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A Differential Geometric Description of the Relationships among Perceptions.

We present a differential geometric method for measuring and characterizing the perceptions of an observer of a continuum of stimuli. Because the method is not based on a model of perceptual mechanisms, it can potentially be applied to a wide variety of observers and to many types of visual and auditory stimuli. The observer is asked to identify which small transformation of one stimulus is perceived to be equivalent to a small transformation of a second stimulus, differing from the first stimulus by a third small transformation. The observer's identification of a number of such transformations can be used to calculate an affine connection on the stimulus manifold. This quantity describes how the observer encodes an evolving stimulus as a perceived sequence of "reference" transformations. This type of encoding is a multidimensional generalization of Fechner's method and reduces to his psychophysical scale when the stimulus manifold is one dimensional and the reference transformation is chosen to be a just noticeable difference. The intrinsic aspects of the nature of the observer's perceptions can be characterized by the curvature and torsion tensors derived from the connection. The multidimensional analogues of psychophysical scale functions exist if and only if these quantities vanish. Differences between the affine connections of two observers characterize differences between their perceptions of the same evolving stimuli. The affine connections of two observers can also be used to map a stimulus perceived by one observer onto another stimulus, perceived in the same way by the other observer. Unlike multidimensional scaling techniques, this method does not assume that the observer has a sense of distance (a metric) or that he/she can otherwise compare stimulus pairs that are oriented along perceptually different directions in the manifold. The method was used to measure the affine connections of observers of a dot moving on different background graphics; e.g., a blank screen, a grid, or two converging lines similar to those used to create the Ponzo illusion. The results comprise quantitative measurements of the background graphic's influence on each observer's perceptions of straightness, parallelism, and distance. The measurements demonstrate differences between the perceptions of the two observers. Copyright 2000 Academic Press.