How does our brain create depth perception from two slightly different images?

[Graeme M]

When we see things, we typically see one image which appears to have depth (if we have two functional eyes of course). I understand this is because the images from each eye are matched, and the slightly different perspective caused by the horizontal displacement of the eyes is used to generate the perception of depth. It seems that when images from both eyes do not match, some kind of contention process causes the separate images to alternate in conscious perception (binocular rivalry).

That seems clear enough.

If however I deliberately cross my eyes, I now see two distinct representations of the scene, for example, a cube on my desk can be made to appear as two distinct cubes. These remain within perception without alternating which shows that the binocular process must still be able to match both images (although I can with a little effort cause one or the other, or both, to fade from perception, but that does require effort - typically, both are distinctly visible).

But what on Earth is going on that I see two images? There is after all no "screen" in my head against which the images are "projected". I can understand that a single conscious scene is generated perceptually via some sort of fusing of the two images. But what are the mechanics behind disassociating the fused image?

 

[BillTre]

When I look at a steropair without a viewer, I first see the two separate images, then I see a third in between which is 3D (or in this case, the perceptual illusion of 3D).
I can then focus in the 3D image and ignore the others.

I can't say what is exactly going on, but it should be noted that there are a several of different cortical visual areas in the brain (several in the cortex, but also in the optic tectum (an important midbrain visual relay area, called the lateral geniculate in humans)). The wikipedia article cites a study of V2 (the second cortial visual area) showing responses to features of binocular disparity.
Which of these can be brought to conscious awareness is not clear to me. Any of them may be the "screen" the consciousness views at different times.
Within the overall visual map in the main visual field are neighboring "ocular dominance columns" which receive input from one eye or the other from the same area of the visual field. This would be input from the two eyes looking at the same external object.

 

[Graeme M]

Yes, you can do the same with a computer keyboard. By crossing your eyes, you can cause the two images to converge in a 3rd space and generate a new "3D" image of the now separated images (if that makes sense). I think this works because of the similarity of the object representation - the repeating block pattern of the keys means you can combine the disassociated images of the key blocks because they are broadly the same ie you've disassociated the previously fused single representation of the keys, and created a new fused image from those now separated images, or something like that - I have no idea of the mechanics of that! What's cool about that is that the new 3D image of a single key has on its surface different letters, for example you can combine the X and C keys into a new 3D object with the letters X and C overlaid, and bingo, you can somehow "focus" on either the X or C and the other fades out...

 

[BillTre]

Looks like I messed up one of my links.
This:

several of different cortical visual areas

should be this:
several of different cortical visual areas
This has the discussion of the binocular disparity sensitive area.