Master Thesis - Fachbereich Informatik

2.3. EDGE DETECTION 25
These operators compute the horizontal and vertical components of a smooth gradient
[21], denoted as gx and gy in the following. The total gradient magnitude g at a pixel
position p in an image can be computed by the following equation:
�
g(p) = g2 x(p)+g2 y(p) (2.21)
An example of the gradient magnitude based on the Sobel operator can be found in
Figure 2.7(b). The following approximations can be used in order to save computational
costs:
g(p) ≈ |gx(p)| + |gy(p)| (2.22)
g(p) ≈ max(|gx(p)|, |gy(p)|) (2.23)
These approximations yield equally accurate results on average [22]. Beside the gradient
magnitude it is possible to compute the angle of the gradient as:
� �
gy(p)
φ(p) =arctan
(2.24)
gx(p)
Although there is a certain angular error with the Sobel gradient [36], it is used very
often in practice, since it provides a good balance between the computational load and
orientation accuracy [16].
The Equations 2.21-2.24 are defined not only for the Sobel operator, but for every other
operator that computes the horizontal and vertical gradient components.
Canny Edge Detector Today, the Canny edge detector [13] is probably the most used
edge detector, and is proven to be optimal in a precise, mathematical sense [65]. It is
designed to detect noisy step edges of all orientations and consists of three steps:
1. Edge enhancement
2. Nonmaximum suppression
3. Hysteresis thresholding
The first step is based on a first-order Gaussian derivative as introduced before. For
fast implementations, the separability of the filter kernel can be used to improve the performance.
Gradient magnitude and orientation can be computed as in Equation 2.21 and
2.24, or using the approximations. The standard deviation parameter σ of the Gaussian
function influences the scale of the detected edges. A lower σ preserves more details (highfrequencies),
but also noisy edges, while a larger σ leaves only the strongest edges. The
appropriate σ depends on the image content and what kind of edges should be detected.
The goal of the nonmaximum suppression step is to thin out ridges around local maxima
and return a number of one pixel wide edges [65]. The dominant direction of the gradient
calculated in step one determines the considered neighbors of a pixel. The gradient magnitude
at this position must be larger than both neighbors, otherwise it is no maximum
and its position is set to zero (suppressed) in the edge image.