Research

An Empirical Explanation: Mach Bands
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)
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 / 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 / (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.










