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Lightness/Brightness

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Simultaneous brightness contrast

Mach bands

Chubb illusion

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An Empirical Explanation: Mach Bands

As with simultaneous brightness and Cornsweet effects, Mach bands can be rationalized in terms of the genesis of visual percepts according to a strategy in which percepts are elicited as reflexes whose network connectivity has been wholly determined by the history of human visual experience.
A particularly challenging problem for this way of explaining the anomalous relationship between luminance and brightness is Mach bands - the name given to the light and dark zones seen at the onset and offset, respectively, of luminance gradients that lack any photometric basis for this effect (Figure 1). It is at first difficult to imagine what empirical (or historical) facts about human interactions with the sources of luminance gradients could explain this gratuitous addition of light and dark bands to the percepts elicited by these stimuli.
Figure 1
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Figure 1 / Mach bands. A) Diagram of the painted disk used by Mach to elicit this effect. When the disk is spun, a luminance gradient is established between the uniformly lighter center of the disk and the uniformly darker region at its periphery. B) Blowup of a portion of the spinning stimulus in (A), indicating the nature and position of Mach bands (the curvature has been removed for simplicity of presentation). As can be seen in response to viewing this stimulus, a band of maximum lightness is apparent at position (2), and a band of maximum darkness at position (3), neither of which are present in the photometric measurements shown in (C). C) Because the portion of the black sector between points (2) and (3) in (A) is a segment of an Archimedean spiral, the luminance gradient generated between the corresponding points on the spinning disk is linear, as indicated by photometric measurement along the line in (B). D) A similar graph of the relative lightness/brightness seen by observers, indicating the illusory lightness maximum just before the initiation of the linear gradient (2), and the illusory minimum just after its termination (3). (After Lotto et al., 1999a)

Nevertheless, there is an entirely empirical explanation of Mach bands that closely parallels the accounts of the simultaneous brightness contrast and Cornsweet effects (see Demonstrations) (Lotto et al., 1999a,b). By interacting with the objects that give rise to luminance gradients, observers will have experienced that the underlying source of such stimuli is sometimes differences in the reflectance properties of flat surfaces (as in the Mach stimulus), sometimes penumbras, and sometimes differences in the illumination of curved surfaces (among other less frequent possibilities not considered here).
An important aspect of experience derived from interacting with curved surfaces is that the luminance gradients associated with such surfaces are frequently adorned with photometric highlights and lowlights at the beginning and end of the gradient. Highlights are a consequence of the relatively greater amount of light returned to the observer from curved surfaces that are to some degree specular (Figure 2); lowlights arise because objects on the surface of the earth are typically illuminated by indirect as well as direct light (Figure 3) (see Lotto et al., 1999a and b for a fuller explanation of these phenomena).
Since the source of the luminance gradient in the Mach stimulus in Figure 4 (and Demonstration) could be either a curved surface or a penumbra on a flat surface (or the result of the surface reflectances, which are, in fact, the source of the printed stimulus), the percept elicited incorporates highlights and lowlights in proportion to the frequency of their historical occurrence as accompaniments of luminance gradients. As predicted by this reasoning, proximal stimuli more consistent with a curved surface as the underlying source (which would normally be adorned with highlights and lowlights) elicit a stronger sensation of Mach bands than oppositely biased stimuli, which elicit such sensations weakly or not at all (see Lotto et al., 1999b). This modulation is similar to the enhancement or diminishment of simultaneous brightness contrast or edge effects achieved by manipulating the relative probabilities of the possible sources of the stimulus in the examples given earlier.
This empirical explanation of Mach bands is the same, in principle, as the explanations of simultaneous brightness and and other brightness contrast effects. The common cause is the genesis of visual percepts according to a strategy in which percepts are elicited as reflexes whose network connectivity has been wholly determined by the history of human visual experience. Because the perceptual responses to the several stimulus categories considered here (i.e., the phenomena in this Demonstration) manifest this strategy in superficially different ways, the common basis of these effects is less obvious than it might otherwise be.
Figure 2
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Figure 2 / The generation of highlights by specular surfaces. A) Because light is reflected maximally at the angle of its incidence, a reflectance maximum will occur for any eye position (indicated by the icon) when a curved surface has some degree of specularity. B) Luminance profile for a perfectly specular surface seen from the viewpoint in (A). In this circumstance, the only light that reaches the observer is from the portion of the curved surface that reflects the incident rays in the direction of the eye. C) Luminance profile for a perfectly Lambertian surface , as seen from the viewpoint in (A). D) The luminance profile derived by combining the curves in (B) and (C) (determined from the image of the rendered cube shown here). Because most natural surfaces have specular as well as Lambertian properties, the luminance gradients generated by curved surfaces are typically adorned by a view-dependent highlight at the onset of the gradient from the better lit to the shadowed surface. (After Lotto et al., 1999a)

Figure 3
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Figure 3 / The generation of lowlights by indirect illumination. A) The object considered here is now lit by indirect as well as direct light, the typical condition of illumination in natural settings. B) The luminance profile for a perfectly Lambertian surface when the illumination is from indirect light only; numbers indicate corresponding points, as in Figure 2. C) The complementary fall-off of direct light across such a surface. D) The luminance profile derived by combining the curves in (B) and (C). When a curved surface is illuminated by both direct and indirect sources (the sun and its light reflected from other objects, for instance), the luminance profile will have a lowlight at the offset of the gradient. (After Lotto et al., 1999a)

Figure 4
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Figure 4 / (Left) Photograph of a real-world cube manifesting a photometric highlight and lowlight (see luminance profile beneath the photo). (Right) A computer- generated image of a similar object, but lacking the highlight and lowlight. Despite the objective absence of these adornments, brightness maxima and minima (Mach bands) are apparent in the positions of their photometric counterparts in A. (From Lotto et. al. [1999])

References

Lotto RB, Williams SM, Purves D (1999a) Mach bands as empirically derived associations. Proc Natl Acad Sci USA 96:5245-5250.

Lotto RB, Williams SM, Purves D (1999b) An empirical basis for Mach bands. Proc Natl Acad Sci USA 96:5239-5244.