A research engineer mounts a camera on a ground robot. She drove the robot on a flat planar ground while the camera captures two images I and I' with overlapping field-of- view at two different times. Let the camera reference frames where the images are taken be (x, y, z) and (x',y', z'). As shown in Fig. 1, the relative motion between these two frames is denoted by (tx, tz, y = 0), where (tx, tz) is the translation vector on the xz-plane and by is the rotation angle around the y-axis. tz Z 0,=0 tx Fig. 1 (a) Write the expression for the essential matrix E between I and I' in terms of (tx, tz, Oy), where Oy = 0. (5 marks) (b) The camera focal length and principle point are given by (fr,fy) and (Cx = 0, Cy = 0) respectively. A 3D point P in the scene appears as p = (Px.Py) in image I, write the expression for the corresponding epipolar line L' in image l' in terms of (fx, fy), (Cx, Cy), (Px.Py) and (tx, tz, By = 0). Show your working clearly. (6 marks) (c) Given a pair of point correspondence pp' i.e. (px. Py) (p'xp'y) from I and I'. (i) Show that the relationship between p and p' is given by Pyfxtx Pxtz) py(fxtx-Pxtz) = 0. (6 marks) (ii) What is the minimum number of image correspondences needed to solve for the relative motion (tx, tz, 6y= 0)? (2 marks) (d) Suppose four 3D points lying on a plane in the scene are concurrently seen by I and I'. If the plane is parallel to both the image planes and is located at d distance away from the reference frame of the first camera (x, y, z), find the homography in terms of (t,t,, y = 0) and d that relates I, I' and the 3D plane.