"Learning occupancy grid maps with forward sensor models, S.Thurn,2008" a formula that is difficult to overcome
The vector $c_t$ has exactly one coordinate which is 1, and the rest are 0s. You can check that if, for example, $c_{t,k}=1$, when you substitute $c_{t,k}=1$ and $c_{t,\ell}=0$ for all $\ell \neq k$ in (26) you get (22). The same happens for (24) and (25).