ECE 847
Digital Image Processing
Homework 3
Canny Edge Detection
Sept. 26, 2008
By
Bryan Willimon
Homework 3
Homework 3:
Canny Edge Detection
Introduction
The purpose of this assignment is to demonstrate the Canny algorithm for finding edges
in an image. Three major steps were involved in determining the final result. The three steps
were Convolution with a Gaussian kernel and its derivative, Non-maximal suppression, and
edge-linking with hysteresis (double thresh holding). The Chamfer distance was computed on
the resulting picture and a template to find an object inside of the image.
Theory
Several steps were taken in the design of the Canny edge detector. Calculations had to
be made about the image to find where the pixel values created a step edge and then bring out
the pixels with the higher magnitude to show the edge. Such calculations involve convolution
with a Gaussian kernel with respect to x and y, finding the magnitude and angle of the
convolved images, then use the magnitude and angle values to give an image with non-maximal
suppression and finally use hysteresis to determine where the edges are.
Gaussian Convolution
The idea is to calculate a Gaussian kernel that would find all of the step edges in each
the x-direction and y-direction. To give optimal results, a 5x1 kernel was computed and used in
both directions. A 5x1 kernel was used based on the fact that it will give a standard deviation
and variance of value 1 and capture 98.76% of the area in a Gaussian. When computing the
Gaussian kernel, the formula is used from –x to x and –y to y based on how long the kernel is:
When computed G(x), the y value would go to zero and then compute. The same goes for G(y)
that x value would be zero. This was used to break down a 5x5 kernel to (2) 5x1 kernels to
speed up the process of convolution.
Once the Gaussian kernels were computed, then you were able to compute the
derivative of the Gaussian kernel:
The original Gaussian kernel was used to smooth out the image and derivative was used to look
for the step edges in the x and y direction.
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The following formula was the process of using both of the Gaussian kernels to compute
the new image with convolution.
Each of the formulas below computed the new gradient images for each convolution in
the x and y direction and also calculated the magnitude and angle of the gradients.
Non-maximal suppression
This technique uses the values of the gradient’s magnitude and phase to find where the
edges are location in the image. I uses the process of taking a 3x3 window for each pixel and
uses the gradient phase to determine what direction to be looking at for comparison. When
the angle is determined, then the value in the positive and negative angle location along with
the current pixel is compared to find the maximum value. If the maximum turns out to be the
pixel that you are currently on, then value stays the same. Otherwise, the current pixel value
goes to zero in the event that one of the other pixels that you were looking at was the
maximum. The various angles along with the pixels that you look at are in the following table:
To account for all angle values positive and negative, then you would add π to the negative
angles to give all values in the upper 180 degree range. In the end, the resulting image would
contains all edges with one pixel thickness.
Hysteresis (Double Thresh Holding)
The final step is hysteresis which I have already used and discussed in the last project.
Hysteresis uses the idea of double thresh holding to find the high values of the edges and weed
out all of the weaker values as to give an accurate and cleaner image. The first step in
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Homework 3
hysteresis is to sort all of the non-zero gradient magnitude values and create a histogram with
each pixel value along with the amount of times that it shows up in the image. Then take a predetermined percentage value to evaluate the high thresh hold number. For instance, 10% was
suggested for this project, but I used 20% based on what looked better for me. Once that value
is computed, then take the high thresh hold and divide it by the ratio between high and low and
compute the low thresh hold value. After having your two values, you would scan through the
image to find all pixel values that had a number higher than the high thresh hold value and turn
them to 255 and everything else to 0. Then take a second pass through the image and find all
of the pixels that have the value of 255 and then check each of their 8-neighbors to determine
which of those pixels have the value above the low thresh hold and then turn them on. I also
took another pass in the opposite direction to make sure that I didn’t miss any edges from the
first pass. The resulting should have all viable edges marked as foreground and the rest as
background.
Chamfering
Chamfering is used to create a probability map based on how close each pixel is from
the closest edge. The second part of this assignment required that we find an object in an
image. Chamfering was used to create a probability map on the image and on the template.
When each new image was computed, then the new template map is taken throughout the
image to calculate the distance from each pixel in the template to the corresponding pixel in
the image. All of the distances were added up and store for that area of the image. Next, you
would search through your database of areas and find the lowest value and that would give the
location of the object with the highest probability. The following images show the algorithm
used to find the value of the probability map:
Before the map is created, you would first go through the image and make each value equal to
the (height*width)+1 to give a reasonable value to represent infinity in the image. Then the
algorithm is computed throughout the image to calculate the values.
Algorithm and Methodology
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Homework 3
The first sets of steps were to calculate the 5x1 Gaussian kernel and then compute the
derivative of the 5x1 kernel. Once the kernels were calculated, I went through the entire image
and used convolution on a 5x1 window of the image and the Gaussian to compute the same
image with a smoothing factor and create a new image. I then used the 5x1 Gaussian derivative
kernel to move throughout the new image to compute the final X and Y gradients. Each
gradient image requires a different function to calculate the gradient for the x and y direction,
but with the kernel being the same value for the x and y direction, then the numbers in the
kernel will look the same for x and y.
Once the gradient images are created, then on to calculating the magnitude and angle
of the two gradients to determine where the edges are located in the image. Each pixel value
for the magnitude is calculated by taking the square root of the combination of each gradient
pixel value squared. Then that gives you the magnitude of pixel value from the original image,
and the higher the value, the better that it is an edge. Next, the phases are calculated for each
pixel value by taking the tangent inverse of (y/x). Once the phases are computed, they will tell
you at which angle that the edge would be pointing, if there is one there.
The next step would deal with non-maximal suppression. Non-maximal suppression would look
at the values for the phase and magnitude to determine if there is an edge at that particular
pixel and how to calculate that. First, the phase would indicate at which angle to be focusing
your view at, either it be somewhere between 0 -> 2π. To narrowing down the search, I added
the value of π to all negative angles to rotate all of the negative angles 180 degrees to match
the positive correspondents. Once I knew which group of angles to look at, I then compared
the pixel that I was currently on and the two pixels either (above and below) or (left and right)
or some combination of the two to determine which magnitude was greater. If the magnitude
that I was currently on was higher, then that pixel value stays the same; otherwise if one of the
two values around the current pixel were higher, then that current pixel value would suppress
to zero (0).
Now that a new image with all of the edges being one pixel thick and represented all of
the step edges in the original image, I then use hysteresis to compute which edges are greater
in value to bring them out and get rid of the rest based on percentages. The first part of
hysteresis is to make a histogram of the pixel values and how many of each value are there in
the suppressed image. Once the graph is made, I calculated that the high thresh hold would be
the top 20% of the group and then used a ratio of 5 between the high thresh hold and low
thresh hold. Once I had the two thresh holds, I just repeated the process from the last
assignment of using double thresh holding. That would then give me my final image with all of
the good edges showing up.
Finally, as an add-on to the project, Chamfering is used to match a template of an object
to the object within a larger image. Chamfering is used to calculate the distance from the
nearest edge in both the template and larger image. I then take the probability image from
both and scan through the larger image with the template as a window and take each individual
pixel value within the template and subtract it from the corresponding pixel value within the
larger image and sum up all of the distances within the template window to give a total value
with each area. I then determine which area out of the whole image is the lowest and consider
that area to be the most likely that contains the template.
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Results
Below is each group of original images, gradient-x and gradient-y, the magnitude and
angle images of the gradients, the image after non-maximal suppression, resulting image using
hysteresis and the images after using chamfering for special cases. I have included cat.pgm,
cameraman.pgm, hydrant.pgm, and cherrypepsi.jpg along with cherrypepsi_template.jpg. Here
are my results:
cat.pgm
Original Image
Gradient Magnitude
X-Gradient
Gradient Angle
Y-Gradient
Non-maximal suppression
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Resulting Image
cameraman.pgm
Original Image
X-Gradient
Y-Gradient
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Gradient Magnitude
Gradient Angle
Resulting Image
Hydrant.pgm
Non-maximal suppression
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Original Image
Y-Gradient
Gradient Angle
X-Gradient
Gradient Magnitude
Non-maximal suppression
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Resulting Image
Chamfering Results:
Finding a Cherry Pepsi
Original Image
Original Template
Chamfered Image
Chamfered Template
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Resulting Image with location of Cherry Pepsi
The results above show the Canny edge detection being used with a 5x1 Gaussian kernel with a
sigma value of 1. Each test resulted in success in finding each of the edges for all images. Also,
the chamfering gave back a successful run finding the Pepsi can in the image.
Conclusion
With the above results shown, the Canny edge detector turned out to be a very useful
and successful tool that was easy to implement to find all viable edges in an image. Based on
the tests that were run, the Canny edge detector picked up more edges in some instances that
were not considered a boundary, but still was calculated as an edge. Different scenarios call for
different values of sigma or maybe a different size kernel to accurately pick up all of the correct
boundaries in an image and not all of the edges. The image that worked well to determine the
boundaries was the cameraman because the background contrasted with him very well. Any
images that are cluttered and shared the same color as the foreground and background tend to
blend together and you can’t find the step edge too well. Also when shadows are involved, the
edge detector will depict the shadow and the object as too separate objects and won’t be able
to discern between the two. In conclusion, the Canny edge detector works well with any image
just as long as the objects have a distinct difference in contrast and with the least amount of
shadows.
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