CSCI 497P/597P - Homework 3

Spring 2020

You may collaborate freely on this homework; the one rule is that you have to write up your own solutions.

Complete the following problems and submit your solutions to the HW3 assignment on Canvas. For all questions, justify your answers either or showing your work or giving a brief explanation. Please typeset your solutions using latex or similar; you may include neatly hand-drawn figures so long as the scan quality is good. You may work with your classmates on these problems, but you must write up your own solutions individually. This is largely new material, so please let me know if you think anything is unclear or ambiguous and I’ll make corrections or clarifications as needed.

  1. 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 minimla 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.

    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.)

      ​ P(chose at least one outlier in one iteration) =

    2. What is the probability that the above event - at least one outlier is chosen - happens in every one of \(K\) iterations of RANSAC?

      ​ P(never chose only inliers in any of K iterations) =

    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?

      ​ P(at least one iteration chose all inliers) =

    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?

  2. In this problem, we will unsuccessfully attempt to break math by finding the intersection point of parallel lines.

    1. Given the homogeneous coordinates of a line \(\ell = [a, b, c]^T\), derive an expression for the slope of the line. You may assume the slope is not undefined.
    2. Give a criterion for determining whether two homoegeneous lines \(\ell_1 = [a_1, b_1, c_1]^T\) and \(\ell_2 = [a_2, b_2, c_2]^T\) are parallel. Again, assume their slopes as derived in part (a) are not undefined.
    3. It turns out that in projective space, the intersection point of two parallel lines is well-defined. Using what we learned about point-line duality, calculate the intersection of \(\ell_1\) and \(\ell_2\) above.

    4. Using the intersection point’s coordinates and the parallel criterion from part (2), what can you say about the homogeneous coordinates of the intersection point of two parallel lines?

    5. A homoegeneous point with the above property is called a “point at infinity”. Describe the meaning of a “point at infinity” using the more intuitive 3D interpretation of 2D projective space.

  3. In lecture, we talked about the epipolar geometry of a pair of cameras. Suppose we have a camera rig that has three cameras. Camera 0 is in canonical pose, centered at the origin and looking down the \(-z\) axis. It has a known intrinsics matrix \(K_0\). Cameras 1 and 2 have known intrinsics and extrinsics: \(K_1, R_1, t_1, K_2, R_2, t_2\).

    1. In terms of the values given, write down the fundamental matrices \(F_1\) and \(F_2\) that relate each of Camera 1 and Camera 2 to Camera 0, such that:

      • For a point \(q\) in Camera 0 and a corresponding point \(p_1\) in camera 1, \(q^T F_1 p_1= 0\).

      • For a point \(q\) in Camera 0 and a corresponding point \(p_2\) in camera 2, \(q^TF_2p_2 = 0\).

    2. Suppose that a point \(p_1\) in homogeneous Camera 1’s homogeneous image coordinates is known (via a very high-accuracy feature matching scheme) to correspond to the same point in the world as the point \(p_2\), in Camera 2’s homogeneous image coordinates. Give an expression for the homogeneous coordinates of the point \(q\) in Camera 0 where that point in the world must project to. You can give your answer in terms of any of the values given in the problem or computed in the prior part.