Probabilities in two slit interference

I'm reading Higher-order interference in quantum physics by Rozema et al. Their Figure 1 is a simple setup

There are two paths. The upper path $$|u\rangle$$ is open and the lower path $$|d\rangle$$ is blocked. U is a unitary interaction between the two modes after the blocker, and there are N input states $$\rho$$. Each $$\rho$$ is spanned by $$|u\rangle$$ and $$|d\rangle$$. Here's what they say the intensities are:

$$I_{00}=NP_{00}=N\langle u|U\rho U^\dagger|u\rangle$$

is intensity when both paths are open.

$$I_{01}=N\rho_{uu}\langle u|U|u\rangle \langle u|U^\dagger|u\rangle$$

with $$\rho_{uu}=\langle u|\rho|u\rangle$$, is intensity when upper path is open.

$$I_{10}=N\rho_{dd}\langle u|U|d\rangle \langle d|U^\dagger|u\rangle$$ is intensity when lower path is open.

I think I understand $$I_{00}$$. We are just projecting the state after the unitary (i.e. $$U\rho U^\dagger$$) onto the upper path. I'm not sure why the other two are written the way they are. I know that this has a rather simple explanation, yet I cannot seem to find it. Please let me know why we can write intensities this way.