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

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.

Mach edges: a critical test of the nonlinear 3rd derivative model for edge-detection

Feature detection is a crucial stage of visual processing. In previous feature-marking experiments we found that peaks in the 3rd derivative of the luminance profile can signify edges where there are no 1st derivative peaks nor 2nd derivative zero-crossings (Wallis and George 'Mach edges' (the edges of Mach bands) were nicely predicted by a new nonlinear model based on 3rd derivative filtering. As a critical test of the model, we now use a new class of stimuli, formed by adding a linear luminance ramp to the blurred triangle waves used previously. The ramp has no effect on the second or higher derivatives, but the nonlinear model predicts a shift from seeing two edges to seeing only one edge as the added ramp gradient increases. In experiment 1, subjects judged whether one or two edges were visible on each trial. In experiment 2, subjects used a cursor to mark perceived edges and bars. The position and polarity of the marked edges were close to model predictions. Both experiments produced the predicted shift from two to one Mach edge, but the shift was less complete than predicted. We conclude that the model is a useful predictor of edge perception, but needs some modification.

Third-derivative filters predict edge locations in spatial vision

Edge detection is crucial in visual processing. Previous computational and psychophysical models have often used peaks in the gradient or zero-crossings in the 2nd derivative to signal edges. We tested these approaches using a stimulus that has no such features. Its luminance profile was a triangle wave, blurred by a rectangular function. Subjects marked the position and polarity of perceived edges. For all blur widths tested, observers marked edges at or near 3rd derivative maxima, even though these were not 1st derivative maxima or 2nd derivative zero-crossings, at any scale. These results are predicted by a new nonlinear model based on 3rd derivative filtering. As a critical test, we added a ramp of variable slope to the blurred triangle-wave luminance profile. The ramp has no effect on the (linear) 2nd or higher derivatives, but the nonlinear model predicts a shift from seeing two edges to seeing one edge as the ramp gradient increases. Results of two experiments confirmed such a shift, thus supporting the new model. [Supported by the Engineering and Physical Sciences Research Council].

Added luminance ramp alters perceived edge blur and contrast:A critical test for derivative-based models of edge coding

In many models of edge analysis in biological vision, the initial stage is a linear 2nd derivative operation. Such models predict that adding a linear luminance ramp to an edge will have no effect on the edge's appearance, since the ramp has no effect on the 2nd derivative. Our experiments did not support this prediction: adding a negative-going ramp to a positive-going edge (or vice-versa) greatly reduced the perceived blur and contrast of the edge. The effects on a fairly sharp edge were accurately predicted by a nonlinear multi-scale model of edge processing [Georgeson, M. A., May, K. A., Freeman, T. C. A., & Hesse, G. S. (in press). From filters to features: Scale-space analysis of edge and blur coding in human vision. Journal of Vision], in which a half-wave rectifier comes after the 1st derivative filter. But we also found that the ramp affected perceived blur more profoundly when the edge blur was large, and this greater effect was not predicted by the existing model. The model's fit to these data was much improved when the simple half-wave rectifier was replaced by a threshold-like transducer [May, K. A. & Georgeson, M. A. (2007). Blurred edges look faint, and faint edges look sharp: The effect of a gradient threshold in a multi-scale edge coding model. Vision Research, 47, 1705-1720.]. This modified model correctly predicted that the interaction between ramp gradient and edge scale would be much larger for blur perception than for contrast perception. In our model, the ramp narrows an internal representation of the gradient profile, leading to a reduction in perceived blur. This in turn reduces perceived contrast because estimated blur plays a role in the model's estimation of contrast. Interestingly, the model predicts that analogous effects should occur when the width of the window containing the edge is made narrower. This has already been confirmed for blur perception; here, we further support the model by showing a similar effect for contrast perception. © 2007 Elsevier Ltd. All rights reserved.

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

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.