SIGNAL SHAPE d dx DIFFERENTIATING

1 A lter for edge detection In graylevel images boundaries are often associated to step-like discontinuities. In its simplest form the problem of nding boundaries in 1D images may therefore be thought of as the problem of nding step-discontinuities in a signal. The naive approach to this problem is to use a diierentiator and a threshold in series. The derivative of the signal ought to present largèspikes' at the location of the step-like discontinuities. One could declare an edge where the derivative of the signal exceeded some reasonable threshold. Unfortunately images are noisy and this strategy leads to detecting a large number of spurious edges. Therefore the image has to be ltered before taking its derivative. Figure 1 illustrates the operation as well as the shape of the image signal along the lter chain. As shown on the gure the output of the edge detector is a pulse of non-zero width. Is it possible to locate the edge more precisely? One solution is to modify the threshold lter by setting the above-threshold region to a linear rather than constant value. Within each edge region we may localize edges to pixel accuracy by picking the maxima of the ouptut of the detector. 1.1 The Gaussian low-pass lter Why use a Gaussian rather than any other low-pass lter? It has been proven that the Gaussian lter is the only low pass lter that does not generatèripples'. This property is called`causality': as the nominal resolution of the image is decreased by lowpass ltering one does not want to increase the number of`visual events' that are present in the image. Another reason: In \Design of optimal lters for edge detection", Canny 1] shows that the derivative of the Gaussian lter, shown as the \combined" lter of gure 1, is nearly optimal. In what follows we summarize his argument. Refer to the original paper for the rigorous derivation. be the convolution of a zero-mean, independent Gaussian random vector : the noise { n x , and a lter k. R is a linear combination of zero-mean Gaussian random variables, it is 1