Master Thesis - Fachbereich Informatik

2.3. EDGE DETECTION 23
N × 1 kernel. This increases the performance significantly for large images and N. More
information about convolution and filter separation can be found for example in [64].
The general procedure of edge enhancement in common derivative-based edge detectors
can be summarized into two steps:
1. Smoothing of the image by convolving with a smoothing function
2. Differentiation of the smoothed image
Mathematically, this can be expressed as follows (here with respect to x):
Iedge(x, y) = K∂/∂x ∗ (S ∗ I(x, y)) (2.18)
= (K∂/∂x ∗ S) ∗ I(x, y)
= ∂S
∗ I(x, y)
∂x
where K∂/∂x indicates the filter kernel approximating the partial derivative with respect
to x. S represents the kernel of the smoothing function. Again, the associativity of the
convolution can be used to optimize processing. Thus, instead of first smoothing the
image with kernel S and then calculating the partial derivative, it is possible to reduce
the problem to a single convolution with the partial derivative of the smoothing kernel
∂S
∂x . Hence, the first-order derivative of a Gaussian is suited as an edge detector which is
less sensitive to noise compared to finite difference filters [24]. The response of the edge
detector can be parametrized by the standard deviation of the Gaussian to control the
scale of detected edges, i.e. the level of detail. A larger σ suppresses high-frequency edges
for example.
2.3.3. Common Edge Detectors
Due to the large number of approaches in this section only a selection of common edge
detectors can be presented. Figure 2.7 visualizes the edge responses of different edge
detectors that will be introduced in the following in more detail.
Sobel Edge Detector A very early edge detector that is still used quite often in the
present is the Sobel operator. It was first described in [51] and attributed to Sobel. It is
the smallest difference filter with odd number of coefficients that averages the image in
the direction perpendicular to the differentiation [36]. The corresponding filter kernel for
x and y are:
⎡
SOBELX = ⎣
⎡
SOBELY = ⎣
1 0 −1
2 0 −2
1 0 −1
⎤
1 2 1
0 0 0
−1 −2 −1
⎦ (2.19)
⎤
⎦ (2.20)