel algorithm ) th e presence of a second event does not affect the subpixel into which the first event is centroided, provided the centroiding resolution is < ^ of a CCD pixel. If the centroiding resolution is increased then th e event separation at which the profile of one event starts to change the subpixel into which the other event is placed, increases.
Secondly, because the events are asymm etrical, even when they do not overlap (in the region between A and B in F ig 5.4) the centroiding error is occasionally greater th an ^ of a CCD pixel. This is due to the way in which the energy associated w ith an event profile is binned into CCD pixels. The centroiding error for bo th the 3-pixel and th e 5-pixel algorithm s are very similar in this region of the graph.
Between regions B and F in F ig 5.4, where the event profiles are partially but not fully overlapping, the 3-pixel algorithm centroids to a greater accuracy th a n the 5-pixel algorithm . This is because for partially overlapping events the 5-pixel algorithm samples a high proportion of the d ata from the other event. The 3-pixel algorithm only samples one pixel either side of the d a ta peak and these pixels contain less of the d a ta associated with the other event. This is why for overlapping events the 3-pixel algorithm is m ore accurate th an the 5-pixel algorithm although when centroiding single events to a high accuracy the 5-pixel algorithm is better.
As the event separation decreases between the region m arked F and the point at which events completely overlap the centroiding error associated w ith bo th events starts to decrease. A t the point when they completely overlap the centroiding accuracy is very close to if not b etter than 1 subpixel. This is because where one event profile lands directly on top of another, the combined event profile is similar to th a t of a single event and so these simulations suggest th a t a narrow point source image would be centroided to a high accuracy.
In conclusion, although the 5-pixel algorithm centroids single events to a higher accu racy th an the 3-pixel algorithm , the 3-pixel algorithm is sufficient for centroiding events to I of a pixel. If there are a high num ber of coincidence events th e 3-pixel algorithm centroids to a much higher accuracy than the 5-pixel algorithm because the algorithm
samples fewer pixels containing contam inating d a ta from the second event. A t the point where the two events completely overlap, the theoretical centroiding error approaches 1
subpixel.
Simulations were then carried out using only a 5-pixel centroiding algorithm and com parisons made between the following cases;
• Events having a FWHM of 1.2 CCD pixels, centroided to ^ of a CCD pixel. F ig 5.4.
• Events having a FWHM of 1.4 CCD pixels, centroided to ^ of a CCD pixel. F ig 5.6.
These simulations were carried out in an identical fashion to th e first set and were intended to observe the affect of event w idth and subpixel size on the centroiding resolution.
5
C e n tro id e d to 1 / 2 of e p ix el. E v e n t FWHU-1.4 Pixels.
E v e n t 1 - S ta tio n a ry . 4 E v e n t 2 - Moving. 3 2 1 0 5 0 1 2 4 Event Separation
Fig 5.6. Centroiding errors associated with events having a 1.4 CCD pixel FWHM
The results of this second set of simulations show th a t there is very little difference between centroiding events w ith a FWHM of 1.4 CCD pixels, and centroiding events with a FW HM of 1.2 CCD pixels. From the simulations carried out by Dick et al the ideal event w idth to which the 5-pixel algorithm is best suited, lies in th e region of 1.5-1.75 CCD pixels and so it is expected th a t events w ith a FW HM of 1.4 CCD pixels would be
more accurately centroided. This however, only applies to isolated events and so if we only examine th e region between A and B then the events w ith a FW HM of 1.4 CCD pixels were centroided only slightly more accurately th an those events whose FW HM were 1.2 CCD pixels.
The most obvious thing to notice about th e results of this sim ulation is th a t when centroiding single events to an accuracy of ^ a CCD pixel the centroiding error is sometimes > 1 subpixel, but when centroiding to ^ a CCD pixel, th e centroiding error for single events never increases above 1 subpixel. This is merely due to th e inherent variation in subpixel sampling size. Varying the event w idth between 1.2 and 1.4 CCD pixels does not change this fact.
In these simulations, where there are no other event distortions ap art from the inherent event asymmetry, any fixed p attern noise is due simply to the event asym m etry itself. They suggest th a t when centroiding to only | a CCD pixel, no fixed p a tte rn noise is present. However, in practice a small degree of fixed p attern noise does appear when centroiding to I a pixel showing th a t it is the combined effect of all the event distortions, not ju st its inherent asymmetry, th a t affects the real level of fixed p attern noise.
Simulations were then carried out in a similar way as F ig s 5.4 and 5.6 bu t this time w ith the coincidence correction circuit (C hapter 2) was enabled.
• F ig 5.7 shows events having a FWHM of 1.2 CCD pixels, centroided to ^ of a CCD pixel.
• F ig 5.8 shows events having a FW HM of 1.4 CCD pixels, centroided to ^ of a CCD pixel.
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C e n tro id e d to 1 / 4 o f a p ix el. E v e n t FWHU- IJt p ix els. E v e n t 1 - S ta tio n a ry . 'q3 § (0
5
E v e n t 2 - Moving.I
OO cn
/ 1 o is 0 1 2 3 4 5Event Separation (Pixels)
Fig 5.7. Affect o f the coincidence correction circuit on centroiding resolution
5 C e n tro id e d to 1 / 2 a p i x e l E v e n t FWHM-1.4 Pixels. E v e n t 1 - S ta tio n a ry . 4 E v e n t 2 — Moving. 3 2 1 0 0 1 2 Event Separation 4 5
The differences between F ig s 5.4 and 5.7 and those between F ig s 5.6 and 5.8 are best described by looking at different regions of the plot, where moving from A to F the second event is getting closer to the first.
1. In the region between A and B bo th graphs are very similar because there is no overlap between the two event profiles. The Coincidence Correction circuit is not used at these event separations as the events are not overlapping.
2. Point B is an anomalous condition, where due to the asym m etrical natu re of the event profile the binned centre of gravity of Event 2 is not even in the same subpixel as the event peak. This leads to large centroiding errors.
3. From B to C only Event 2 is centroided using the Coincidence Correction, no cor rection being associated w ith Event 1. This situation arises when there are 2 pixels between the event peaks.
4. From C to D a similar situation to B arises where th e centre of gravity of Event 2 is not in the same subpixel as the event peak, leading to relatively large errors except where the Coincidence Correction is applied.
5. From D to F Coincidence Correction is applied to b oth events w ith consequent improvement in resolution.
6. From point F onwards the two events completely overlap and hence the conditions for Coincidence Correction are not fulfilled.
These simulations show th a t the Coincidence Correction circuit is of great im portance for optim um resolution where there is a high likelihood of resolved coincidences occur ring. Such situations could, for example, occur when observing extended sources or point sources which are very close together. For bo th sets of simulations it can be seen th a t where Coincidence Correction is applied (prim arily between B and C and between D and F ) there is a large improvement in centroiding accuracy, typically greater th an 2 subpixels.