The Ordered Manifold of Depth in Phenomenology

Metzger (1957) says that the phenomenological study of visual space is in accord with the naive realism of everyday life, though this claim should not be taken as an ontological commitment. It means that if the space of expe­rience is studied solely on phenomenological grounds, the experimental evi­dence will show that it is primarily a space of things, which are perceived in a visual field where height, width and depth are equally presented. This claim also has important implications for the interpretation of well-known mecha­nisms underlying depth perception, such as binocular disparity, whose func­tion is specifying the spatial features of appearances rather than recovering the third dimension from two-dimensional retinal images. For instance, Metzger (1966a) claims that if monocular vision is considered in this phenomenologi­cal sense it is shown in reality to provide fundamental clues for experiencing depth, but it has often been neglected when intended only as a mechanism to recover three-dimensional properties. Moreover, these clues form an ordered manifold that can be represented in table 2.

The clues and their efficacy are considered at the level of the observable con­nection and the change they induce on appearances. Therefore, the relation with the stimulation is taken into account as a model that approximates what the distribution of stimuli should be like given the phenomenal properties – hence as a mapping and not as a causal representation. The clues specify the spatial properties of figures and of qualities as a function of the connections and changes brought about by a state or “process,” the latter term denoting movements, rotations, deformations and qualitative alterations that the con­figurations, which are endowed with form, extension and localization, may undergo in visual space (1966b: 719). The clues regard the relations in space, like perceived distance, or the inherent properties of appearances that allow them to display a visual body, that is, a solid object. Thus, even though these clues enable the perceiver to extract from appearances a space that could be represented on a two-dimensional plane, this space has to be taken as the two­dimensional boundary determined by a third dimension – as Brentano would have it.

Intersection is one of these clues. It occurs when parts of at least two dis­tinctly bounded units are seen to lie in the same visual direction so that they coincide with each other.

In 24(a), the larger and the included smaller trapezium have two segments in common. Each common segment has a double phenomenal value in the sense that it appears either as the coincident upper or bottom edge of two distinct unitary figures. They are seen as boundaries of two distinct surfaces, like the two parallelograms in figure 24(b) were they to be moved in the direction of the ar­rows. This brings about the phenomenal stratification of the surfaces delimited by these boundaries so that figure 24(a) might also appear as a roof, as in 24(c).

Stratification is a distribution in depth of surfaces, which in most cases is not arbitrary but is ruled by the factors shown by Petter (1956). Though not arbitrary, the stratification due to phenomenal coincidence may not be sufficiently univocal to determine a unique ordering of in-depth distribution. Instead, the in-depth order is always determined for the intersection with oc­clusion that occurs in amodal completion, because the surface that appears in­complete lies behind as occluded by the complete surface. Intersection occurs also for visual objects that move toward one another in the field and overlap each other. An object can come in contact with another so that some of its parts interpenetrate the parts of the latter. An object can slide over another so that some of its parts occlude the parts of the latter. Intersection cannot spec­ify the distance between stratified surfaces in any of its variants, even though the iteration of intersection may provide a compelling appearance of relative distance like in the artwork in figure 25.

Size alteration regards the properties of the figure and in particular its dis­tance. The correlation between size and distance is well known. Elements with perceptual constant size that are distributed at different distances, so that both size and distance appear to increase steadily, provide strong textural gradients for their in-depth localization as well as for the orientation of the surface on which they lie. Besides, different distances determine different perceptual size of objects at rest. In addition, Metzger emphasizes the function of the alteration of perceptual size, which can determine the variable appearances of distance and depth. Consider the alteration in which one and the same ob­ject changes its size through various phases. Such a change usually appears as either an expansion (contraction) or a forward (backward) movement. How­ever, Calavrezo (1934) has shown that if the object size alteration consists of a sequence of appearances that occur in different positions, so that the center of the object is laterally displaced for each appearance, the phenomenal expan­sion and contraction are seen as different in-depth localizations of the object.

Distortion regards visual bodies. It conveys information on the relative in­depth localization of the parts of an object from patterns of lines and surface that would look distorted were they perceived as such on a plane rather than displaying at least one aspect of that object. Metzger remarks that distortion is not a character of the single appearance, because each appearance is just what it looks to be. Distortion is rather brought about in a series of appear­ances and forced upon subjects if the parts of appearances are compressed in a two-dimensional plane. Of course, in order to avoid distortion and be per­ceived as a non-distorted view of a three-dimensional object, the considered appearance should be one of the infinitely many projections that this object geometrically admits. To be sure, every two-dimensional projection of a three­dimensional object is geometrically equivalent to any other, provided that no essential part of the object is lost because of its complete occlusion. However, not every geometrically equivalent projection has the same phenomenologi­cal value taken as an aspect of the object. Kopfermann (1930) has shown that this phenomenological value depends on the magnitude of the distortion that would be perceived in the boundaries and surfaces of figures if they appeared separated and confined to a two-dimensional plane rather than related to one another as parts of a solid object. To find the conditions in which subjects per­ceive this relation, which he had supposed to be grounded on factors like Wert­heimer’s grouping mechanisms or Rubin’s boundary function, Kopfermann arranged several glass panes, on which distinct lines and figures were drawn, one behind the other into a box at a mutual distance of 2 cm. Subjects could observe the lines and the figures by looking at the panes, which were placed on the same line of sight, from a binocular standpoint. Figure 26 is an example of one among the tested conditions.

The panes A, B and C present respectively a slanted parallelogram, two oblique segments (down on the left and up on the right) along with two an­gular sectors on the remaining sides, and finally an irregular hexagon whose vertices may be visually coincident with the ends of the segments and the sec­tors. In cases like this, Kopfermann reported that such visual elements appear related as the edges and the facets of a cube being thus ordered in depth (small irregularities in the cube are intentionally left to let its elements be no­ticed). Metzger contends that this kind of result is due to the connection that pops up when the ordering in depth removes the distortions of edges and surfaces, which would occur were they perceived as projected on the two­dimensional plane.

This reasoning can be generalized to other systems of deformation. Metzger (1934, 1935) shows how a sequence of appearances in an orthographic projec­tion are perceived as the change of places of a solid object in depth rather than as deformations in a two-dimensional plane. Sets of rods are randomly placed on a disc that rotates at as near a distance as possible from a translucent screen, while light is projected on the rods from such a distant source that it casts the shadow of the rods on the screen without perspective cues to depth. Since the shadows are nearly orthographically projected, projection lines are parallel to one another as well as orthogonal to the screen plane. Hence, identi­cal objects at various distances project identical images on the plane. Subjects are asked to report what they see when looking at the screen, which appears as a bright quadrangular surface with dark lines or bands on it. If the disc per­forms a moderately fast rotation, all the shadows actually wander back and forth on the plane of the screen. If subjects get closer to it, they can see this movement for a slow rotation of the disc. If the movement of the disc becomes faster, subjects see the screen as a sort of window and the dark lines connected in a rigid spatial pattern so that the lines appear to be rotating as the edges of an irregular prism around a vertical axis behind the window.

Metzger (1966a: 579) states a general rule about the role of deformation in depth perception. When a spatial configuration composed of many connected parts rotates, the two-dimensional projection changes in many respects. Dis­tances, lengths, directions, angles, and curvature are altered, some elements go past and become coincident with one another, and in some circumstances even the intersection points of surfaces become shifted. The greater the num­ber of elements that would be independently deformed at the same time, the more compellingly they appear connected and endowed with in-depth spati- ality. The orthographic projection of rods on the rotating disc is a particular case of the rule. Instead of perceiving a variable, relative motion of rods, the movement appears to be uniform as soon as the rods no longer appear as inde­pendent shadows but rather as edges of connected surfaces.

This rule has an autonomous phenomenological nature. Metzger (1935: 199) reports that there can be perceived movements in depth although they do not correspond to the actual distribution of the rods. If two rods are obliquely and eccentrically placed on the disc, subjects see one shadow line wander­ing forward and back on the frontal-parallel plane and the other line circling around it along a wider path, whose radius is equal to the greatest actual dis­tance between the shadows. Subjects can also see other movements. The sec­ond line appears to swing behind the first as an appendix thereof, so that the form of the movement resembles the numeral “8.” The two lines sometimes appear to circle or oscillate around a common axis lying between them, which at the same time shifts forward and back along a straight path. The radius of this apparent rotation is equal to half the greatest distance between the two shadows. Metzger (1934, 1935) points out the minimal deformation factor that satisfies the rule that minimizes the simultaneous alteration of more than one element. The critical factor for depth to appear is the change of distance among shadow lines. If three rods are placed near to one another on a concentric arc of a circle and the disc is made to rotate forward and backward with a 120° angle, so that the shadows at the outer places on the screen do not meet, the dark lines still get out of the window plane and realize a movement that repro­duces the actual motion of the rods. Metzger concludes that the characteristic feature of depth appearance is that the distances between the dark lines are preserved through the perceived movement of the rigid body they delimit as edges, so that the alterations are limited to its relative position with respect to the surrounding field. The preservation of distance between the dark lines is perceptually meaningful because the lines no longer appear as dark figures on a bright ground, rather as surface boundaries consistent to the constantly delimited space. For this reason the change of distance between the outer dark lines and the window margins does not induce subjects to see the lines rotating around the margins. The margins are perceived as discontinuities of the surrounding field, so that the changes in their distance from the lines are per­ceived as a change in the orientation of the rigid body. Metzger (1934) suggests that this kind of deformation does not obey the same rule as the distortions that were studied by Kopfermann. In the latter case, the three-dimensionality is perceived because the depth ordering of points, sides and angles gives rise to regularly arranged parts of visual objects, rather than to two-dimensional configurations that show tangled intersection points and coincident sides as well as distorted lengths and angles. In the case of deformations, on the other hand, it can be observed that a highly irregular wire configuration does not pro­vide any clue for depth at rest, while it brings about a compelling appearance of depth by means of rotation. Moreover, regular configurations may yield a compelling depth appearance by rotation just through those projections that Kopfermann found to give only the least conspicuous depth appearance.

Source: Calì Carmelo (2017), Phenomenology of Perception: Theories and Experimental Evidence, Brill.