Write down the \(x\) and \(y\) residuals for a pair of corresponding points \((x, y)\) in image 1 and \((x', y')\) in image 2 under a homography (projective) motion model. Assume the homography matrix is parameterized as \[ \begin{bmatrix} a & b & c\\ d & e & f\\ g & h & 1 \end{bmatrix} \]
In the inner loop of RANSAC, how many points are used to fit a candidate model if you are fitting a line to a set of 2D points?
In the inner loop of RANSAC, how many pairs of corresponding points are used to fit a candidate model if you are fitting a translation to a set of correspondences?
In the inner loop of RANSAC, how many pairs of corresponding points are used to fit a candidate model if you are fitting a homography to a set of correspondences?
In this problem, we’ll analyze the RANSAC algorithm to help us understand how to decide how many iterations to run (\(K\)). Suppose we are fitting some model that requires a minimal set of \(s\) points to fully determine (e.g., \(s=4\) matches for a homography, \(s=2\) points for a line). We also know (or have assumed) that the data has an inlier ratio of \(r = \frac{\text{\# inliers}}{\text{\# data points}}\); in other words, the probability that a randomly sampled point from the dataset as a probability of \(r\) of being an inlier.
5.1 What is the probability that in a single iteration (i.e., one model hypothesis), at least one of the points chosen is an outier? (Hint: It may be helpful to start with the probability that no point is an outlier.)
5.2 What is the probability that the above event - at least one outlier is chosen - happens in every one of \(K\) iterations of RANSAC?
5.3 What is the probability that the above event doesn’t happen? In other words, what is the probability that in at least one of the \(K\) trials, all \(s\) points were inliers?
5.4 Suppose that you want to be reasonably sure that RANSAC will succeed: you’d like the probability of success (i.e., the probability computed in the prior part) to be at least \(\mathbf{P}\). What \(K\) should you choose?