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Categorization and Object Shape

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Towards a Theory of Thinking

Part of the book series: On Thinking ((ONTHINKING))

Abstract

Categorization is essential for perception and provides an important foundation for higher cognitive functions. In this review, I focus on perceptual aspects of categorization, especially related to object shape. In order to visually categorize an object, the visual system has to solve two basic problems. The first one is how to recognize objects after spatial transformations like rotations and size-scalings. The second problem is how to categorize objects with different shapes as members of the same category. I review the literature related to these two problems against the background of the hierarchy of transformation groups specified in Felix Klein’s Erlanger Programm. The Erlanger Programm provides a general framework for the understanding of object shape, and may allow integrating object recognition and categorization literatures.

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Notes

  1. 1.

    In addition, the basic level is the most inclusive level at which we tend to interact with objects in a similar way (Rosch et al. 1976), indicating the importance of knowledge about motor interactions for categorization (see Helbig et al. 2006).

  2. 2.

    In mathematics, a group is a set that has rules for combining any pair of elements in the set, and that obeys four properties: closure, associativity, existence of an identity element and an inverse element. The mathematical concept of group has been used in cognitive psychology (e.g., Bedford 2001; Chen 2005; Dodwell 1983; Leyton 1992; Palmer 1983, 1989; Shepard 1994).

  3. 3.

    The Euclidean similarity group can be further subdivided (e.g., Bedford 2001), but this is not important for present purposes. A more detailed description of the hierarchy of transformation groups can be found in Michaels and Carello (1981, p. 30-37), Palmer (1983), or Cutting (1986).

  4. 4.

    There are also transformations which go beyond point transformations (see Ihmig 1997). However, these do not seem to be of primary importance for an understanding of object shape.

  5. 5.

    Nonlinear transformations seem to play a role also within the visual system. The retina is not flat but curved, and projections onto the retina are therefore distorted in a nonlinear way. Moreover, the projection from the retina to the primary visual cortex is highly nonlinear.

  6. 6.

    Note that invariants need not necessarily be defined in relation to mathematical groups; invariants can be defined also regarding perspective transformations (e.g., Pizlo 1994), which do not fulfill the requirements of a mathematical group.

  7. 7.

    However, Edelman (1998) and Cutzu and Edelman (1996, 1998) do not interpret these results in terms of nonrigid transformations of pictorial representations, but regard the data as evidence for the existence of a low-dimensional monotonic psychological space, in which the similarity relations of the high-dimensional distal shape-space are represented. For a discussion of Edelman’s account of recognition and categorization see Graf (2002).

  8. 8.

    In a variation of this approach, Shepard (1994) claimed that the linearity of transformation time (in mental rotation tasks and apparent motion tasks) is an invariant.

  9. 9.

    Not only the extent but also the type of topological distortion might be relevant (in analogy to the affine transformation group, cp. Wagemans et al. 1996). However, possible influences of the type of topological transformations are not in the focus of this work (for a discussion of the types of transformation see Bedford 2001).

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Author Note

The work has been supported by a grant from the Max Planck Institute for Biological Cybernetics, Tübingen, and from the Max Planck Institute for Human and Cognitive Brain Sciences, Leipzig. I thank Christoph Dahl for substantial help in creating the 3D morph objects, some of which are depicted in Fig. 6.

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  1. Department of Psychology, Max Planck Institute for Human Cognitive and Brain Sciences, Stephanstraße 1a, 04103, Leipzig, Germany

    Markus Graf

  2. Max Planck Institute for Biological Cybernetics, Tübingen, Germany

    Markus Graf

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  1. Parmenides Foundation, Kardinal Faulhaber Str. 14 a, Munich, 80333, Germany

    Britt Glatzeder

  2. Dept. Psychology, York University, Keele Street 4700, Toronto, M3J 1P3, Canada

    Vinod Goel

  3. Parmenides Foundation, Kardinal Faulhaber Str. 14 a, Munich, 80333, Germany

    Albrecht Müller

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Graf, M. (2010). Categorization and Object Shape. In: Glatzeder, B., Goel, V., Müller, A. (eds) Towards a Theory of Thinking. On Thinking. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03129-8_6

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