Projective invariance and the measurement of visual shape

Abstract

The problem of measurement in psychology is as old as experimental psychology. One narrow domain of psychology is privileged, in that it admits an absolute scale: a ratio scale of measurement with a point of origin. That domain is the assessment of shape or form through vision. Poincaré, Cassirer, and Gibson approached this topic by studying invariants, specifically projective invariants. In the study of vision, shapes should be measured in projective terms. As a convention or a strategy, measurement in projective terms may be conducted without hindrance. Many objections have been posed to the measurement of shape in projective terms; nearly all misrepresent basic geometric ideas. Projective invariants provide the right tools to evaluate goodness of shape constancy, and it provides these tools in its own right (in contrast to measures like distance and angle). Under conditions that allow shape constancy in human vision, sometimes stable estimates of projective invariants by observers do not match projective invariants at the eye. Some examples are given for projective invariants of coplanar conics, the computation of the measures is discussed, current objections to the approach are addressed, and consequences are drawn for the study of visual shape constancy.