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Chapter 2 ■ Edge-Detection Techniques 33

In addition, it would be nice to have a numerical measure of how successful
an edge-detection scheme is in an absolute sense. There is no such measure in
general, but something usable can be constructed by thinking about the ways
in which an edge detector can fail or be wrong. First, an edge detector can
report an edge where none exists; this can be due to noise, thresholding, or
simply poor design, and is called a false positive. In addition, an edge detector
could fail to report an edge pixel that does exist; this is a false negative. Finally,
the position of the edge pixel could be wrong. An edge detector that reports
edge pixels in their proper positions is obviously better than one that does
not, and this must be measured somehow. Since most of the test images will
have known numbers and positions of edge pixels, and will have noise of a
known type and quantity applied, the application of the edge detectors to the
standard images will give an approximate measure of their effectiveness.
One possible way to evaluate an edge detector, based on the above discus-
sion, was proposed by Pratt [1978], who suggested the following function:

1
I A
i=1 1 + αd(i) 2
E 1 = (EQ 2.8)
max(I A , I I )
where I A is thenumberofedgepixels found by theedgedetector, I I is the
actual number of edge pixels in the test image, and the function d(i) is
the distance between the actual ith pixel and the one found by the edge
detector. The value a is used for scaling and should be kept constant for any
set of trials. A value of 1/9 will be used here, as it was used in Pratt’s work.
This metric is, as discussed previously, a function of the distance between
correct and measured edge positions, but it is only indirectly related to the
false positives and negatives.
Kitchen and Rosenfeld [1981] also present an evaluation scheme, this one
based on local edge coherence. It does not concern itself with the actual position
of an edge, and so it is a supplement to Pratt’s metric. It does concern how well
the edge pixel fits into the local neighborhood of edge pixels. The first step
is the definition of a function that measures how well an edge pixel is continued
on the left; this function is:

kπ π


a(d, d k )a , d + if neighbor k is an edge pixel

L(k) = 4 2 (EQ 2.9)
0 Otherwise

where d is the edge direction at the pixel being tested, d 0 is the edge direction
at its neighbor to the right, d 1 is the direction of the upper-right neighbor, and
so on counterclockwise about the pixel involved. The function a is a measure