[Solved] EECE5639 Computer Vision I Homework 6

Homographies, Stereo and Motion

1. Explain the differences between a planar homography between two images, the Essential and theFundamental matrices in stereo.
2. Where are the epipoles in the case when the two cameras have parallel optical axes (the canonicalconfiguration)?
3. Show how the projection of a point in a planar scene at world coordinates (X,Y ) to pixel coordinates (u,v) in an image plane can be represented using a planar affine camera model. Under what conditions is the use of an affine transformations appropriate when viewing a planar scene? How many degrees of freedom are there in the model and what is the minimum number of calibration points needed to estimate the transformation? What effects can a planar affine transformation have on parallel lines?
4. Consider the convergent binocular imaging system shown below. The cameras and all the points arein the y = 0 plane. The image planes are perpendicular to their respective camera axes. Find the disparity corresponding to the point P. Hint: The perpendicular distance between any point (x_o,y_o)

an a line given by ax + by + c = 0 is (ax_o + by_o + c)/ a² + b².

x

5. Determine the matrices H_l and H_r needed to normalize the entries of the fundamental matrix before estimating the Fundamental matrix using the Eight Point Algorithm. Hint: given a set of points

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p_i = [x_i,y_i,1]^T with i = 1,,n define x = 1/n^P_i x_i, y = 1/n^P_i y_i and

Then find a 3 3 matrix H such that

Hp_i = p_i with p_i = [(x_i x_i)/d,(y_i y_i)/d,1]^T with i = 1,,n

6. Use the method of least squares to derive a linear system of equations to estimate the affine transformation that maps a set of points (x_i,y_i) into new points (x⁰_i,y_i⁰). Show that it is not necessary to solve a 6 6 system all at once, since the problem can be decomposed into two smaller sets of equations.
7. (Old Exam) Two identical security cameras are mounted in a room as shown in the figure below. The world coordinate system W is at one corner of the room, and each camera has its own coordinate system C₁ and C₂. In the following, P^w, P¹ and P² represent the coordinates of a point P with respect to the world coordinate system W, the camera 1 coordinate system C₁ and the camera 2 coordinate system C₂, respectively. The world coordinates of the centers of projectionand are (2,2,4) and (4,3,3), respectively. The focal length of the cameras is 1 and their image planes are located at zⁱ = 1, i = 1,2, respectively.
   • Let E₁ and E₂ be the epipoles in camera 1 and 2 respectively. Find the camera coordinates and of the epipoles expressed in their respective camera systems.
   • The cameras capture images of a fly in the room. Let f₁ and f₂ be the images of the fly in the first and second camera, respectively. The camera 1 coordinates of the image in the first camera are 1). Find the equation of the epipolar plane containing the fly, expressed in the camera 2 coordinate system. Hint: find
   • The fly flies following a straight line with constant velocity with respect to the world coordinate system(3,2,1). Find the camera coordinates of the FOE in camera 1.
8. (Old Exam) Consider the Hankel matrix:
   • What is the complexity of the underlying dynamics?
   • Find a regressor of the formexplaining the data in the given matrix.
   • Find the values of x and y.

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