Mach edges: local features predicted by 3rd derivative spatial filtering

Stuart A. Wallis, Mark A. Georgeson

Research output: Contribution to journalArticlepeer-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 (1st derivative) filter, or as zero-crossings in the 2nd derivative (ZCs). We tested those ideas using a stimulus that has no local peaks of gradient and no ZCs, at any scale. The stimulus profile is analogous to the Mach ramp, but it is the luminance gradient (not the absolute luminance) that increases as a linear ramp between two plateaux; the luminance profile is a blurred triangle-wave. For all image-blurs tested, observers marked edges at or close to the corner points in the gradient profile, even though these were not gradient maxima. These Mach edges correspond to peaks and troughs in the 3rd derivative. Thus Mach edges are inconsistent with many standard edge-detection schemes, but are nicely predicted by a recent model that finds edge points with a 2-stage sequence of 1st then 2nd derivative operators, each followed by a half-wave rectifier.
Original languageEnglish
Pages (from-to)1886-1893
Number of pages8
JournalVision Research
Issue number14
Publication statusPublished - Jul 2009

Bibliographical note

NOTICE: this is the author’s version of a work that was accepted for publication in Vision Research. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Wallis, Stuart A. and Georgeson, Mark A. (2009). Mach edges: local features predicted by 3rd derivative spatial filtering. Vision Research, 49 (14), pp. 1886-1893. DOI 10.1016/j.visres.2009.04.026


  • human vision
  • psychophysics
  • edge localisation
  • spatial derivatives
  • Mach bands


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