CSCI 497P/597P - Homework 4

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 HW4 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.

1. (10 points) 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.

2. (10 points) 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 \(F_{10}\) and \(F_{20}\) 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, \(q^T F_{10} p_1= 0\).

      • For a point \(q\) in Camera 0 and a corresponding point \(p_2\) in Camera 2, \(q^TF_{20}p_2 = 0\).

   2. Suppose that a point \(p_1\) in Camera 1’s homogeneous pixel 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 pixel coordinates. Give an expression for the homogeneous pixel 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.