Recovery of 3-D structure from motion is neither Euclidean nor affine

The relationship between simulated and judged depth separations for pairs of probe dots on planar surface patches was examined in a series of 6 experiments. The simulated slant of the patches was varied without varying the simulated depth separation of the probe dots by varying the depth gradient orthogonal to the direction determined by the probe dots on the image plane. Judged depth separation varied with mean slant for constant simulated depth separations. When observers judged depth separations along a closed path, the integral of the signed depths did not sum to zero, as would be required in Euclidean geometry. These results are inconsistent with the view that the mapping between simulated and perceived 3-D structure is affine and indicate that, in general, the perceived structure cannot be represented in either a Euclidean space or an affine space. Moreover, these results are consistent with a first-order temporal analysis of the optic flow.