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4.2 Edges 211

Figure 4.31 Human boundary detection (Martin, Fowlkes, and Malik 2004) c 2004 IEEE. The darkness of the
edges corresponds to how many human subjects marked an object boundary at that location.

Qualitatively, edges occur at boundaries between regions of different color, intensity, or
texture. Unfortunately, segmenting an image into coherent regions is a difficult task, which
we address in Chapter 5. Often, it is preferable to detect edges using only purely local infor-
mation.
Under such conditions, a reasonable approach is to define an edge as a location of rapid
intensity variation. 3 Think of an image as a height field. On such a surface, edges occur
at locations of steep slopes, or equivalently, in regions of closely packed contour lines (on a
topographic map).
A mathematical way to define the slope and direction of a surface is through its gradient,
∂I ∂I
J(x)= ∇I(x)=( , )(x). (4.19)
∂x ∂y

The local gradient vector J points in the direction of steepest ascent in the intensity function.
Its magnitude is an indication of the slope or strength of the variation, while its orientation
points in a direction perpendicular to the local contour.
Unfortunately, taking image derivatives accentuates high frequencies and hence amplifies
noise, since the proportion of noise to signal is larger at high frequencies. It is therefore
prudent to smooth the image with a low-pass filter prior to computing the gradient. Because
we would like the response of our edge detector to be independent of orientation, a circularly
symmetric smoothing filter is desirable. As we saw in Section 3.2, the Gaussian is the only
separable circularly symmetric filter and so it is used in most edge detection algorithms.
Canny (1986) discusses alternative filters and a number of researcher review alternative edge
detection algorithms and compare their performance (Davis 1975; Nalwa and Binford 1986;
Nalwa 1987; Deriche 1987; Freeman and Adelson 1991; Nalwa 1993; Heath, Sarkar, Sanocki
et al. 1998; Crane 1997; Ritter and Wilson 2000; Bowyer, Kranenburg, and Dougherty 2001;
Arbel´ aez, Maire, Fowlkes et al. 2010).
Because differentiation is a linear operation, it commutes with other linear filtering oper-

3 We defer the topic of edge detection in color images.