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44 Chapter 2 ■ Edge-Detection Techniques

The derivative with respect to x is therefore

x 2
x
− 2σ 2
G (x) = − e (EQ 2.22)
σ 2
In two dimensions, a Gaussian is given by

2 2
x +y
− 2σ 2
2
G(x, y) = σ e (EQ 2.23)

and G has derivatives in both the x and y directions. The approximation to
Canny’s optimal filter for edge detection is G’, and so by convolving the input
image with G’,we obtain animage E that has enhanced edges, even in the pres-
ence of noise, which has been incorporated into the model of the edge image.
A convolution is fairly simple to implement, but is expensive computation-
ally, especially a two-dimensional convolution. This was seen in the Marr
edge detector. However, a convolution with a two dimensional Gaussian can
be separated into two convolutions with one-dimensional Gaussians, and the
differentiation can be done afterwards. Indeed, the differentiation can also
be done by convolutions in one dimension, giving two images: one is the x
component of the convolution with G’ and the other is the y component.
Thus, the Canny edge-detection algorithm to this point is:

1. Read in the image to be processed, I.

2. Create a one-dimensional Gaussian mask G to convolve with I.The
standard deviation(s) of this Gaussian is a parameter to the edge detector.
3. Create a one-dimensional mask for the first derivative of the Gaussian in
the x and y directions; call these G x and G y . The same s value is used as
in step 2.

4. Convolve the image I with G along the rows to give the x component
image I x , and down the columns to give the y component image I y .
5. Convolve I x with G x to give I x ’, the x component of I convolved with the
derivative of the Gaussian, and convolve I y with G y to give I y ’.
6. Compute the magnitude of the edge response (i.e., if you want to view the
result at this point) by combining the x and y components. The magnitude
of the result can be computed at each pixel (x,y) as:

2

M(x, y) = I x (x, y) + I (x, y) 2
y