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Mach bands explained by response normalization.

Kingdom FA - Front Hum Neurosci (2014)

Bottom Line: More recently, the dominant idea has been that Mach bands are a consequence of a visual process that generates a sparse, binary description of the image in terms of "edges" and "bars".Another recent explanation is that Mach bands result from learned expectations about the pattern of light typically found on sharply curved surfaces.I show that a simple one-dimensional model of response normalization explains the range of conditions under which Mach bands occur, and as importantly, the conditions under which they do not occur.

View Article: PubMed Central - PubMed

Affiliation: McGill Vision Research, Department of Ophthalmology, McGill University Montreal, Quebec, Canada.

ABSTRACT
Mach bands are the illusory dark and bright bars seen at the foot and knee of a luminance trapezoid. First demonstrated by Ernst Mach in the latter part of the 19th century, Mach bands are a test bed not only for models of brightness illusions but of spatial vision in general. Up until 50 years ago the dominant explanation of Mach Bands was that they were caused by lateral inhibition among retinal neurons. More recently, the dominant idea has been that Mach bands are a consequence of a visual process that generates a sparse, binary description of the image in terms of "edges" and "bars". Another recent explanation is that Mach bands result from learned expectations about the pattern of light typically found on sharply curved surfaces. In keeping with recent multi-scale filtering accounts of brightness illusions as well as current physiology, I show however that Mach bands are most simply explained by response normalization, whereby the gains of early visual channels are adjusted on a local basis to make their responses more equal. I show that a simple one-dimensional model of response normalization explains the range of conditions under which Mach bands occur, and as importantly, the conditions under which they do not occur.

No MeSH data available.


Related in: MedlinePlus

Model applied to a trapezoidal edge, shown as the black line. The green lines show the responses of four of the filter scales selected at 2 octave intervals, while the purple line shows the sum of all 9 filter responses. Left: without response normalization the summed response is close to veridical. Right: with response normalization Mach bands are produced.
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Figure 4: Model applied to a trapezoidal edge, shown as the black line. The green lines show the responses of four of the filter scales selected at 2 octave intervals, while the purple line shows the sum of all 9 filter responses. Left: without response normalization the summed response is close to veridical. Right: with response normalization Mach bands are produced.

Mentions: where as is the peak-to-mean amplitude of the filter response to the edge, ā is the mean amplitude across all filter scale responses and k is a constant that determines the strength of normalization. In all the simulations here k is fixed at 0.5 and was chosen to provide a good visual fit to the data from Figure 4 in Wallis and Georgeson (2012), shown here in Figure 7. Besides k, which is fixed throughout, there are no free parameters in the model. The effect of the response normalization stage is to reduce the response amplitudes of all filters, but more so for ones with relatively high amplitude, resulting in more equal filter responses across scale. The factor ā (mean amplitude) causes the normalization to be invariant with respect to stimulus contrast and results in Mach bands whose contrasts are invariant with stimulus contrast, as reported by Wallis and Georgeson (2012). It is important to note that the response normalization term used here is not the same as a saturating point-wise nonlinearity, such as the well-known Naka-Rushton equation (Naka and Rushton, 1966). The response normalization term here is a constant applied multiplicatively to each filter’s response, a constant that alters the amplitude of each response but does not “distort” its shape by introducing new Fourier components as does a point-wise saturating nonlinearity. In the final stage of the model the normalized responses of all 9 filters are summed to produce the predicted brightness profile. For illustration purposes the summed responses have been multiplied by another constant in order to scale them to match the amplitude of the stimulus luminance profile (0.4 in all cases), in order that the physical stimulus profile and its predicted appearance can be directly compared.


Mach bands explained by response normalization.

Kingdom FA - Front Hum Neurosci (2014)

Model applied to a trapezoidal edge, shown as the black line. The green lines show the responses of four of the filter scales selected at 2 octave intervals, while the purple line shows the sum of all 9 filter responses. Left: without response normalization the summed response is close to veridical. Right: with response normalization Mach bands are produced.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4219435&req=5

Figure 4: Model applied to a trapezoidal edge, shown as the black line. The green lines show the responses of four of the filter scales selected at 2 octave intervals, while the purple line shows the sum of all 9 filter responses. Left: without response normalization the summed response is close to veridical. Right: with response normalization Mach bands are produced.
Mentions: where as is the peak-to-mean amplitude of the filter response to the edge, ā is the mean amplitude across all filter scale responses and k is a constant that determines the strength of normalization. In all the simulations here k is fixed at 0.5 and was chosen to provide a good visual fit to the data from Figure 4 in Wallis and Georgeson (2012), shown here in Figure 7. Besides k, which is fixed throughout, there are no free parameters in the model. The effect of the response normalization stage is to reduce the response amplitudes of all filters, but more so for ones with relatively high amplitude, resulting in more equal filter responses across scale. The factor ā (mean amplitude) causes the normalization to be invariant with respect to stimulus contrast and results in Mach bands whose contrasts are invariant with stimulus contrast, as reported by Wallis and Georgeson (2012). It is important to note that the response normalization term used here is not the same as a saturating point-wise nonlinearity, such as the well-known Naka-Rushton equation (Naka and Rushton, 1966). The response normalization term here is a constant applied multiplicatively to each filter’s response, a constant that alters the amplitude of each response but does not “distort” its shape by introducing new Fourier components as does a point-wise saturating nonlinearity. In the final stage of the model the normalized responses of all 9 filters are summed to produce the predicted brightness profile. For illustration purposes the summed responses have been multiplied by another constant in order to scale them to match the amplitude of the stimulus luminance profile (0.4 in all cases), in order that the physical stimulus profile and its predicted appearance can be directly compared.

Bottom Line: More recently, the dominant idea has been that Mach bands are a consequence of a visual process that generates a sparse, binary description of the image in terms of "edges" and "bars".Another recent explanation is that Mach bands result from learned expectations about the pattern of light typically found on sharply curved surfaces.I show that a simple one-dimensional model of response normalization explains the range of conditions under which Mach bands occur, and as importantly, the conditions under which they do not occur.

View Article: PubMed Central - PubMed

Affiliation: McGill Vision Research, Department of Ophthalmology, McGill University Montreal, Quebec, Canada.

ABSTRACT
Mach bands are the illusory dark and bright bars seen at the foot and knee of a luminance trapezoid. First demonstrated by Ernst Mach in the latter part of the 19th century, Mach bands are a test bed not only for models of brightness illusions but of spatial vision in general. Up until 50 years ago the dominant explanation of Mach Bands was that they were caused by lateral inhibition among retinal neurons. More recently, the dominant idea has been that Mach bands are a consequence of a visual process that generates a sparse, binary description of the image in terms of "edges" and "bars". Another recent explanation is that Mach bands result from learned expectations about the pattern of light typically found on sharply curved surfaces. In keeping with recent multi-scale filtering accounts of brightness illusions as well as current physiology, I show however that Mach bands are most simply explained by response normalization, whereby the gains of early visual channels are adjusted on a local basis to make their responses more equal. I show that a simple one-dimensional model of response normalization explains the range of conditions under which Mach bands occur, and as importantly, the conditions under which they do not occur.

No MeSH data available.


Related in: MedlinePlus