Binocular disparity · Random dot stereograms · Horopter · Panum's fusional area
Left eye view (cross-fuse with right)
Right eye view
Julesz random-dot stereograms (1960): Two images look like pure noise individually. When fused (cross-eye or parallel fusion), the brain extracts a 3D shape purely from binocular disparity — with zero monocular cues. This proved that stereopsis is a global, not local, computation. The floating square appears to pop out or recede depending on disparity sign. Cross-fuse by crossing your eyes until the two images overlap.
Disparity formula:
δ = I·(1/d − 1/D)
where I = interocular distance, d = object distance, D = fixation distance.
Crossed disparity (δ>0): object is nearer than fixation → seen in front.
Uncrossed disparity (δ<0): object is farther → seen behind.
Stereo depth threshold: ~5–10 arcsec (= ~0.003°) — the most precise spatial sense we have. Corresponds to ~1mm depth at arm's length.
Vieth-Müller Horopter: The locus of points in 3D space whose images fall on corresponding retinal points. Points on the horopter have zero disparity and are fused perfectly.
The geometric horopter is a circle passing through both nodal points and the fixation point.
Panum's fusional area: A region around the horopter within which objects are seen singly (though with depth). Objects outside Panum's area cause diplopia (double vision).
Width of Panum's area: ~6–12 arcmin centrally, increases to ~1° in periphery.
Color coding: Green band: Panum's fusional area (single vision + stereopsis) Red region: diplopia zone Yellow line: geometric horopter (VMC)