Topology Underlying GIBSON'S Echological Theory of Perception
During the exploration of the surrounding environment, the brain links together external inputs in a complex of sensations, giving rise to the perception of a persisting object. During imaginative processes, the same object can be recalled in mind even if it is out of sight. Here, Borsuk's theory of shape and the Borsuk Ulam theorem provide a mathematical foundation for Gibson's notion of persistence perception. Real scene visual signals are collectively the umbra of physical shapes impinging on the optic nerves, which we map to similar shape representations. We show how Gibson's ecological theory of perception accounts for our knowledge of world objects by borrowing a concept of invariance in topology. A series of transformations can be endlessly and gradually applied to a pattern, in particular to the shape of an object, without affecting its invariant properties, such as connectedness and boundedness of parts of a visual scene. In sum, high level representations of objects in our environment are mapped to simplified views (our interpretations) of the objects, in order to construct a symbolic representation of the environment. The fact that our perception of an object continues even when it is out of sight, can be explained by viewing regions on a sphere surface as multiple representations of object shapes. The representations can be projected continuously to an ecological object that we have seen and continue to see, thanks to the mapping from shapes in our memory to shapes in Euclidean space.