**Properties of Projection Matrix**

**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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