Mach bands are a common visual phenomenon and can be observed both on reflectance edges (as presented on Fig. 1) and on illumination edges, where a penumbra is created. They served in a number of vision paradigms because they are a robust phenomenon easily created in the laboratory and because they demonstrate important theoretical issues. Certainly the most general issue is the easily measurable difference between the physical stimulation and an emerging percept.
History of Research
It became clear very early that this simple display yields a physiological explanation. The first neural mechanism offered as an explanation was lateral inhibition. Mach himself envisioned a similar mechanism, a potential neural response that would enhance one side of the edge and inhibit the other. When the actual retinal cells performing lateral inhibition were found in the Limulus eye , the connection seemed obvious. This elegant explanation was dominant for about a hundred years. During that time experimental studies in perception mostly covered the role of physical characteristics of the gray regions: their reflectance, slope, width of the bands, etc. Mathematical models were developed based on these psychophysical data. They were essentially based on distance-dependent excitation and surround inhibition, which takes place on the retina (presumably in ganglion cells).
The gradient of the slope determines the size and intensity of the bands: steeper slopes produce brighter and thinner light bands and darker dim bands .
The Dark-adapted eye does not perceive bands, and the appearance of light bands depends on light adaptation .
Edge orientation is not important.
Bands produced by luminance ramps are thinner .
Visibility of bands is altered if luminance bars and ramps are placed next to the bands .
The shape of the gray regions is not important, while ramps of intermediate width produce most visible bands. Sensitivity is lower for high spatial frequencies .
Phase relationships among Fourier components of the underlying waveforms of the gray regions influence the appearance of the bands .
The results on symmetry of light and dark bands are inconsistent.
The acumulated data revealed that, though important, retinal mechanisms alone cannot account for the phenomena. Primary visual cortex contained cells that proved to be another good candidate for a possible neural foundation of the described phenomenon . In cat BA 17, there are cells that better respond to luminance ramps (even-symmetric cells) and cells that better respond to abrupt luminance steps (odd-symmetric cells).
Tolhurst , based on his adaptation experiments, postulated “edge” detectors (odd-symmetric operators) and “bar” detectors (even-symmetric operators). Their spatially limited mutual inhibition would explain Mach bands on the ramps and the lack of bands in luminance steps. This theory was neat and simple but could not explain all the conditions. For example, it cannot explain why the width of the adjacent stimulus is not important.
The proposal offered by Morrone and Burr  was that the appropriate combination of the outputs for even-symmetric and odd-symmetric operators might better account for the experimental data. This, however, requires two sets of operators to indicate the position of “salient features” such as lines, edges, and bars. If the position of so-called maximum local energy of the feature corresponds to an odd-symmetric filter, the discovered feature is an edge. In the case of an even-symmetric filter, it is a bar (line). The model very precisely and quantitatively matched a large amount of existing data and predicted some later obtained results. However, the model could not account for intensity differences, i.e., for the change in appearance due to the change in contrast.
A Completely different approach was taken, at about the same time, by Watt and Morgan  who turned to rule-based rather than feature-based solutions. There are three rules (edge rule, bar rule, and null rule, corresponding to a luminance plateau) that interpret the output obtained by five computational stages, starting with the filtering by even-symmetric operators. The results are categorized into zero-bounded responses and regions of inactivity. After the rules are applied, it is possible to infer about luminance levels that would influence brightness. The model well predicts everything the previous models did (ramp vs. step, width of bands). Unfortunately, the model has a problem with spatial frequency that dictates the appearance of Mach bands.
A simpler model postulated only a single filter, with two channels, one selective for high- and one selective for low spatial frequencies . The first channel does not respond to the ramp and only signals the existence of dark and light bars on the edge of the ramp. The second channel responds to the ramp and to the luminance edge. The two channels compose the final percept signaling the ramp and the two bars (i.e., Mach bands). The problem is the lack of a formalized rule specifying stimuli conditions that would elicit certain responses (bar or edge).
The next model tried to mimic the responses of V1 cells to lines and edges . In the first phase the model produces Gaussian lines and error-function-shaped edges. Based on these, the reconstruction process recovers lines and edges in the scene. The model nicely predicts most of the data but was not developed enough to allow for proper testing.
The model proposed by Kingdom and Moulden  was designed to explain brightness perception in general. It starts with light adaptation, continues with multi-spatial scale filtering and a power-law transformation. After that, interpretation rules are used to calculate brightness for each spatial scale. Finally, outputs from all the scales are averaged. This advanced model can now properly deal with the spatial frequency issue as well as most other experimental results from the brightness domain.
The latest attempts are filling-in models. They stress the importance of contours but without special interest in surface brightness. The model by Pessoa  can still explain Mach bands using special boundary computations responsive to luminance gradients and steps.
All of the presented models are concerned with identification of bars, lines, and edges mimicking only early vision processes. Nevertheless, the set of primitives even in early stages should be richer and able to include intensity changes. This would allow models to account more specifically for the already obtained data and enable more sophisticated predictions (including gradients of brightness, differences in intensities, temporal effects, etc.).
Filling-in type come closest to an explanation of Mach bands. Newer versions include the allocation of the edges followed by the filling-in process. However, it is still the edge that determines the outcome. The remaining issue for these models is the brightness of the areas within the edges.
The Chevreul illusion was introduced a few years before Mach bands (in 1839) and is nowadays presented in most textbooks (and our Fig. 1) as Mach bands. The original Mach illusion (Fig. 2a) has a ramp instead of uniformly gray surfaces, which are typical only for the Chevreul illusion.
Vision research the demonstrated that the Chevreul illusion is just a special case of a much broader phenomenon known as Mach bands.
Brightness/lightness contrast. The two effects were often linked together, probably because lateral inhibition was the first explanation offered in both cases. Later it became clear that lateral inhibition performed by retinal cells cannot be an explanation for either of these phenomena.
Craik-O’Brien-Cornsweet illusion (Fig. 2b). The same, more central mechanisms, such as simple V1 cells (BA 17), were proposed as an explanatory neural mechanism for both illusions.
- 3.Davidson, M.: A perturbation analysis of spatial brightness interaction in flashed visual fields. Ph.D. dissertation (unpublished). University of California, Berkeley (1966)Google Scholar
- 4.Fiorentini, A.: Foveal and extra foveal contrast threshold data point of a non uniform field. Atti delta Fondazione Giorgio Ronchi 12, 180–186 (1957)Google Scholar
- 5.Heinemann, E.: Simultaneous brightness induction. In: Jameson, D., Hurvich, L. (eds.) Handbook of Sensory Physiology, vol. VII-4, pp. 146–169. Springer, Berlin (1972)Google Scholar
- 9.Syrkin, G., Yinon, U., Gur, M.: Simple cells may be the physiological basis of the Mach band phenomenon: evidence from early monocularly deprived cats. Soc. Neurosci. Abstr. 20, 312 (1994)Google Scholar
- 13.Fiorentini, A., Baumgartner, G., Magnussen, S., Schiller, P., Thomas, J.: The perception of brightness and darkness: relations to neuronal receptive fields. In: Spillmann, L., Werner, J. (eds.) Visual Perception: The Neurophysiological Foundations, pp. 129–161. Academic, San Diego (1990)CrossRefGoogle Scholar