We wish to test whether the visual system errs in using these pseudocues across changes in viewpoint, resulting in the failures of roughness constancy observed in the data. We assume that cues and pseudocues are scaled and combined by a weighted average (Landy, Maloney, Johnston, & Young,
1995). In viewing a surface of roughness
r from a given viewpoint
ϕ _{v}, the observer forms the roughness estimate
where the values
w _{ i } combine the scale factors and weights and thus need not sum to 1 as weights usually do. In this study, observers compared this roughness estimate with that for a second surface patch with roughness
r′ viewed from a different viewpoint,
ϕ _{v}′,
to decide which one was rougher. Consider the situation in which two surfaces were perceived to be equally rough; that is,
R =
R′. Subtracting
Equations 8 and
9 yields
where Δ
R _{ i } =
R _{ i }′ −
R _{ i }. We assume that
w _{d} was nonzero; therefore, we can rearrange
Equation 10 as
where
a _{p} = −
w _{p}/
w _{d}, and so forth. We define Δ
r _{p} =
E[Δ
R _{p}] =
r _{p} −
r _{p}′, Δ
r _{m} =
E[Δ
R _{m}] =
r _{m} −
r _{m}′, and so forth.
Equation 11 effectively expresses the effect of the pseudocues in terms of the viewpoint-invariant cues. If we take expected values of both sides of
Equation 11, we have
This equation expresses the relationships between the systematic errors in viewpoint-invariant and viewpoint-dependent cues. If an observer were roughness constant across viewpoint, Δ
r _{d} should be 0, as
r _{d} =
r _{d}′ =
r, the physical roughness of the surface. Otherwise, Δ
r _{d} is the observer's systematic error in matching surfaces in roughness across viewpoints: The systematic deviations from the identity line for each condition and observer in
Figure 9 are estimates of Δ
r _{d} for that observer and condition. Consequently, we can treat
Equation 12 as a regression equation,
where
$\Delta r \xaf d $
is the systematic error that the observer makes in a condition, taken from
Figure 9, and the remaining values are the estimates of the pseudocue values that we computed for each stimulus condition, plotted in
Figure 12. For each observer, we regressed all of the failures across conditions versus the pseudocue values for that condition to determine which pseudocues could account for the observer's systematic failures in roughness constancy across all comparisons.