Issue regarding the list-mode negative Poisson log-likelihood function about parallelproj HOT 1 CLOSED

Hi Kun,

1. The equation from our paper is the negative Poisson logL using sinogram data (y and \bar(y)) are sinograms and the sum runs over all sinogram bins "i".
2. The definition of Ote is equivalent except for:

• a global minus sign (they define the logL, we the negative logL)
• they ignore additive contaminations such as scatter and randoms

To see that they are equivalent, consider this:

$$logL = \sum_i y_i \log \bar{y_i} - \sum_i \bar{y_i} $$

now rewrite $y_i$ as

$$y_i = \sum_1^{y_i} 1$$

(the counts in sinogram bin "i" are a sum of ones $y_i$ times)

$$logL = \sum_i \sum_1^{y_i} \log \bar{y_i} - \sum_i \bar{y_i} $$

the first two sums can be rewritten as the sum over all events (with event index $t$ and corresponding sinogram bin $i_t$)

$$logL = \sum_{events \ t} \log \bar{y}_{i(t)} - \sum_i \bar{y_i} $$

If we now insert the forward model (ignoring scatter and randoms)

$$\bar{y_i} = \sum_j a_{ij} x_j$$

we get the expression of Ote et al.

from parallelproj.