Homework 4
Due Friday, Nov. 17, 9 a.m.

Written: (To do some of these, you will probably want to use Matlab, not compute by hand).

1. Compute the F-matrix for a non-verged stereo camera system with 10
    micron pixels and a 12.5mm focal length lens. The camera is 600x800
    and the center of projection is in the middle of the image. The
    baseline is 100mm. Assume all external (metric) values are to be in
    mm.

2. Consider a verged stereo system where the baseline is a distance
    d along the x axis and the angle of vergence for the right camera
    is a known angle theta.

    a) Compute the homogeneous transformation that relates points in
    left camera coordinates to points in right camera coordinates.

    b) Compute the form of the E-matrix for this family of stereo rigs.

3. Show the form of the homography relating points on a plane parallel
    to the image plane, imaged under perspective projection.

CS461:

4. The *horopter* is defined as the point of zero disparity (or
    equivalently, their coordinates in the left and right image are the
    same).

    Consider a symmetrically verged stereo rig: that is, one in which
    both cameras are rotated about the y axis toward each other by the
    same angle, but with opposite sign. Compute an expression for
    points on the horopter for this system for all points in the plane
    y=0. (Hint: just work out projection for a stereo system in the x-z
    plane with the given geometry and solve for z in terms of x)

Implementation:


1. a) Implement a function that given a rotation axis n (a unit vector)
       and angle a (in radians), computes a 3x3 rotation matrix.  You can
       either implement the Rodrigues form (see
       http://mathworld.wolfram.com/RodriguesRotationFormula.html) or the
       equivalent form given in the slides and in Trucco in Verri. Call
       your function

          rMat = Rodrigues(axis,angle);

    b) Implement a function that creates a 4x4 homogeneous transformation
       from a rotation matrix and a translation vector.  Call your function

	hTrans = HomogeneousTransform(rMatrix,tVector)

    c) Implement a full perspective projection model (including camera
        intrinsic parameters).  Call your function:

    pts2D = Perspective(pts3D, intrinsic)

    where   pts2D is a 3xn array of projected points in homogeneous  
coordinates
            pts3D is a 4xn array of 3D points in homogeneous coordinates
	   intrinsic is a 3x3 intrinsic parameter matrix

    === Note that the returned projected points are expected to be in
    pixel coordinates and are  homogenous --- that is, the final
    coordinate is a 1.

2. Create a simulation of a stereo camera system. To do this, create a
    stereo camera projection function:

    [leftPts2D,rightPts2D] = StereoProject(pts3D,intrinsicParms,hTrans)

    where: leftPts2D and rightPts2D are the projected points
	  pts3D are 3D points to be projected in *left camera coordinates*
	  intrinsicParms is are the intrinsic parameters for both cameras
	  hTrans is the homogeneous transformation that takes points
	  	  in left camera coordinates to right camera coordinates.


3. Write a function to perform E-matrix (or F-matrix) estimation for a
    pair of stereo cameras.  Call your function

    eMatrix = EMatrixEstimate(leftPts2D,rightPts2D)

    where eMatrix is a 3x3 essential matrix
	 leftPts is a 3xn array of points in the left camera
	 rightPts is a 3xn array of corresponding points in the right
	 camera


For CS 461 only:

4. Implement homography estimation. Call your function

    hMatrix = HMatrixEstimate(leftPts2D,rightPts2D)

    where hMatrix is a 3x3 homography matrix
	 leftPts is a 3xn array of points in the left camera
	 rightPts is a 3xn array of corresponding points in the right
	 camera


Submit the following scripts:

Problem1.m:

	 a) Create and print the rotation matrix that is a rotation
	 about the vector parallel to v = (1,2,1) and a rotation of .2
	 radians. Call this matrix R.

	 b) Create and print the homogeneous transform that is formed
	 when combining the rotation in part a) with a translation of
	 1000 units in the z direction.

          c) Generate 20 random 3D points in homogeneous coordinates
	    such that abs(x) < 200, abs(y) < 200, abs(z) < 200.  Call
	    the points pts3D. Plot these points in figure(1) (use
	    plot3).

	 d) Create a homogeneous transform that moves the points
	 1000mm along the z direction and rotates them pi/4 radians
	 about the z axis. Computed the transformed points and call
	 them pts3Da. Plot these points in figure(2).

	 e) Project the points in pts3Da using the intrinsic
	 parameters specified for written problem 1. Call the
	 projected points pts2D. Plot these points in figure(3) (using
	 plot).

Problem2.m


        Create two stereo camera systems: a) one that has a baseline
        translation of 100mm along the x axis and no rotation, and b)
        one that the same as a, but the right camera is also turned 5
        degrees toward the left camera.

        Now, project the points in pts3Da to produce two pairs of
        projections [lpts2Da, rpts2Da] and [lpts2Db, rpts2Db]. Print
        the resulting projections.

Problem3.m

	Estimate and print the F-Matrices for both projections in
	Problem 2. Compute the corresponding E-Matrices and print
	them.


Problem4.m

	Repeat the generation of points in problem 1, but now fixing
	z=0 for all points. Image the points as described in problem
	2, and print out the two homographies that result.