David Corfield
invariants

Idea

Cassirer in The concept of group and the theory of perception (1944)

If perception is to be compared to an apparatus at all, the latter must be such as to be capable of “grasping intrinsic necessities.” Such intrinsic necessities are encountered everywhere. It is only with reference to such “intrinsic necessity” that the “transformation” to which we subject a given form is well defined, inasmuch as the transformation is not arbitrary and executed at random but proceeds in accordance with some rule that can be formulated in general terms. (p. 26) (“grasping intrinsic necessities” due to Max Wertheimer)

Goes on to say in geometry this involves group invariance.

The concept of factors of constancy in the face of variation of both external and internal conditions of perception is the realization, in modern form, of that which in principle…was known to Kant, the analyst, and which he stated in terms of mediating schemata. (Buehler)

Area of a Euclidean triangle.

[3, 2]E(2)[3, \mathbb{R}^2] \sslash E(2) \to \mathbb{R}

BE(2)[3, 2]B E(2) \vdash [3, \mathbb{R}^2] \to \mathbb{R}

Co-shape in

Triangle is shape in 2\mathbb{R}^2; area is co-shape in [3, 2][3, \mathbb{R}^2]

it remained for Gibson to adopt the radical hypothesis of what he called the ecological approach to perception (Gibson, 1961, 1979), namely, the hypothesis that under normal conditions, invariants sufficient to specify all significant objects and events in the organism’s environment, including the dispositions and motions of those objects and of the organism itself relative to the continuous ground, can be directly picked up or extracted from the flux of information available in its sensory arrays.

In the case of the modality that most attracted Gibson’s attention—vision—the invariants generally are not simple, first-order psychophysical variables such as direction, brightness, spatial frequency, wavelength, or duration. Rather, the invariants are what J. Gibson (1966) called the higher order features of the ambient optic array. (See J. Gibson, 1950, 1966, 1979; Hay, 1966; Lee, 1974; Sedgwick, 1980.) Examples include (a) the invariant of radial expansion of a portion of the visual field, looming, which specifies the approach of an object from a particular direction, and (b) the projective cross ratios of lower order variables mentioned by J. Gibson (1950, p. 153) and by Johansson, von Hofsten, and Jansson (1980, p. 31) and investigated particularly by Cutting (1982), which specify the structure of a spatial layout regardless of the observer’s station point.

For invariants that are significant for a particular organism or species, Gibson coined the term affordances (J. Gibson, 1977). Thus, the ground’s invariant of level solidity affords walking on for humans, whereas its invariant of friability affords burrowing into for moles and worms. And the same object (e.g., a wool slipper) may primarily afford warmth of foot for a person, gum stimulation for a teething puppy, and nourishment for a larval moth. The invariants of shape so crucial for the person are there in all three cases but are less critical for the dog and wholly irrelevant for the moth. (Shepard 1984, p. 418)

There are good reasons why the automatic operations of the perceptual system should be guided more by general principles of kinematic geometry than by specific principles governing the different probable behaviors of particular objects. Chasles’s theorem constrains the motion of each semirigid part of a body, during each moment of time, to a simple, six-degrees-of-freedom twisting motion, including the limiting cases of pure rotations or translations. By contrast, the more protracted motions of particular objects (a falling leaf, floating stick, diving bird, or pouncing cat) have vastly more degrees of freedom that respond quite differently to many unknowable factors (breezes, currents, memories, or intentions). Moreover, relative to a rapidly moving observer, the spatial transformations of even nonrigid, insubstantial, or transient objects (snakes, bushes, waves, clouds, or wisps of smoke) behave like the transformations of rigid objects (Shepard & Cooper, 1982).

It is not surprising then that the automatic perceptual impletion that is revealed in apparent motion does not attempt either the impossible prediction or the arbitrary selection of one natural motion out of the many appropriate to the particular object. Rather, it simply instantiates the continuing existence of the object by means of the unique, simplest rigid motion that will carry the one view into the other, and it does so in a way that is compatible with a movement either of the observer or of the object observed.(Shepard 1984, p. 426)

Putting the considerations concerning preference for the simplest transformation that preserves rigid structure together with those concerning the conducive conditions for impletion of such a transformation, I have posited a hierarchy of structural invariance (Shepard, 1981b). At the top of the hierarchy are those transformations that preserve rigid structure but that require greater time for their impletion. As the perceptual system is given less time (by decreasing the SOA [stimulus onset asynchrony]), the system will continue to identify the two views and hence to maintain object conservation, but only by accepting weaker criteria for object identity. Shorter paths that short-circuit the helical trajectory will then be traversed, giving rise to increasing degrees of experienced nonrigidity (Farrell & Shepard, 1981). Likewise, if the two alternately presented views are incompatible with a rigid transformation in three-dimensional space, the two views will still be interpreted as a persisting object, but again a nonrigid one. (Shepard 1984, p. 430)

Instead of saying that an organism picks up the invariant affordances that are wholly present in the sensory arrays, I propose that as a result of biological evolution and individual learning, the organism is, at any given moment, tuned to resonate to the incoming patterns that correspond to the invariants that are significant for it (Shepard, 1981b). (Shepard 1984, p. 433)

…although J. Gibson (1970) held that perceiving is an entirely different kind of activity from thinking, imagining, dreaming, or hallucinating, I like to caricature perception as externally guided hallucination, and dreaming and hallucination as internally simulated perception. Imagery and some forms of thinking could also be described as internally simulated perceptions, but at more abstract levels of simulation. (Shepard 1984, p. 436)

Foreshadowing the commutative diagram that I much later proposed (Shepard, 1981b, p. 294), Heinrich Hertz succinctly stated that “the consequents of the images must be the images of the consequents” (Hertz, 1894/1956, p. 2). (Shepard 1984, p. 441)

References

  • Shepard, R. N. (1981b). Psychophysical complementarity. In M. Kubovy & J. R. Pomerantz (Eds.), Perceptual organization (pp. 279-341). Hillsdale, NJ: Erlbaum.
  • Shepard, Roger N., 1984 ‘Ecological constraints on internal representation: resonant kinematics of perceiving, imagining, thinking, and dreaming’, Psychological Review 91:417-47 (pdf)
  • R. Shepard, Perceptual– cognitive universals as reflections of the world. Psychonomic Bulletin & Review, 1, 2–28. 1994
  • Eloise H. Carlton, Roger N. Shepard, Psychologically simple motions as geodesic paths I. Asymmetric objects, Journal of Mathematical Psychology Volume 34, Issue 2, June 1990, Pages 127–188.

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