A DIMENSIONLESS MEASURE OF EDGE
Department of Computer Science
The University of Western Australia
Nedlands, W.A. 6009
email: [email protected]
Most edge detection methods require a thresholding operation to eliminate noise. Finding a good method of automatically choosing a threshold remains an unsolved problem. It is argued that the main cause of this difficulty is that the edge strength calculated by most edge detection methods is a dimensional quantity (usually related to the intensity gradient). If one uses a measure of edge significance that is dimensionless, and is hence invariant to image scale or brightness then thresholding difficulties are considerably reduced. This paper presents a dimensionless measure of edge significance derived from the Local Energy model of feature detection. It is also argued that high-pass filtering should be used to obtain image information at different scales, instead of the more usually applied low-pass filtering. Using this approach, the choice of scale only affects the relative significance of features without degrading their localisation.
Traditional gradient-based edge detection methods require a thresholding operation to decide the acceptance or rejection of a point as an edge or not. The choice of this threshold is one of the most critical parameters in producing a good edge map. Finding a good method of automatically choosing a threshold remains an unsolved problem though some efforts have been made. Kundu and Pal  used human psychophysical data to devise a thresholding scheme. Canny  introduced the idea of thresholding hysteresis which has proved to be a useful heuristic.
The result of performing an edge detection using gradient based techniques such as Sobel or Canny is an edge strength proportional to the intensity gradient. This is a dimensional quantity having units lumens=radian (pixel coordinates represent viewing directions, hence have units of radians). Second order differential operators have units lumens=radians2. The gradient of intensities in an image depends on many factors, including illumination, blurring and scale. For example doubling the size of an image while leaving its grey values unchanged will halve all the intensity gradients. Any gradient based edge detection process will need to use a threshold modified appropriately. However, a priori one does not know the size and scale of an image.
Indeed choosing to retain a point as an edge or not solely on its 'strength' may not be a desirable thing to do. For example consider a black square and a grey square against a white background. The edges around the black square will be stronger, but one would consider the edges of the grey square to be just as significant and important as those around the black one. What one should use to decide whether an edge is significant or not is not clear. However some parameters that one could use might include its strength, sharpness, continuity and so on.