Explain why it is possible to estimate the 3D pose of the camera relative to the road from knowledge of the parameters 𝑎𝑎𝑖𝑖, 𝑏𝑏𝑖𝑖, and 𝑐𝑐𝑖𝑖. If needed, you may assume that the road is locally flat and straight, and that the camera is mounted with its optical axis parallel to the road.

Problem 1. Assume that 2 images have been captured using a single camera at 2 different time instants, t’ and t. Between these 2 points in time, the camera has translated as illustrated in the birds-eye view that appears below: in the direction x’ by the distance f , and also in
the direction z’ by the distance f. There is no translation in the direction y’.
Virtual image plane at time t’
a) Find the essential matrix that relates these two camera positions.
b) Solve for the epipole in the image at time t. Clearly explain your choice of image coordinates.
Problem 2.  Consider a calibrated stereo camera arrangement, and assume that you have been given the translation vector t and rotation matrix R that relates the two cameras:
3 cos( 45°)
𝑡𝑡 = 2 𝑅𝑅 = sin( 45°)
1 0
−sin( 45°)
cos (45°)
0
0
0.
1
a) Solve for the essential matrix.
b) Compute the rank of your answer to part (a) using singular value decomposition, SVD. If you use computational tools, cut-and-paste your code as part of your answer.
x
z
Virtual image plane
at time t
x’
z’
Problem 3. (10 points.) Consider the simple stereo imaging geometry that was introduced in class, as shown below. Both optical axes are parallel, and both cameras have the same focal length. In this view from above, the overall coordinate reference frame (x, y, z) is centered at the left camera.
Assume that all distances are given in units of meters. Let f = 0.025 and B = 0.15. Suppose that you are given the following corresponding pair of points from the two images:
(𝑥𝑥𝐿𝐿′, 𝑦𝑦𝐿𝐿′) = (0.005, 0.0) and (𝑥𝑥𝑅𝑅′ , 𝑦𝑦𝑅𝑅′ ) = (0.003, 0.0).
a) Solve for the 3D point (x, y, z) that is associated with these two image points.
b) Suppose that the system designer is considering a slightly larger baseline distance of B+∆B. Give a clear argument explaining why (or why not) the designer might expect that computed values of z will be higher
in accuracy.
Problem 4. Suppose that an automobile has been equipped with a single
forward-looking camera. The intrinsic parameters of the camera are known, including the focal length f.
Also suppose that designers have developed a good technique for detecting highway lane lines using this camera. Intuitively, the detected lane lines are indicated in green in the figure below.
Suppose that 2 lane lines have been detected as follows, where u and v represent the image coordinate system:
Line 1: 𝑎𝑎1𝑢𝑢 + 𝑏𝑏1𝑣𝑣 + 𝑐𝑐1 = 0
Line 2: 𝑎𝑎2𝑢𝑢 + 𝑏𝑏2𝑣𝑣 + 𝑐𝑐2 = 0
Explain why it is possible to estimate the 3D pose of the camera relative to the road from knowledge of the parameters 𝑎𝑎𝑖𝑖, 𝑏𝑏𝑖𝑖, and 𝑐𝑐𝑖𝑖. If needed, you may assume that the road is locally flat and straight, and that the camera is mounted with its optical axis parallel to the road.

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