To address these issues, we asked whether search for pairs of hierarchical stimuli can be explained in terms of shape differences and interactions at the global and local levels. To build a quantitative model, we drew upon our previous studies in which the dissimilarity between objects differing in multiple features was found to be accurately explained as a linear sum of part-part dissimilarities (
Pramod & Arun, 2014; Pramod & Arun,
2016;
Sunder & Arun, 2016). Consider a hierarchical stimulus AB, where A represents the global shape and B is the local shape. Then, according to the model (which we dub the multiscale part sum model), the dissimilarity between two hierarchical stimuli AB and CD can be written as a sum of all possible pairwise dissimilarities among the parts A, B, C, and D as follows (
Figure 5A):
\begin{eqnarray*}{\rm{d}}\left( {{\rm{AB,CD}}} \right) &=& {{\rm{G}}_{{\rm{AC}}}} + {{\rm{L}}_{{\rm{BD}}}} + {{\rm{X}}_{{\rm{AD}}}} + {{\rm{X}}_{{\rm{BC}}}}\\
&& + \,{{\rm{W}}_{{\rm{AB}}}} + {{\rm{W}}_{{\rm{CD}}}} + {\rm{constant}}\end{eqnarray*}
where G
AC is the dissimilarity between the global shapes, L
BD is the dissimilarity between the local shapes, X
AD and X
BC are the across-object dissimilarities between the global shape of one stimulus and the local shape of the other, and W
AB and W
CD are the dissimilarities between global and local shape within each object. Because there are seven possible global shapes, there are
7C
2 = 21 pairwise global-global dissimilarities corresponding to G
AB, G
AC, G
AD, etc., and likewise for L, X, and W terms. Thus, in all, the model has 21 part-part relations times 4 types + 1 constant = 85 free parameters. Importantly, the multiscale part sum model allows for completely independent shape representations at the global level, local level, and even for comparisons across objects and within object. The model works because the same global part dissimilarity G
AC can occur in many shapes where the same pair of global shapes A and C are paired with various other local shapes.