Description

Summary:
1. Hough transform for circle detection and Canny edge detection (programming).
2. RANSAC edge proposal (programming)
3. RANSAC number of trials (theoretical)
Submission instructions:
• For each programming task, submit a .py source code file. Do not need to include any images or data files you used.
• For each pen & paper task, submit a .pdf file. Type your solution in LaTeX, Word, or another text editor of your choice and convert it to PDF – do not submit photographs or scans of handwritten solutions!
• In all files, include at the top names of all students in the group
• Choose exactly one person in your group that submits your solution via Moodle, it will count for the entire group.
1 Circle detection
Figure 1 shows a selection of the last Finnish pre-euro coins (10, 5, and 1 marks; and 50 and 10 pennis, respectively). We apply the Hough transformation to detect the coins.

Figure 1: A selection of Finnish coins.
• Download the image coins.jpg from Moodle. Read it and convert to grayscale.
• The mint specifies that the diameter of the 5 mark coin is 24.5 millimetres . The resolution of the image is approximately 0.12 mm/pixel. Calculate the radius r of the coin in pixels. • Apply the Canny edge detector to find edges in the grayscale image. Use the built-in function skimage.feature.canny. Visualize the edges and check that the outlines of the coins are detected.
• Use skimage.transform.hough circle to calculate the Hough transform of the edge detection result. Use the radius you calculated above. Draw the result. You should obtain something similar to Figure 2 that peaks strongly around the center of the 5 mark coin.
• Using skimage.transform.hough circle peaks, select the two highest peaks from the Hough transform. Get all outputs from the function, i.e., your call should look like: accums, cx, cy, radii = hough_circle_peaks(…).
• Apply matplotlib.patches.Circle to draw the circles at the coordinates found superimposed on the original image. See https://matplotlib.org/api/_as_gen/matplotlib. patches.Circle.html for more help. Use from matplotlib.patches import Circle to import the circle tool to your code.

Figure 2: Circle Hough transform result. The strongest peaks highlight the 5 mark coin centers (compare to Figure 1).
2 Fitting a line using RANSAC
with open(‘noisyedgepoints.npy’, ‘rb’) as f: X = np.load(f) y = np.load(f)
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After fitting the line, plot the outliers that got filtered by the RANSAC algorithm and the inliers respectively. The result should be similar to Figure 3.

Figure 3: Plot of results for task 2
3 Pen and paper task
This task expands on Question 2 of the ”Hough Transform and RANSAC” quiz:
• Suppose the fraction of outliers .
• The minimum number of samples to fit your model is m.
• k is the number of running trials so that with probability p, at least one set of samples is free from outliers.
Prove that the number of trials is .