To examine the impact of local disparity variation on half-occlusion detection in natural scenes, we first sampled 10,000 stereo-image patches from the natural scene database using the procedure discussed above. We found that 86.5% of the sampled stereo-image pairs were centered on binocularly visible scene points, and that 13.5% were centered on half-occluded scene points. We determined which patches had half-occluded centers directly from the range measurements by determining which patches had centers where the horizontal disparity gradient (
Display Formula\(DG = \Delta \theta /\Delta X\)) with any other point was 2.0 or higher (a disparity gradient of 2.0 corresponds to Panum's limiting case; see Bülthoff, Fahle, & Wegmann,
1991). The disparity gradient in the half-occlusion scenario depicted in
Figure 2A is somewhat larger than 2.0. Second, to quantify local depth variation, we computed the disparity contrast of all patches with binocularly visible centers. For all analyses, disparity contrast was computed over a local integration area of 1.0° (0.5° full-width at half-height; see
Equation 1); results are robust to this choice (see
Supplementary Figure S1). Third, the similarity of the left- and right-eye image patches was quantified with the correlation coefficient
\begin{equation}\tag{2}{\rho _{LR}} = {{\sum\limits_{{\bf{x}} \in A} {{\bf c}_L^W\left( {\bf{x}} \right){\bf c}_R^W\left( {\bf{x}} \right)} } \over {\left\| {{\bf c}_L^W\left( {\bf{x}} \right)} \right\|\left\| {{\bf c}_R^W\left( {\bf{x}} \right)} \right\|}}\end{equation}
where
Display Formula\({\bf c}_L^W\)and
Display Formula\({\bf c}_R^W\) are windowed left- and right-eye Weber contrast images (see
Methods) and where
Display Formula\(\left\| {\;{\bf{c}}\left( {\bf{x}} \right)\;} \right\| = \sqrt {\sum\nolimits_{{\bf{x}} \in A}^N {{\bf{c}}{{\left( {\bf{x}} \right)}^2}} } \) is the L2 norm of the contrast image in a local integration area
A. The integration area is determined by the size of a cosine windowing function
Display Formula\(W\) (see
Methods); the windowing function determines the size of the spatial integration area within which binocular correlation is computed. Fourth, under the assumption that the correlation coefficient is the decision variable, we used standard methods from signal detection theory to determine how well half-occlusions can be detected in natural images. Specifically, we determined the conditional probability of the decision variable given (a) that the center pixel was binocularly visible for each disparity contrast
Display Formula\(p\left( {{\rho _{LR}}|bino,{C_\delta }} \right)\) and (b) that the center pixel was half-occluded
Display Formula\(p\left( {{\rho _{LR}}|mono} \right)\) (
Figure 5C), swept out an ROC curve (
Figure 5D), computed the area underneath it to determine percent correct, and then converted to
d′. Finally, we repeated the steps for different spatial integration areas. Half-occlusion detection performance (
d′) changes significantly as a function of the spatial integration area for each of several disparity contrasts (
Figure 5E). Clearly, local depth variation reduces how well binocularly visible points can be discriminated from half-occluded points. (Note that the same procedure could be adapted to work in the retinal periphery with one straightforward extension. For any given patch in one eye's image, a cross-correlation could be performed to determine the peripheral locations in the other eye to compare. The correlation of the two patches yielding the maximum correlation could then be used as input to the procedure described above.)