Perceiving geometry

In general, the complex discrepancies between the physical world and the perceptual spaces that define lightness, brightness, and color can be understood in terms of an empirical strategy that has evolved to circumvent the inverse problem. If this really is the way vision operates, the same scheme should also explain perceptual phenomena pertinent to all the other visual qualities we see, preferably in quantitative terms. One of the most important of these additional qualities is the perception of geometry, the way we see spatial intervals, angles, shapes, and distances. Obviously, perceiving these fundamental aspects of physical geometry in a manner that enables appropriate behavior is critical. And here again biological vision must deal with the direct unknowability of the world by means of light from the environment falling on the retina.

In some ways, understanding the inverse problem and its consequences in the context of geometry is easier than understanding the similar quandary in the domains of lightness, brightness, and color. For instance, it's easy to appreciate that objects of different sizes with different orientations and at different distances from observers can all produce the same retinal image (Figure 11.1). It was these facts about projective geometry that Berkeley had used as a basis for his arguments about vision in the early eighteenth century (see Figure 7.8).

The perception of geometry offers an abundance of perceptual weirdness. Numerous discrepancies between measurements made with rulers or protractors and the corresponding perceptions have been described, providing plenty of challenges for any explanation of this aspect of vision (Figure 11.2). If the circuitry of the visual brain is determined by responses to projections on the retina whose sources can't be known directly, these phenomena should be also accounted for by the frequency of occurrence of different geometrical configurations in retinal images generated by the geometry of the world.

Figure 11.1 The inverse problem as it pertains to geometry. The same projection on the retina can be generated by many physically different objects in three-dimensional space. This inherent uncertainty applies to the geometry of any retinal image. (Courtesy of Bill Wojtach)

Figure 11.1 The inverse problem as it pertains to geometry. The same projection on the retina can be generated by many physically different objects in three-dimensional space. This inherent uncertainty applies to the geometry of any retinal image. (Courtesy of Bill Wojtach)

Catherine Qing Howe, another superb product of the Chinese educational system, was the person who met this challenge head on. Qing had been a graduate student for about three years in the Department of Neurobiology at Duke when she came to see me one day in 2000 in obvious distress. She had begun a doctoral dissertation on ion channels with a molecular biologist in the department but had become increasingly disenchanted with her project, her mentor, and molecular biology as a way to pursue her interests. She indicated that she wanted to start over by joining my lab. Such shifts in direction were unusual, and switching from the molecular biology of ion channels to perception was about as radical a change as one could imagine in neuroscience. But she eventually convinced me that this was really what she wanted to do.

Figure 11.2 Discrepancies between measured geometry and what we see. A) Two parallel lines appear bowed when presented on a background of converging lines. B) Segments of a continuous line obscured by a bar appear to be vertically displaced. C) The same line appears longer when it is vertical than horizontal. D) A line terminated by arrowheads looks longer than the same line terminated by arrow tails. E) The same line appears longer when presented in the narrower region of the space between diverging lines. F) The shape and size of the surfaces with the same dimensions (see the key below) can look very different. (After Purves and Lotto, 2003)

At age 10, Qing had been chosen by the Chinese Academy of Sciences as one of 30 intellectually gifted children to receive an individualized curriculum in the Beijing Middle School system. She had entered Peking Union Medical College at age 15, the youngest student they had ever taken. There she had became increasingly interested in cognition and behavior, and had decided to pursue these topics in the context of psychiatry. Her idea (much as mine had been as a first-year medical student with similar intentions) was that the best way to understand these issues was through molecular pharmacology. During her last year in med school, she was elected to represent her class in an exchange program with the University of California San Francisco Medical School. The experience convinced her to immigrate to the United States, and after receiving her medical degree in 1997, she entered the neurobiology graduate program at Duke.

Qing quickly recognized that if we were going to explain the phenomena in Figure 11.2 in empirical terms, a first step would be to determine the frequency of occurrence of geometrical images projected onto the retina, much as we had begun to assess the frequency of luminance and spectral distributions in images generated by sources in the natural world. Happily, there was a relatively easy way to acquire this information. Laser range scanning is a technique routinely used to monitor the geometrical conformance of an architectural plan to the progress of the building under construction. The device commonly used provides accurate measurements of the distances from the origin of the scanner's laser beam to all the points (pixels) in a digitized scene. By setting the height of the scanner at the average eye level of human observers, we could evaluate the projected geometry of retinal images routinely generated by real-world objects. The summer after Qing joined the lab, we could be seen lugging this machine and its accessories around the Duke campus to scan a variety of scenes ( Figure 11.3).

A database of this sort has some serious limitations. The range of distances analyzed was restricted to a few hundred meters, and the scenes were acquired in a particular locale (the Duke campus) and in a particular season (summer). Moreover, when human observers look at the world, they don't do so in the systematic manner of a laser scanner: They fixate on objects and parts of objects that contain information particularly pertinent to subsequent behavior (such as other people, faces, and object edges). Nonetheless, the database provided a reasonable approximation of the prevalence of different two-dimensional geometries in images generated by 3-D sources in the world. To test the hypothesis that the geometry we see is determined by accumulated information about the frequency of occurrence of various geometrical projections, Qing sampled thousands of images in the database with templates configured in the same form as a stimulus pattern of interest (for example, the geometrical stimuli in Figure 11.2). By sampling a large data set pertinent to a particular stimulus, she could ask whether the way people actually see the geometry of the retinal image is determined empirically.

Figure 11.3 Determining the relationships between images and the physical geometry of objects in the world. A) A laser range scanner. A mirror inside the rotating head directs a laser beam over a scene; the signal reflected back from object surfaces is detected by a photodiode, which, in turn, produces an electrical signal. A quartz clock determines the tiny interval between the transmitted pulse and the signal from the returning beam; based on the speed of light, a microcomputer calculates the distance from the image plane of the scanner to each point on object surfaces. B) Ordinary digital images of a natural scene, and an outdoor scene that contains some human artifacts. C) The corresponding range images the scanner acquired. Color-coding indicates the distance from the image plane of the scanner to each point in the scenes. (From Howe and Purves, 2005. Copyright 2005 Springer Science+Business Media. With kind permission of Springer Science+Business Media.)

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To understand this approach, consider the perceived length of a line compared to its actual length in the retinal image. In human experience, the length of a line on the retina will have been generated by lines associated with objects in the real world that have many different actual lengths, at different distances from the observer, and in different orientations (see Figure 11.1). As a result, it would be of no use to perceive the length of the line in the retinal image as such, just as it would be of little or no use to see luminance or the distribution of spectral energy as such. To deal successfully with the geometry of objects in the world, it would make far more sense to generate perceived lengths empirically. The length seen would be determined by the frequency of occurrence of any particular length in the retinal image relative to all the projected lengths experienced by human observers in the same orientation. In keeping with the argument in Chapter 10, this relationship discovered by feedback from trial-and-error behavior would have shaped the perceptual space for line lengths. As a result, the lengths seen would always differ from the geometrical scale of lengths measured with a ruler ( Figure 11.4).

For instance, if in past human experience 25% of the lines in retinal stimuli generated by objects in the world have been shorter than or equal to the length of the stimulus line in question, the rank of that projected length on an empirical scale would be the 25th percentile. If the length of another line stimulus had a rank of, say, the 30th percentile, then the stimulus at the 25th percentile should appear shorter, to a degree determined by the two ranks. The consequence of this way of perceiving line lengths—or spatial intervals, generally—is routine discrepancies between measurements of lines with rulers and the subjective "metrics" that characterize perception. As with lightness, brightness, or color, seeing visual qualities according to their empirical ranks maintains the relative similarities and differences among physical objects that are pertinent to successful behavior despite the direct unknowability of geometry in the world. The strategy works so well that we imagine that the geometry we see represents the actual geometry of objects, leading to the erroneous idea that the demonstrations in Figure 11.2 are "illusions." In fact, they are signatures of the way the visual system contends with this aspect of the inverse problem.

Figure 11.4 An empirical scale based on the frequency of occurrence of line lengths in retinal images (red), compared to a scale of lengths measured with a ruler (black). Because any given line length (25 units in the example here) has a different rank on these two scales (dotted lines), if what we see is determined empirically, there will always be a discrepancy between the perceived length of a line and its measured length in the retinal stimulus or in the world. (After Howe and Purves, 2005. Copyright 2005 Springer Science+Business Media. With kind permission of Springer Science+Business Media.)

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