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1. Projection matrix has a null vector.

• Projection matrix P is a 3 times 4 matrix, so it must have a null vector. This null vector is the center C of the camera.
• For some point A, there exists a point X on the line passing through points A and C. In other words, there is a lambda in mathbb{R} such that X=C+lambdaA. Let x be the image of X, then x=PX=P(C+lambdaA)=PC+lambdaPA=lambdaPA.
• All the points on the line passing through points A and C are projected on the same image x.
• If A moves, then the image x is also moved.
• C satisfied with this property is the only one, the center of the camera.

2. Each column of projection matrix represents the specific image.

• Let P=[  p_1  |  p_2  |  p_3 | p_4  ] where p_i is a column vector.
• In world coordinate, the points which are located on x, y, and z-axis at infinity are (1,0,0,0)^t, (0,1,0,0)^t, and (0,0,1,0)^t.
• The images of these points are P(1,0,0,0)^t=p_1, P(0,1,0,0)^t=p_2, and P(0,0,1,0)^t=p_3.
• The image of the origin in world coordinate is P(0,0,0,1)^t=p_4.
• To sum up, p_1, p_2, and p_3 are the images of  points on x, y, and z-axis at infinity. p_4 is the image of the origin in world coordinate.

3. Each row of projection matrix represents the specific plane.

• Let P=[  p^{1t}  |  p^{2t}  |  p^{3t}  ]^t where p^{i} is a row vector.
• For the point X on the principle plane which is parallel with the image plane and passes through the camera center, PX=(x,y,0)^t form.
• Since the line passing through points C and X is parallel with the image plane, this line intersects with the image plane at infinity whose image is also at infinity.

• p^{3t}X=0, which means that X is on the plane p^{3}.
• In the same way, the point Y such that PY=(0,y,w)^t is on the plane p^{1} and the point Z such that PZ=(x,0,w)^t is on the plane p^{2}.

• p^{1} and p^{2} are called axis planes, and p^{3} is called the principle plane.

4. Principle point

• Principle axis is the line passing through point C and perpendicular to the principle plane. So p^{3} is the normal vector of the principle plane.
• The point q in the normal direction of the principle plane at infinity is q=(p_{11}^{3t}, p_{21}^{3t} , p_{31}^{3t}, 0)^t where  p_{ij}^{3t} is the (i, j) element of p^{3}.
• The image of q is called principle point.

5. Backprojection

• For an image x, its world points is P^+x where P^+=P^t(PP^t)^{-1}.
• P(P^+x)=P(P^t(PP^t)^{-1}x)=PP^t(PP^t)^{-1}x=x.
• As rank(P)=3, PP^t is full-rank. So, PP^t is invertible.
• For the ray passing through point C and the image x, an arbitrary point X on this ray can be represented as a linear combination such that X=C+lambdaP^+x.

• Reference

[1] Hartley, R. and Zisserman, A. (2003) Multiple View Geometry in Computer Vision. 2nd Edition, Cambridge University Press, Cambridge.

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