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Question 1: Canny Edge Detection Implementation [5 pts]

In 1986, John Canny defined a set of goals for an edge detector and described an optimal method for achieving
them. Canny specified three issues that an edge detector must address:
• Error rate: Desired edge detection filter should find all the edges, there should not be any missing edges,
and it should respond only to edge regions.

• Localization: Distance between detected edges and actual edges should be as small as possible.
• Response: The edge detector should not identify multiple edge pixels where only a single edge exists.
Remember from the lecture that in Canny edge detection, we will first smooth the images, then compute gradients,
magnitude, and orientation of the gradient.

This procedure is followed by non-max suppression, and finally hysteresis
thresholding is applied to finalize the steps. Briefly, follow the steps below for practical implementation of Canny
Edge detector :

1. Read a gray scale image you can find from Berkeley Segmentation Dataset, Training images, store it as a
matrix named 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 (call it σ > 0).
i

3. Create a one-dimensional mask for the first derivative of the Gaussian in the x and y directions; call these Gx
and Gy. The same σ > 0 value is used as in step 2.
4. Convolve Ix with Gx to give I

x
, the x component of I convolved with the derivative of the Gaussian, and
convolve Iy with Gy to give I

y

, y component of I convolved with the derivative of the Gaussian.
5. Compute the magnitude of the edge response by combining the x and y components. The magnitude of the
result can be computed at each pixel (x, y) as: M(x, y) = q
I

x
(x, y)
2 + I

y
(x, y)
2
.

6. Implement non-maximum suppression algorithm that we discussed in the lecture. Pixels that are not local
maxima should be removed with this method. In other words, not all the pixels indicating strong magnitude
are edges in fact. We need to remove false-positive edge locations from the image.
7. Apply Hysteresis thresholding to obtain final edge-map. You may use any existing library function to
compute connected components if you want.

Definition: Non-maximal suppression means that the center pixel, the one under consideration, must have
a larger gradient magnitude than its neighbors in the gradient direction. That is: from the center pixel,
travel in the direction of the gradient until another pixel is encountered; this is the first neighbor. Now,
again starting at the center pixel, travel in the direction opposite to that of the gradient until another
pixel is encountered; this is the second neighbor.

Moving from one of these to the other passes though
the edge pixel in a direction that crosses the edge, so the gradient magnitude should be largest at the edge pixel.
Algorithmically, for each pixel p (at location x and y), you need to test whether a value M(p) is maximal in
the direction θ(p).

For instance, if θ(p) = pi/2, i.e., the gradient direction at p = (x, y) is downward, then
M(x, y) is compared against M(x, y − 1) and M(x, y + 1), the values above and below of p. If M(p) is not
larger than the values at both of those adjacent pixels, then M(p) becomes 0. For estimation of the gradient
orientation, θ(p), you can simply use atan2(I

y
, I′
x
).

Hint: It is assumed that the gradient changes continuously as a function of position, and that the gradient at
the pixel coordinates are simply sampled from the continuous case. If it is further assumed that the change in
the gradient between any two pixels is a linear function, then the gradient at any point between the pixels can be
approximated by a linear interpolation.

For a sample output, please refer to figure below: chessboard image’s X component of the convolution with
a Gaussian (a), Y component of the convolution with a Gaussian (b), X component of the image convolved with the
derivative of a Gaussian (c), Y component of the image convolved with the derivative of a Gaussian (d), resulting
magnitude image (e), and canny-edge image after non-maximum suppression (f) are shown.
ii

Your tasks:

• Choose three example gray-scale images from Berkeley Segmentation Dataset (Training Images) CLICK
HERE. When executed, your algorithm should plot intermediate and final results of Canny Edge Detection
process as similar to the figure illustrated above.
• Please show the effect of σ in edge detection by choosing three different σ values when smoothing. Note that
you need to indicate which σ works best as a comment in your assignment.

What to submit:

• Code
• A short write-up about your implementation with results: 1 ) A figure showing intermediate results as in
Figure above, and 2) Similar Figures showing the effect of σ.
iii