6. Testing the model: The staircase Gelb effect
Important aspects of the proposed model are illustrated by a series of experiments conducted by Cataliotti and Gilchrist (1995; Gilchrist & Cataliotti, 1994), using a display we will call the staircase Gelb effect. A contiguous series of five squares spanning the gray scale from black to white, in roughly equal steps, was suspended in midair in a laboratory room and brightly illuminated by a homogeneous rectangular patch of light projected by an ellipsoidal theatrical spotlight mounted on the ceiling at a distance of 2.8 M. from the squares. The observer viewed this display with no restrictions from a distance of 4 M and indicated the lightness of each square by selecting a matching chip from a brightly illuminated Munsell chart housed in a rectangular chamber that rested on a table immediately in front of the observer. The entire lab room was normally illuminated by fluorescent lighting but the illumination on the five squares was thirty times brighter than the ambient level.
The results are shown in Figure 8. The striking aspect in the results is the dramatic compression in the range of perceived grays. Even though the physical stimulus contains the entire gamut of gray shades from black to white, observers perceive only a range of grays from light middle gray to white.
6.1. Applying the anchoring model
This result makes sense if we apply our anchoring model. We can treat the stimulus squares as members of two frameworks, one local and one global. The five squares form a local group based on their proximity, their coplanarity, and most likely their similar luminance values. In addition each is part of the global framework that includes the entire visual field. The application of the model is shown schematically in Figure 9. The diagonal line (which will be called the L-line) shows the lightness values computed solely within the local framework, using formula (1) from Section 4.5. The horizontal line (which will be called the G-line) shows the lightness values of the target squares in the global framework. They reflect the fact that, without the local group of squares, each square by itself would appear white in the global framework.
Notice that the obtained value for each square lies in between its value in the local framework and its value in the global framework. This compromise lies at the heart of our new theory. In this particular case the compromise is roughly 30% local and 70% global, but we propose that in various other situations the balance of the compromise might shift in favor of either the local framework or the global framework, depending upon the relative strength of these two frameworks.
6.3. Weighting factors in the staircase Gelb effect
Gilchrist and Cataliotti (1994) conducted a series of variations on the staircase Gelb experiment to test the anchoring model by varying a series of factors that might alter the weighting of the local framework. Two of these factors, articulation and field size, were suggested by the early lightness perception literature, especially in the work of David Katz (1935). Two other factors, that we call configuration and insulation, were uncovered in the course of our investigation.
Apparently the squares constitute a stronger framework if they are arranged in a Mondrian pattern than if they are simply arranged in a line. This can be seen in Figure 10. Notice that both in the case of five squares and in the case of ten squares the obtained data fall closer to the L-line than to the G-line, suggesting that the local framework is stronger with a Mondrian configuration. Several additional experiments are needed to further sort out whether the Mondrian configuration produces better constancy than the linear configuration because of the greater number of adjacent ratios, because of the scrambling of the luminance staircase, or because of some other factor.
Three Mondrians, containing 2, 5, and 10 target surfaces respectively, were presented under the same basic conditions. The results, shown in Figure 11, make it clear that the strength of the local framework depends strongly on the number of squares in the group, even when the luminance range within the group is held constant. Our data indicate that the crucial factor is number of different surfaces, not number of different gray levels, but this needs to be confirmed under a wider range of conditions. We call this factor articulation, defined simply as the number of different surfaces in a framework, in deference to Katz (1935). Perhaps this usage fails to capture the richness of Katz's concept of articulation, but our goal is to be more operational than Katz while using his term in order to recognize his contribution.
Katz argued that the greater the articulation within an illumination frame of reference, the higher the degree of lightness constancy. We are modifying Katz's usage of this concept somewhat. Our proposal is that the higher the degree of articulation in a target's framework, the more the lightness of the target is anchored solely within that framework.
According to Katz (1935), the degree of lightness constancy within a given field of illumination depends on the size of the field. But as Rock (1975, 1983; Rock & Brosgole, 1964) has shown so often for various factors in perception, size can be defined in either retinal terms or in phenomenal terms. Katz believed that both of these meanings of size are effective in lightness constancy and hence he offered his Laws of Field Size. The first law holds that the degree of constancy varies with the retinal size of a field. He supported it with the observation that if one looks though a neutral density filter held at arms length and then slowly brings the filter toward the eye, the degree of lightness constancy for surfaces seen within the boundaries of the filter increases as the filter comes to occupy a larger proportion of the visual field.
The second law holds that constancy varies with perceived size. This he demonstrates by keeping the filter at arms length while he slowly walks backward away from a wall containing various surfaces. As the perceived size of the region of wall seen through the filter increases, so does constancy, even though retinal size is now held constant.
Although his second demonstration establishes the effectiveness of perceived size with retinal size held constant, his first demonstration does not establish retinal size with perceived size held constant. When the filter is drawn closer to the observer's eye, the total area of the surfaces seen through the filter grows both in retinal and perceived size.
In several experiments we have found perceived size to be effective but not retinal size (when perceived size is controlled). We repeated the experiment with the five square Mondrian, but used a Mondrian five times larger both in width and height. Results from the small Mondrian and the large Mondrian are shown in Figure 12 (Gilchrist & Cataliotti, 1994). The increase in size produced a significant darkening for only the black square, but there appears to be a trend for the other squares as well. We obtained little or no difference when we increased size simply by moving the observer closer to the Mondrian, which increased the net size of the display with little or no change in its perceived size. Of course, if the observer were moved so close that the five squares display fills the entire visual field, this would constitute a qualitatively different stimulus and perceived lightness values would change substantially.
When Bonato and Gilchrist (1994) found higher luminosity thresholds for targets of larger area, their targets were larger both in visual angle as well as perceived size. In a follow-up experiment (Bonato & Gilchrist, in press) they tested luminosity thresholds for the disk in a disk/annulus display. They varied, in a controlled way, both the retinal size of the display and its perceived size, finding the luminosity threshold to vary with perceived size but not retinal size.
Bonato and Cataliotti (submitted) showed the importance of phenomenal size in a different way. They found a higher luminosity threshold for a region perceived as ground than for a region perceived as figure. Although the area of the two regions was the same in the display (shown in Figure 13), the phenomenal area of the ground region is greater because it is perceived to extend behind the figural (face) region. This hidden part of the background can be called its amodal area (Kanizsa, 1979). A quantitative analysis of the Bonato and Cataliotti results indicates that the functional area of the ground region (as this bears on the luminosity threshold) includes only some of the area behind the figure, not all of it. This finding is consistent with those of Shimojo & Nakayama (1990), who used an apparent motion display to determine how far a ground region is perceived (functionally) to extend behind a figure/ground contour.
Gilchrist and Cataliotti (1994) discovered that a white border surrounding the group of squares seems to insulate it from the influence of the global framework. A border of black paper has no such effect, as can be seen in Figure 14. Thus, when a region of maximum luminance completely surrounds a group of specially lighted surfaces, it seems to dramatically reduce the belongingness between the group and the remainder of the visual field. The same insulation effect applies to the single square used in the basic Gelb effect as well. Cataliotti and Gilchrist (1995) found that when a white square is placed next to the black Gelb square, the darkening of the black square is less than half what would be commensurate with the luminance ratio between them. But both they and McCann and Savoy (1991) found that when the white region completely surrounds the Gelb square, the darkening effect is highly commensurate with the luminance ratio between them.
At this point this insulation effect remains little more than an empirical result; we have no deeper explanation for it. But the effect is apparently not reducible to local contrast. When the five squares with a white border were compared to a window panes arrangement in which contact between each of the darker targets and white was maximized, and a concentric arrangement in which the contact was minimized, no differences were found, as can be seen in Figure 15. the key requirement seems to be merely that the inner framework is completely enclosed by a white border.
