When talking about the double-slit experiment, most physics books consider the wall to be a infinitely high potential, so that the photon is either reflected or transmitted through one of the slits.
(Side-question: is it reasonably correct to imagine the wall as a mirror with two small slits? It either reflects or transmits the photon, with maybe some edge cases like partial reflection).
Now let's replace the mirror by a black body, so that the photon is either absorbed or transmitted through one of the slits (furthermore, let's assume that the screen behind the wall is very far away). Quantum mechanics predicts (I'm talking about Copenhagen interpretation only) that there is a certain probability $P_W$ that the photon is absorbed by the wall, and also a probability $P_T = 1 - P_W$ that it is transmitted.
If that is correct, then this seems to me like some kind of half-measurement. The wave function collapses only if the photon is absorbed, which means that the photon "has to decide" where it hits the wall. If the photon is transmitted, the wave function cannot completely collapse (if it did, there would not be any interference pattern behind the wall).
However, the wave function is not left unchanged too. Those eigenstates of the position operator that correspond to an absorption have been "filtered out", because after the wave has passed the wall, the photon can no longer be detected to be absorbed by the wall. Furthermore, while the photon passed the wall, the wave function has been changed in a way that (as far as I understand it) does not seem to satisfy the Schrödinger equation - it is a partial collapse, not an evolution over time.
If that is correct, what process describes such a partial collapse? In the mathematical formulation of quantum mechanics, there's no mention of such a partial measurement - it's just measurement or no measurement, and it says that a measurement let's the wave function collapse to an eigenstate of the measurement operator. Clearly, if the photon does not hit the wall, it does not collapse to an eigenstate of the position operator.