We have evolved to live in a (spatially) 3D world. For our "natural" understanding no material 2D objects exist. 2D objects can be only conceptual. Drawings on a plane are conceptually defined 2-dimensional but if we measure any line - how thin ever - it is of course 3-dimensional.
I think that no one can really have a percept of a pure 2D image. At least it belongs to some 3D environment. It's drawn on a plane or just floating in a space.
Consequently I think that only humans - having a symbolic language - can define a concept of 2D objects. Thus "perceiving" them is a conceptual illusion. My opinion is that differences in perceiving images depends on person's conceptual world; her thoughts, experiences and metaphysics of understanding. That's why I think that the 2D/3D perceiving is subjective and cannot be objectively defined let alone measured.