The topological approach to perceptual organization

Abstract

To address the fundamental question of “what are the primitives of visual perception”, a theory of topological structure and functional hierarchy in visual perception has been proposed. This holds that the global nature of perceptual organization can be described in terms of topological invariants, global topological perception is prior to the perception of other featural properties, and the primitives of visual form perception are geometric invariants at different levels of structural stability. In Part I of this paper, I will illustrate why and how the topological approach to perceptual organization has been advanced. In Part II, I will provide empirical evidence supporting the early topological perception, while answering some commonly considered counteraccounts. In Part III, to complete the theory, I will apply the mathematics of tolerance spaces to describe global properties in discrete sets. In Part IV, I will further present experimental data to demonstrate the global-to-local functional hierarchy in form perception, which is stratified with respect to structural stability defined by Klein's Erlangen Program. Finally, in Part V, I will discuss relations of the global-to-local topological model to other theories: The topological approach reformulates both classical Gestalt holism and Gibson's direct perception of invariance, while providing a challenge to computational approaches to vision based on the local-to-global assumption.