This software implements a number of edge detection methods. Since locally maximum intensity
gradients (first-derivative intensities) appear as zeros of the second-derivative image intensities, the
zero-crossings of the second derivative image intensities can be used as edges also. In 2-D, second
derivative is defined by the Laplacian. Zero-crossings are much easier to locate than locally maximum
values. Zero-crossings of the second derivative image intensities, however, correspond to locally
maximum as well as locally minimum intensity gradients, and locally minimum gradients do not
correspond to edges. Locally minimum intensity gradients often have small gradients and can be
removed from the detected edges.
To reduce the effect of noise, Gaussian smoothing is often used. Therefore, an image is first convolved
with a Gaussian smoother before determining its Laplacian. It turns out that smoothing an image with a
Gaussian and then determining its Laplacian is the same as convolving the image with the Laplacian of a
Gaussian (LoG). The latter is computationally faster because it requires only one convolution while
smoothing followed by the Laplacian requires two convolutions.
Edge detection by the LoG operator is due to Marr and Hildreth [1]. Clark [2] showed that the
zero-crossing edges correspond to locally minimum intensity gradients if their first and third derivatives
have the same sign. This provides a means to identify and remove the false edges. An example of edge
detection by this method using the LoG operator is shown below. The original image is shown in Fig. 1a
and the Laplacian edges are shown in Fig. 1b. The edges are overlaid with the original image to visually
evaluate the quality of the edges. After removing the false edges, we obtain the edges shown in Fig. 1c.
gradients (first-derivative intensities) appear as zeros of the second-derivative image intensities, the
zero-crossings of the second derivative image intensities can be used as edges also. In 2-D, second
derivative is defined by the Laplacian. Zero-crossings are much easier to locate than locally maximum
values. Zero-crossings of the second derivative image intensities, however, correspond to locally
maximum as well as locally minimum intensity gradients, and locally minimum gradients do not
correspond to edges. Locally minimum intensity gradients often have small gradients and can be
removed from the detected edges.
To reduce the effect of noise, Gaussian smoothing is often used. Therefore, an image is first convolved
with a Gaussian smoother before determining its Laplacian. It turns out that smoothing an image with a
Gaussian and then determining its Laplacian is the same as convolving the image with the Laplacian of a
Gaussian (LoG). The latter is computationally faster because it requires only one convolution while
smoothing followed by the Laplacian requires two convolutions.
Edge detection by the LoG operator is due to Marr and Hildreth [1]. Clark [2] showed that the
zero-crossing edges correspond to locally minimum intensity gradients if their first and third derivatives
have the same sign. This provides a means to identify and remove the false edges. An example of edge
detection by this method using the LoG operator is shown below. The original image is shown in Fig. 1a
and the Laplacian edges are shown in Fig. 1b. The edges are overlaid with the original image to visually
evaluate the quality of the edges. After removing the false edges, we obtain the edges shown in Fig. 1c.
| (a) (b) (c) |
Fig. 1. (a) An image. (b) The zero-crossings of the LoG of the image using a Gaussian of standard
deviation 3 pixels. (c) Zero-crossing edges after removing the false edges.
Many detected edges correspond to small details in an image and need to be removed. The number of the
detected edges can be reduced by using a larger Gaussian smoother. This, however, displaces the edges
from their true positions and the larger the standard deviation of the Gaussian, the larger the
displacement. A method developed by Bergholm [3] can keep the edges well localized without allowing the
weak edges to enter the image. This process, which is known as edge focusing, detects edges at a low
resolution by using a rather large standard deviation of Gaussian. Then, the standard deviation of the
Gaussian is gradually reduced and the edges are tracked from low to high resolution. Fig. 2a below shows
edges obtained in this manner starting from standard deviation of 5 pixels down to 3 pixels with half pixel
steps. To remove the edges further, one can simply remove those that have gradients below a threshold
value. The threshold value can be specified interactively while viewing the edges. Weak edges of Fig. 2a
removed in this manner are shown in Fig. 2b.
A method that directly finds locally maximum intensity gradients to avoid the detection of false edges is
proposed by Canny [4]. At each image point the gradient direction is determined and in that direction a
check is made to see whether or not pixels on both sides of the point have lower gradient magnitudes. If
they do, the pixel under consideration is taken as the edge. This method should provide edges that are
the same as those obtained by a combination of the Marr-Hildreth and Clark method. But due to
approximations in computation of edge directions by the Canny method, obtained edges are slightly
different from those obtained by Clark. Edges of Fig. 1a determined by the Canny method are shown in
Fig. 2c.
deviation 3 pixels. (c) Zero-crossing edges after removing the false edges.
Many detected edges correspond to small details in an image and need to be removed. The number of the
detected edges can be reduced by using a larger Gaussian smoother. This, however, displaces the edges
from their true positions and the larger the standard deviation of the Gaussian, the larger the
displacement. A method developed by Bergholm [3] can keep the edges well localized without allowing the
weak edges to enter the image. This process, which is known as edge focusing, detects edges at a low
resolution by using a rather large standard deviation of Gaussian. Then, the standard deviation of the
Gaussian is gradually reduced and the edges are tracked from low to high resolution. Fig. 2a below shows
edges obtained in this manner starting from standard deviation of 5 pixels down to 3 pixels with half pixel
steps. To remove the edges further, one can simply remove those that have gradients below a threshold
value. The threshold value can be specified interactively while viewing the edges. Weak edges of Fig. 2a
removed in this manner are shown in Fig. 2b.
A method that directly finds locally maximum intensity gradients to avoid the detection of false edges is
proposed by Canny [4]. At each image point the gradient direction is determined and in that direction a
check is made to see whether or not pixels on both sides of the point have lower gradient magnitudes. If
they do, the pixel under consideration is taken as the edge. This method should provide edges that are
the same as those obtained by a combination of the Marr-Hildreth and Clark method. But due to
approximations in computation of edge directions by the Canny method, obtained edges are slightly
different from those obtained by Clark. Edges of Fig. 1a determined by the Canny method are shown in
Fig. 2c.
| (a) (b) (c) |
Fig. 2. (a) Edges determined through edge focusing. (b) Edges reduced further by removing
small-gradient edges. (c) Edges determined by the Canny method.
The edge detection methods described so far use intensity differences as the metric to detect edges.
Edges can also be determined by using intensity ratios as the metric also. Intensity ratios are
independent of scene illumination [5]. This means edge strength in dark and bright areas in an image is
the same if it corresponds to the same change in surface reflectance. Edges of Fig. 1a determined by
intensity ratios using the LoG operator are shown in Fig. 3a. By removing the weak edges, we obtain Fig.
3b.
small-gradient edges. (c) Edges determined by the Canny method.
The edge detection methods described so far use intensity differences as the metric to detect edges.
Edges can also be determined by using intensity ratios as the metric also. Intensity ratios are
independent of scene illumination [5]. This means edge strength in dark and bright areas in an image is
the same if it corresponds to the same change in surface reflectance. Edges of Fig. 1a determined by
intensity ratios using the LoG operator are shown in Fig. 3a. By removing the weak edges, we obtain Fig.
3b.
| (a) (b) |
This software can also find edges in color images. Locally sharp color changes are detected and considered
edges. An example of color image segmentation by this software is given in Fig. 4. Color edges in Fig. 4a
are shown in Fig. 4b after the removal of weak and noisy edges.
edges. An example of color image segmentation by this software is given in Fig. 4. Color edges in Fig. 4a
are shown in Fig. 4b after the removal of weak and noisy edges.
(a)
| Fig. 4. (a) A color image and (b) its edges. Weak and noisy edges have been interactively removed. |
To obtain a license for this edge detector, place an order here =>
 | |  |
Fig. 3. (a) Edges determined by the LoG operator when using intensity ratios instead of intensity
differences. (b) Intensity ratio edges after removing the weak edges.
References
[1] Marr D. and E. Hildreth, Theory of edge detection, Proc. R. Soc. Lond., 207:187-217 (1980).
[2] Clark, J. J., Authenticating edges produced by zero-crossing algorithms, IEEE Transactions on Pattern
Analysis and Machine Intelligence, 11(1):43-57 (1989).
[3] Bergholm, F., Edge focusing, IEEE Trans. Pattern Analysis and Machine Intelligence, 19:726--741
(1987).
[4] Canny, J., A computational approach to edge detection, IEEE Trans. Pattern Analysis and Machine
Intelligence, 8:679-714 (1986).
[5] Goshtasby, A. 2-D and 3-D Image Registration, Wiley Press, 2005.
differences. (b) Intensity ratio edges after removing the weak edges.
References
[1] Marr D. and E. Hildreth, Theory of edge detection, Proc. R. Soc. Lond., 207:187-217 (1980).
[2] Clark, J. J., Authenticating edges produced by zero-crossing algorithms, IEEE Transactions on Pattern
Analysis and Machine Intelligence, 11(1):43-57 (1989).
[3] Bergholm, F., Edge focusing, IEEE Trans. Pattern Analysis and Machine Intelligence, 19:726--741
(1987).
[4] Canny, J., A computational approach to edge detection, IEEE Trans. Pattern Analysis and Machine
Intelligence, 8:679-714 (1986).
[5] Goshtasby, A. 2-D and 3-D Image Registration, Wiley Press, 2005.
(b)
| 2-D edge detection |
| Image Fusion Systems Research |
____________________________________________________________________________________