algebraic calculation Assignment Help

algebraic calculation 1   Prove that the vector from the viewpoint of a pinhole camera to the vanishing point (which is a point on the image plane) of a set of 3D parallel lines in space is parallel to the direction of that set parallel lines. Please show steps of your proof.

Hint: You can either use geometric reasoning or algebraic calculation. 

If you choose to use geometric reasoning, you can use the fact that the projection of a 3D line in space is the intersection of its “interpretation plane” with the image plane.  Here the interpretation plane (IP) is a plane passing through the 3D line and the center of projection (viewpoint) of the camera.  Also, the interpretation planes of two parallel lines intersect in a line passing through the viewpoint, and the intersection line is parallel to the parallel lines.

algebraic calculation Assignment Help

If you select to use algebraic calculation, you may use the parametric representation of a 3D line: P = P0 +tV, where P= (X,Y,Z)T is any point on the line (here  T denote for transpose),   P0 = (X0,Y0,Z0)T is a given fixed point on the line, vector V = (a,b,c)T represents the direction of the line, and t is the scalar parameter that controls the distance (with sign) between P and P0.

algebraic calculation 2. Show that relation between any image point (xim, yim)T  (in the form of (x1,x2,x3)T in projective space ) of a planar surface in 3D space and its corresponding point (Xw, Yw, Zw)T on the plane in 3D space can be represented by a 3×3 matrix. You should start from the general form of the camera model (x1,x2,x3)T = MintMext(Xw, Yw, Zw, 1)T, where the image center (ox, oy), the focal length f, the scaling factors( sx and sy),  the rotation matrix R and the translation vector T are all unknown. Note that in the course slides and the lecture notes, I used a simplified model of the perspective project by assuming ox and oy are known and sx = sy =1, and only discussed the special cases of a plane. So you cannot directly copy those equations I used.  Instead you should use the general form of the projective matrix, and the  general form of a plane nx Xw + ny Yw + nz Zw  = d. 

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