One of the central tasks of the human visual system is to derive shape information about the environmental object from the optical array that is available to the eye. Psychologists have made many attempts to understand how this task is solved, including purely phenomenological analyses (e.g. Gestalt theory), direct approaches, and indirect approaches (e.g. the theory of unconscious inference proposed by Helmholtz (1894) and the computational theory by Marr (1982)). This paper investigates the hypothesis that the visual system performs a topological analysis of the image it encounters. The notion of a topological analysis cuts across the above mentioned approaches and, if found valid, it has the potential to reconcile computational and phenomenological frameworks. Before introducing the experiments, we shall attempt to position the concept of a topological perceptual analysis within the bewildering variety of current theories relating to perceptual organization. We shall also summarize the scant existing studies that suggest a topological function for the visual system.

Any given pattern can be described in an indefinite number of ways; in terms of its contrast or spatial contrast distribution, in terms of the elements that make up the pattern, or with focus on structural aspects such as symmetry or goodness. Following Palmer (1978) and Palmer (1982), these descriptions can be grouped into analytic descriptions, which are based on the constituent features or elements of a pattern, and into structural descriptions that refer to global aspects of the pattern.

The analytic level: Analytic theories attempt to explain pattern perception in a reductionist fashion. They adhere to the underlying assumption that patterns of arbitrary degrees of complexity can be understood in terms of their elements, including their description and location within a given spatial framework. Consequently, proponents of such analytic theories propose operators that work more or less exclusively on a local level. They focus on simple local properties of visual patterns, such as contour changes or other perceptual primitives, guided by the hope that a more or less hierarchical order of elements of increasing complexity can be found to explain perceptual grouping in a bottom-up manner. A search for abstract structural properties is considered to be computationally too expensive. In this framework, the litmus test for a potential perceptual primitive are short reaction times in search paradigms that suggest rapid, parallel, and automatic or pre-attentive processing. Many perceptual primitives, such as line terminations or T-junctions have thus been identified. With the help of local operators or feature detectors, a large number of perceptual phenomena can be explained, as has for example been demonstrated by Julesz (1981) within the framework of texton theory. It assumes a variety of different feature detectors that identify basic representational elements or textons, such as line terminators. However, texture segregation is often not compatible with texton theory but rather based on other features such as local inhomogeneities.

The analytic level of analysis is usually adopted whenever percepts are analyzed in a bottom-up fashion. However, after the promising start provided by Marr (1982) a bottom-up theory of perceptual organization is not only still lacking, but the quest for perceptual primitives may not be able to solve the problem of perceptual organization. There is also evidence that the visual system performs global analyses previously deemed computationally too complex. Enns and Rensink (1991) and Rensink and Enns (1995) found that rapid and parallel visual search can directly access global aspects of patterns that are more sophisticated than previously thought.
The structural level: In contrast, holistic theories share the notion that representations of patterns cannot be achieved solely by feature detectors, but require some higher-order analyzers. These analyzers process relational properties between the elements of a given pattern. The closedness of a pattern (e.g. three line segments forming a triangle) is an example of a structural aspect or emergent property which is not equivalent to location and orientation information about the line segments that make up the pattern. The analytic description has to be supplemented with relational properties. Gestalt theorists considered global aspects of the stimulus to be fundamental for the percept and attempted to relate perceptual organization to brain processes. However, even when not rooted in physiological evidence, Gestalt laws, such as proximity, closure, or symmetry, can be considered to constitute grouping principles used by the visual system to organize the visual field has elaborated the notion of a symmetry analyzer based on the Gestalt notion of symmetry as a grouping factor. Many structural aspects of symmetry have been documented and, such as its relative power compared to other grouping factors, its pervasiveness, and the predominance of vertical over oblique symmetry axes. However, it is very difficult to combine the findings into one theory of symmetry. Since symmetry appears to be a prominent structural analyzer that can be varied orthogonally to topology, it will be considered and compared to topology in Experiments 1 and 2.