True X-rays may appear in a finite number of charge patterns (Fig. 1):

- 1.
- Monopixels, with no other pixel above threshold around.
- 2.
- Bipixels, with two adjacent pixels above threshold.
- 3.
- Tripixels, with two pixels above threshold beside the maximum. They are necessarily L-shaped (aligned tripixels are not produced by true X-rays) and the corner pixel always receives the largest charge.
- 4.
- Quadripixels, forming a closed square (elongated four-pixel patterns are not produced by true X-rays).

For a relatively large incoming flux, two X-rays may materialize as monopixels in adjacent pixels and be counted as an apparent split event. This comes in addition to the true pile-up (two X-rays in the same pixel). In the following I compute the apparent pattern distribution (after pile-up), neglecting the possibility that small amounts of charge (below threshold) in peripheral pixels of two close-by charge patterns may combine to exceed threshold.

I call the incoming X-ray flux/pixel/frame (assumed uniform
and random) and the expected (expectation value of the)
count rate/pixel/frame in pattern *j* (to avoid ambiguities, I note *i*
charge patterns of incoming photons and *j* patterns of detected events).
Except in Appendix C the paper deals only with ideal (as opposed to
noisy) quantities. The actual measured count rate would be with Poisson fluctuations.

I call the expected count rate/pixel/frame in pattern *j* of clean
(not piled-up) events. does not correspond to a measurable
quantity, since one has no way to tell a clean event from a piled-up one
*a posteriori*, but it is useful to know what it is.
is called the flux loss. It is the loss
in detection efficiency due to pattern overlap.
is called the pile-up fraction. It is the fraction
of measured events whose energy will be wrong.
Note that in this paper I always refer to the flux loss and pile-up fraction
*per pattern*. This is appropriate since the pattern is known in the data.
Global quantities would be obtained by summing over the patterns.

In the next subsection and 6 I derive the relationship between , , and .The formulae appear as a product of the probability to get such a pattern times the probability not to destroy it by receiving another photon in the exclusion zone around the "target'' event. For , probability to get clean (not piled-up) events, the first (production) term is simply .

Since the probability not to receive a charge pattern of type *i* in a pixel
is always of the form , the second (suppression) term
is always of the form , where is a sum over all pixels in the exclusion zone (including the target event).
I note here the coefficient associated to (clean events).
For the exclusion zone would not include the target event,
since additional charge patterns there would preserve the event's geometry.
The exclusion zone is larger for larger patterns
(because not one of their *i* pixels must be close to the original pattern),
so that will appear as ,
where *n*_{ji} is the area of the exclusion zone for incoming pattern *i*
and measured pattern *j*.
Mathematically, one may start by convolving
the area which must be below threshold around the measured pattern *j*,
plus the pattern itself (black, grey and white pixels in Fig. 1),
with the area of the incoming pattern *i* for a given orientation
(black and grey pixels in Fig. 1).
The exclusion zone (Fig. 2) may then be obtained by setting
all pixels above 0 in the convolution to 1, multiplying by the fraction
of patterns with that orientation, and summing over all orientations.

For , the production term must include all possibilities of pile-up preserving the geometrical shape of the pattern, and the exclusion zone does not include the target event.

The formulae have been checked against simulations (Ferrando et al. 1996) for various pattern distributions and incoming flux. Except for which is known to be approximate anyway (6.3) I have seen no evidence of discrepancy. Early comparison to calibration data is presented in Appendix C.

I first consider the simplest case of monopixel events. For such an event to remain clean (not piled-up) there must be no other pixel above threshold in the exclusion zone (Fig. 1, top left), nor in the target pixel. Therefore there must be no other monopixel pattern in the 9 corresponding pixels, whereas bipixels must be avoided over a larger area: the same 9 pixels, plus 3 pixels outside (on the left, top, right or bottom depending on the bipixel's orientation) from which the bipixel would spill over into a pixel immediately next to the event. The same reasoning leads to 15 pixels to be avoided for tripixels and 16 for quadripixels. Fig. 2 shows graphically the exclusion zone for all orientations of the secondary charge patterns. Its area is the same for all orientations because the target (monopixel) is symmetrical.

To get the production term for , one can first say that monopixels can appear only from one or more superposed monopixels. The probability to have zero monopixels in a given pixel is . Therefore the probability to have at least one is .Finally the target pixel must be removed from the exclusion area for monopixels.

Finally, using ,one obtains:

(1) | ||

(2) | ||

(3) |

(4) |

Because the expected count rate for a flat spatial distribution is not strictly increasing as a function of incoming photon flux, there is an ambiguity in reconstructing the incoming photon flux. This ambiguity can be overcome by considering at the same time the fraction of diagonal events over truly isolated monopixels (dashed line in Fig. 3). This is also by itself ambiguous, but matching both is not. At this kind of count rate the statistics (even in a single frame) is enough to distinguish solutions unambiguously, except at very high incoming flux, which is not a very interesting case.

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