World enough and time: The Erlangen program and perception

Determining whether two patterns of neural activity describe the same objec t is a general problem that serves as siren call to philosophers as well as psychologists and neuroscientists. In the accompanying article, Bedford ar gues that, despite many different guises, different instances of the object identity problem can be understood in the same abstract framework. Conside red are apparent motion and stereo matching among many others. The proposal is that there is a one to one relationship between the hierarchy of geomet ries in Felix Klein's Erlangen program and the hierarchy in the organizatio n of the brain's perceptual systems. The principal consequence of this idea is that the hierarchy of geometries provides an ordering on the likelihood of perceiving correspondence in two sensory patterns. This provides an emp irical prediction, but one with plenty of wiggle room as we shall see. Bedf ord's paper compels us to examine whether all sensory processes are isomorp hic in being describable in terms of common mathematical, information theor etic, computational, or biological principles. If so, how rich are these pr inciples in suggesting experiments, and is Klein's geometric hierarchy a se nsible candidate for understanding shape perception, object correspondence, identity, and binding? I shall focus on apparent motion and stereopsis bec ause, while stereo and motion are formally similar, much experimental work demonstrates that they are subject to very different limitations. I shall e xamine constraints on stereo correspondence, and explore an apparently para doxical aspect of stereo matching when viewed from the perspective of the K lein hierarchy. Since our visual system is willing to tolerate great differ ences between the first and second stimuli in apparent motion, it is diffic ult to assess the idea that a hierarchy of geometries imposes an ordering o n the likelihood of perceptual correspondence. To circumvent this difficult y, some experiments have employed a competitive motion paradigm in which a single object is succeeded by two objects and a choice of motion direction is required. However, some scrutiny suggests that the interpretation of the se experiments is not straightforward, and hence offers less than compellin g evidence with respect to the hierarchy of geometries. Finally, while it m ay be true that the purpose of the theorists is to polarize the experimenta lists, a good theory should be fecund in suggesting new questions as well a s being open to refutation. The value of the Erlangen program for understan ding perception is not yet clear on either count.