School of Life & Health Sciences, Aston University, Birmingham, B4 7ET, UK.
Mark Georgeson, Stuart Wallis; Mach bands and models of spatial vision: the role of 1st, 2nd and 3rd derivative operators in encoding edges and bars. Journal of Vision 2012;12(9):319. doi: 10.1167/12.9.319.
In the 1860s Ernst Mach observed that light or dark bands could be seen at abrupt changes of luminance gradient, in the absence of peaks or troughs in luminance. Mach bands have been important for theories of visual feature coding, because they suggest that bars or lines are found at peaks and troughs in the output of even-symmetric spatial filters, such as the Gaussian second-derivative. Our experiments showed that the probability of seeing Mach bands was nearly independent of the contrast (20-80%), duration (50-300 ms), and spatial scale of the ramp-edge luminance waveform, but increased with the relative sharpness of its ‘corners’. These results rule out the idea that Mach bands depend simply on the amplitude of the second derivative, but we develop a multi-scale model, based on Gaussian first- and second-derivative filtering incorporating automatic scale selection (Lindeberg, 1998), that can account accurately for both the perceived structure of the bands and the probability of seeing them. A key idea is that Mach band strength (probability) depends on the ratio of second- to first-derivative responses at the peaks in the second-derivative scale-space map. This ratio is contrast-invariant, nearly scale-invariant, and increases with the sharpness of the ‘corners’ of the ramp – all as observed. Our observers also marked the perceived positions of the bands and the edges of the bands. We find that the edges of Mach bands pose a difficult challenge for models of edge detection. But if the second-derivative output is subject to a smooth, threshold-like nonlinearity (power function, exponent p=3) then differentiated again, the peaks in this nonlinear third-derivative response predict the locations of Mach band edges strikingly well. Mach bands thus shed new light on the multiscale filtering systems of human spatial vision.