July 1962
Four Cases of the Perception of Size in Space
J. J. Gibson, Cornell University
(Preliminary draft, for criticisms)
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The explanation of size-perception seems to be in a theoretical muddle. I have argued that it must be different in the window-situation (e.g., the puppet-theater, the scale-model, the picture, and the typical laboratory setup) than it is in the terrestrial situation. It also must be different for the size of a single object and the sizes of all objects in a field of view. In the full-view terrestrial situation there exist stimulus gradients in an array, with the nose at one extreme and the horizon.i.horizon; at another. We shall treat the earth abstractly as an illuminated plane with a structure or texture which is regular , or at least stochastically so. Let us consider four cases of size perception in this general situation.
I. First note that the amount or quantity of optical texture that is intercepted (occluded) by an object resting flat on the ground does not change with any change in the distance of the object from the observer. Accordingly two objects of the same size at different distances will intercept the same amount of optical texture (although having different “visual angles” or “projected sizes”). The density of optical texture becomes infinite at the horizon of a textured surface and the angular projection of any object resting on the surface becomes vanishingly small as the object goes to a very great distance.
The following hypotheses are proposed. The size of an object on the ground is specified by the amount of optical texture occluded. The distance of that object is specified by the density of the optical texture occluded. For a surface of uniform physical texture, the amount of optical texture corresponding to a given object remains constant as the object recedes from O, while the density of the texture (amount per unit of solid angle) approaches infinity. If these variables can be registered, it should be theoretically possible to perceive an object as having constant size out to an indefinitely great distance. I believe that this is possible (Visual World, p. 186).
Note that the angular size of each optical texture element becomes infinitely small at the theoretical horizon of an environmental surface while the density of the texture elements becomes infinitely great. But the quantity of texture occluded by a given object on the surface is invariant at any distance. For example, a one-foot paving-stone will cover 144 one-inch mosaic tiles, and this holds equally for the corresponding perspectives of the stone and the tiles. (It does not have to be assumed that the texture elements must be counted, for the invariant ratio holds for the whole optic array corresponding to the surface of the earth. The pickup of the potential information we are talking about is another question).
II. Next consider the quantity of texture included in a given visual solid, angle, as distinguished from that occluded by a given object on the ground. I mean a fixed angular extent delimited by a contour in the visual field, such as a negative afterimage of color. A “form” of this sort includes increasingly more texture as it is projected on a physical background surface at increasingly greater distances. The amount of texture included becomes infinite when the given angular extent is at the optical horizon of the surface, just as density of optical texture approaches infinity at the horizon. Consequently the apparent size of this “form”, if it is taken to be on the surface, should become indefinitely great as it approaches the horizon, just as its apparent distance should become indefinitely great. The experimental facts of the apparent size of negative afterimages, Emmert’s Law, are consistent with this hypothesis, although the experiments seem to have been performed indoors only, and with relatively nearby backgrounds.
Note that in this theory a visual solid angle is as such, i.e., an “angular size” or a “retinal size”, is empty of stimulus information. It may yield a sensation of “extensity”, as the analytic introspectionists have claimed, but it determines neither the perception of the size nor the distance of an object unless it is combined with a textured optic array having gradients of stimulus information with respect to the layout of physical surfaces.
It may, however, more or less determine a sort of linkage of the “ideas” of size and distance. We can now evaluate the “size-distance invariance hypothesis”. It states (in Ittelson’s formulation) that a given retinal size determines a unique ratio of two subjective impressions, phenomenal size and phenomenal distance. The impressions are taken to be linked, psychologically. Note that the present hypothesis asserts a stimulus invariant (amount of texture occluded in a graded array) not a psychological invariant (Cf. Beck and Gibson). The proposed invariant is external, not internal. The hypothesis of reciprocal subjective impressions does seem to explain certain experimental results of judging apparent size and apparent distance of a presumptive object when only a figure is given in an empty field. It seems to apply to situations where only clues to distance (or size) of a single entity are available. But it is irrelevant if applied to natural terrain situations where (I suggest) stimulus information for both physical size and physical distance of an object is available. There do occur, without question, various kinds of “reduced” situations which are intermediate between these extremes, and for which assumptions, inferences, and even guesses will occur.
III. Next consider the problem of the apparent size of the moon. The moon, like an afterimage, always has a fixed angular extent. But it is never seen on the ground, as terrestrial objects are, in a context of texture gradients. It is always seen in the sky. It never occludes part of a graded distribution of optical texture and hence its absolute phenomenal size and its absolute phenomenal distance should both be more or less vague and indefinite, if they can only be specified, respectively, by amount and density of occluded optical texture. Something can be predicted, however, about the relative phenomenal size of the moon at the horizon and the zenith – the “moon illusion”. A fixed angular extent intercepted by the horizon itself, as in a rising or setting moon, yields the information for an immensely large object and the more it becomes a vacant and meaningless sensation of angular extensity, that is, an object of indefinite size. It might therefore be predicted that the horizon moon would appear larger than the zenith moon or that, of two artificial optical moons superposed on a natural optic array, that near the horizon will appear to be the larger (Kaufman and Rock).
There is also the possibility that the zenith moon is taken to be “on” a sort of presumptive ceiling (or a flattened dome, if one prefers) and that hence it is closer than the horizon moon, and therefore smaller by the hypothesis of the reciprocity of the ideas of size and distance (e.g., Gruber). However, the apparent distance of the zenith moon is less than that of the horizon moon only to many observers, not to all (e.g., Boring). Its phenomenal distance, in truth is undetermined. When it appears farther and smaller to some Os, the reciprocity hypothesis fails.
IV. Finally, consider the problem of the perception of the size and distance of an object not resting on the ground, and not seen against the sky, but invisibly supported at a short distance above the ground (Perception of the Visual World, p. 178). We must now take into account the gradients which arise because of parallax, i.e., because of two eyes and the observer’s movement. With no binocular or motion parallax, such an object is seen at a greater distance and with a greater size than it actually possesses. Its distance is given by the density of the texture where the ground is occluded, and its size by the amount of texture occluded, as here predicted (unpublished data). The phenomenal object has receded within its cone of rays, as it were, to make optical contact with the ground. When binocular or motion parallax is permitted in this situation, however, the phenomenon of depth-at-an-edge appears and the object appears to come forward and upward, decreasing in size as it does so. The discontinuities of texture-disparity and texture-motion at the contour, together with the continuous gradients of disparity and motion for the ground surrounding it, now yield information as to the air-distance between object and ground together with the ground-distance between the observer and the background-locus. The phenomenon of the air-distance between surfaces, or depth-at-an-edge, has been studied in the special situation of an “optical cliff”, or drop-off of the ground itself (E. Gibson and Walk).
The apparent size of the object in the second situation described is not given by the amount of optical texture occluded. How accurately the size is perceived is not known, and research is called for. It may be that the hypothesis of a psychological linkage between phenomenal size and phenomenal distance applies in this case. But it should be remembered that an object invisibly supported above the ground is an atypical case for object perception. Processes of associative learning and inference can be expected to occur in such cases.
I have previously suggested that the size of an object on the ground is given by the “scale” of the optical ground at that point (Visual World, p. 181). Later, my formula was that the size of such an object is given by the “angular size of its projection relative to the angular size of the elements of texture in the adjacent optic array” (Chapter in Koch, Vol. 1, p. 479). This ratio is essentially the same as the quantity of optical texture occluded by the optical form of the object – the formula as proposed. It is not a natural number, but a relational number, or what is sometimes called a “scaled-coefficient.” It seems to be analogous to a “dimensionless number.” I believe that an eye is sensitive to stimulus variables of this sort.
Conclusions: (1) In certain stimulus arrays the size of an object is specified; in others it is not. When it is, size is perceived correctly. (2) When it is not, the size-distance reciprocity hypothesis will often explain the observations, but sometimes will not. (3) In any case, “retinal size” does not seem to be the sensory basis of size-perception.