3 Limitations of the null-sum test

The data quantization on $\log _2 K$ bits may cause $a+b+c \not= 0$ for a given photo-event. Misdirected projections due to bad optical alignment, or small errors in the CCD spot ranging may have the same effect. A tolerance $\varepsilon$must be set, replacing Eq. (4) by the double inequality:

$$-\frac{\varepsilon}{ 2}\leq a+b+c\leq +\frac{\varepsilon }{2}\cdot$$ (6)

This may cause the null-sum test to fail and create incorrect photo-events. Suppose a photon having $\Delta$-projections (a₁,b₁,c₁) which comply with the inequality:

$$\vert a_1+b_1+c_1\vert\leq\varepsilon/2.$$ (7)

There may exist lists $L_{\rm A}$, $L_{\rm B}$, and $L_{\rm C}$, such that changing one $\Delta$-coordinate among a₁, b₁, or c₁ by another one in the lists leaves Eq. (7) unchanged. The same thing may happen by changing two $\Delta$-coordinates (either a₁ and b₁, or a₁ and c₁, or b₁ and c₁). The back-projection process will therefore generate an extra photon. As this pseudo-photon is a crossover between two or three existing photons, we call it "cross-photon''. There are two types of cross-photons. The first type regards those made by crossover of two existing photons (two $\Delta$-coordinates from the same existing photon). The second type is a cross-photon having its $\Delta$-coordinates originating from three different existing photons.

Type-1 cross-photons are due to the non-zero tolerance $\varepsilon$. Type-2 cross-photons occur with an increasing probability when three or more photons are present in the same frame.

In order to assess the frequency of occurence of cross-photons, one of us (SM) wrote a simulation software generating projections from random numerical "photons''. The number of generated photons per frame is variable and complies with Poisson's law. Given $\bar N$, the average number of photons per frame, this software draws a variable number N of photons for each frame, such as:

$${\rm prob}(N=k)=\frac{\bar N^k}{ k!}\exp (-\bar N).$$ (8)

We considered $\bar N\leq 5$ and we set K=1024, as in the prototype to be built (see Sect. 5). We measured the quantity of type-1 and type-2 cross-photons generated, normalized by the number of input photons. Figure 2 plots the results for several values of $\bar N$ and $\varepsilon$. It clearly appears that type-2 cross-photons dominate for large $\bar N$.

  

Figure 2: Results of numerical simulations with a "Poisson'' photon generator. Percentage (normalization by the number of incoming photo-events) of type-1 (solid line) and type-2 (dashed line) cross-photons generated, for different average numbers of photons per frame: $\bar N$, and null-sum test tolerance in pixels: $\varepsilon$