ABSTRACT
Consider a situation in which, say, elliptically shaped visual stimuli continuously vary in the lengths of their radii, a and b, all other parameters being held fixed. This is a simple example of a two-dimensional continuous stimulus space: Each stimulus can be described by two coordinates, (x1, x2), taking their values within a region of Re×Re.1 The dimensions can be chosen in an infinity of ways. One can put x1=a, x2=b, or x1=a:b, x2=ab (aspect ratio and size), or one can even choose dimensions for which one has no conventional geometric terms, say, x1=exp(2a+3b), x2=log(3a+2b). The number of dimensions, in this case two, is a topological invariant (i.e., it is constant under all-continuous one-to-one transformations of the space), but the choice of the dimensions is arbitrary: With any given choice of one obtains other representations by arbitrarily transforming these dimensions into , provided the transformations are one-to-one and smooth. If one imposes a certain
“subjective” (computed from perceptual judgments) metric on the stimulus space, this metric must be invariant with respect to the choice of stimulus dimensions. In multidimensional Fechnerian scaling (MDFS), on which this work is based, this invariance is achieved automatically by the procedure of computing Fechnerian distances.
