Depth rendition of three-dimensional displays

Geometry of imaging with a stereocamera whose optics are a distance $a_C$ apart and whose axes are converged to an object plane at a distance $z_C$ Two targets are in the midsagittal plane, Q at a distance $dy$ above the horizontal, P in the horizontal plane a distance $d{z}_C$ in front of the object plane, where $dy=d{z}_C$ , small compared with $z_C$ . The left and right image planes, here shown noninverted, each contain representations of the two targets with coordinates $d\theta =dy/z$ and $d\phi =\pm a_C\times d{z}_C/2\times {z_C}^2$ along their $\phi ,\theta$ axes, ignoring a single magnification constant. Using the subscript C when referring to the recording situation, the parameter $R_C$ characterizes the rendition of the line element $dy=d{z}_C$ and is given by the ratio of the image-plane representations of the binocular separation on the left and right sides ( $a_C\times d{z}_C/{z_C}^2$ ) and the spatial interval in the object plane ( $dy/z_C$ ) It evaluates to $a_C/z_C$ and, once fixed in the recording session, remains independent of the magnification with which the right and left images are subsequently displayed on the viewing screen, superimposed with controlled separate exposure to the two eyes.

Schema showing a display screen at a distance $z_O$ from the observer and how the differentiated lateral positions of the right and left eye images $C_R$ and $C_L$ , generated in the manner shown in Fig. 1, represent the virtual third dimensional localization of a target C at a distance $d{z}_O$ . For an observer with interocular separation $a_O$ , the separation $C_L{C}_R$ in the screen plane is related to distance $d{z}_O$ by $t=a_O\times d{z}_O/(z_O-d{z}_O)$ or $d{z}_O=t\times z_O/(a_O-t)$ . The depth rendition for this situation $R_O=a_O/z_O$ is relative to that with which the record had been acquired, $R_C$ . With respect to the original two target objects separated by equal distances in depth and height, the observer is presented with a visual scene whose depth rendition, i.e., disparity/visual size ratio, is given by $R_O/R_C=a_O\times z_C/a_C\times z_O$ . In the experiments comparing the perceived depth of C with the apparent separation of two markers shown binocularly on the screen itself, the measures for both are read off as spatial intervals on the screen and hence share physical dimension, though perceptually a comparison is made between intervals in two submodalities with quite different processing.

Plan view of the observations, not to scale. (right) The observer’s task was to set marker C so that the depth angle $ACB$ appears to be $90\mathrm{deg}$ , i.e., as if the view was that of a perfect cube in depth, its corner pointing to the observer. (left) The observer’s task was to set a marker C so that its apparent distance in front of the fixation plane equals the distance separating markers A and B within the fixation plane. The 3D localization of C was achieved by differential target positioning of the right and left eyes’ views of displays on a computer monitor. Experiments were also performed with target C behind the frontal plane containing A and B.

Data, in two subjects, for two experiments in which the observer adjusted the disparity of feature C (see Fig. 3) according to two instructions: (a) match the magnitude of the seen depth to the comparison distance marked off on the screen or (b) place C so that $ABC$ forms a right angle in depth. Data are plotted in angular measure, abscissa the visual angle subtended at the eye of the comparison interval, ordinates the disparity, distributed equally between the right and left eyes’ views. Observation distance $90\mathrm{cm}$ .

Results, for four observers, in an experiment in which the observer set the disparity of point C (see inset) to match its depth to a range of comparison intervals $AB$ in the frontal plane. Distances are linear in object space, converted from angles in the manner shown in Fig. 2. Viewing the simple dot stereograms without secondary clues on a monitor screen at $90\mathrm{cm}$ , all observers required greater stimulus depth than prescribed by geometry for veridical depth rendition (thin straight line), i.e., they saw the configuration foreshortened in depth.

Replication of experiment in Fig. 4 for one observer at three observation distances, all other stimulus conditions, including visual angles and convergence, remaining constant. Results have been normalized for the intermediate distance. The straight line is the expected disparity if the relative depth rendition in the reconstruction were 1.0, i.e., if the observer’s view were veridical. The actual values, 6.0, 6.0, and 4.8, respectively, indicate that considerably more disparity is required in the display for the observer’s report of a match between depth and lateral distances than if the display were geometrically true.