Comparing Measurements of Pictorial Depth: Conditions for Placing Methods, Observers and Depiction Styles on One Scale

Two observers can agree about what lies in front in a picture yet report different amounts of apparent depth. People report pictorial depth by judging which of two points looks nearer, setting a probe to match a surface, adjusting relative sizes, or rating three-dimensionality. Two such methods can agree closely without sharing a unit. We ask when different methods, observers, depiction styles and viewing conditions can share one scale. Each set of scores, a measurement block, is a graph node. Two blocks are linked only if their shared observations pass six checks fixed in advance, among them outcome type, removal of declared offsets and position effects, proportionality and covariance. If the adjusted shared patterns are exactly proportional, a connected graph fixes every block's gain, its relative scale factor. The checks accept proportionality only within a set margin, so a link supports the gains without proving exact proportionality. One unit is still not one depth coordinate. Suppose each block scores shared stimuli with its own origin and positive unit. A matrix-rank test then decides when a common coordinate is identified: when only one overall origin and unit remain free. If each shared stimulus, an anchor, is scored by exactly two blocks, this needs two disjoint sets of anchors, each connecting every block, their compatibility assumed, not tested. They also suffice, except for anchor values satisfying exact polynomial relations (probability zero for anchor values drawn from any distribution with a density). We adapt network gain estimation from test linking, with a loop-consistency check. A reanalysis of published probe settings on 3-D renderings passes one of 171 candidate depiction-style links through all six checks, each judged by its own 95% bound, unadjusted for the 171 pairs. Links between tasks remain untested; they need shared stimuli and a joint uncertainty model.