Mach edges: a key role for 3rd derivative filters in spatial vision

Stuart A. Wallis, Mark A. Georgeson

Research output: Unpublished contribution to conferencePosterpeer-review


Edges are key points of information in visual scenes. One important class of models supposes that edges correspond to the steepest parts of the luminance profile, implying that they can be found as peaks and troughs in the response of a gradient (first-derivative) filter, or as zero-crossings (ZCs) in the second-derivative. A variety of multi-scale models are based on this idea. We tested this approach by devising a stimulus that has no local peaks of gradient and no ZCs, at any scale. Our stimulus profile is analogous to the classic Mach-band stimulus, but it is the local luminance gradient (not the absolute luminance) that increases as a linear ramp between two plateaux. The luminance profile is a smoothed triangle wave and is obtained by integrating the gradient profile. Subjects used a cursor to mark the position and polarity of perceived edges. For all the ramp-widths tested, observers marked edges at or close to the corner points in the gradient profile, even though these were not gradient maxima. These new Mach edges correspond to peaks and troughs in the third-derivative. They are analogous to Mach bands - light and dark bars are seen where there are no luminance peaks but there are peaks in the second derivative. Here, peaks in the third derivative were seen as light-to-dark edges, troughs as dark-to-light edges. Thus Mach edges are inconsistent with many standard edge detectors, but are nicely predicted by a new model that uses a (nonlinear) third-derivative operator to find edge points.
Original languageEnglish
Publication statusPublished - 2006
Event11th Applied Vision Association Christmas Meeting - Aston University, Birmingham (UK), United Kingdom
Duration: 18 Dec 2006 → …


Other11th Applied Vision Association Christmas Meeting
Country/TerritoryUnited Kingdom
CityAston University, Birmingham (UK)
Period18/12/06 → …

Bibliographical note

Abstract published in Eleventh Applied Vision Christmas Meeting, Perception, 2007, 36(2), pp. 314-315, ISSN 0001-4966. If you have discovered material in AURA which is unlawful e.g. breaches copyright, (either yours or that of a third party) or any other law, including but not limited to those relating to patent, trademark, confidentiality, data protection, obscenity, defamation, libel, then please read our Takedown Policy and contact the service immediately.


  • edges
  • visual scenes
  • luminance profile
  • gradient filter
  • first-derivative
  • zero-crossings
  • second-derivative
  • stimulus
  • third-derivative operator
  • edge points


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