  We perform a parametric study of the newly developed time-of-flight (TOF)
image reconstruction algorithm, proposed for the real-time imaging in
total-body Jagiellonian PET (J-PET) scanners. The asymmetric 3D filtering
kernel is applied at each most likely position of electron-positron
annihilation, estimated from the emissions of back-to-back $\gamma$-photons.
The optimisation of its parameters is studied using Monte Carlo simulations of
a 1-mm spherical source, NEMA IEC and XCAT phantoms inside the ideal J-PET
scanner. The combination of high-pass filters which included the TOF filtered
back-projection (FBP), resulted in spatial resolution, 1.5 $\times$ higher in
the axial direction than for the conventional 3D FBP. For realistic $10$-minute
scans of NEMA IEC and XCAT, which require a trade-off between the noise and
spatial resolution, the need for Gaussian TOF kernel components, coupled with
median post-filtering, is demonstrated. The best sets of 3D filter parameters
were obtained by the Nelder-Mead minimisation of the mean squared error between
the resulting and reference images. The approach allows training the
reconstruction algorithm for custom scans, using the IEC phantom, when the
temporal resolution is below 50 ps. The image quality parameters, estimated for
the best outcomes, were systematically better than for the non-TOF FBP.
