Answer the following, considering the concrete set of points shown below. Justify your answer for each section:
What class of models is appropriate for fitting this data (e.g., line, circle, ...)? Write the equation for the model in the form $y=f(x; a, b)$, with parameters $a$, $b$, ... How many degrees of freedom does your model have?
Sketch what the (approximate) best fit of the model might look like. How many inliers and outliers are there?
If you were to use RANSAC to fit this model to the data, what values would you use for the initial number of points and the consensus set size ($s$ and $d$ in the lecture slides)?
If you were to use RANSAC on this data with this chosen parameter $s$, what is the probability that during a single iteration all $s$ sampled points will be inliers? Hint: Use $\binom{n}{k}$ notation to justify the answer, and the scipy.misc.comb function in Python to compute the numerical probability (If haven't, you may follow the instruction in Part 2 to install SciPy; run import scipy.misc before you use the function).
If you were to use RANSAC on this data with this chosen parameter $s$, how many iterations will it take to converge with probability > 99%? In other words, how many iterations would you have to run to be 99% sure at least one of the iterations contained only inliers?
Suppose the above plot now has 100 random outliers (with the same number of inliers). What is the answer to questions d and e in this case?
Taking a step back, suppose you're using a Hough transform instead to fit the model. How many dimensions does the Hough space have and what do they correspond to? What does a single point in the Hough space correspond to?
Would you expect Hough transform and RANSAC to find the same optimal model? (Both 'yes' and 'no' answers are valid, but you have to justify why)
If you were to use the Hough transform under the conditions in f, with 100 outliers, would you expect it to be faster or slower than RANSAC?
