Properties of Projection Matrix

1. Projection matrix has a null vector.

NullVector

  • Projection matrix PP is a 3×43 × 4 matrix, so it must have a null vector. This null vector is the center CC of the camera.
  • For some point AA, there exists a point XX on the line passing through points AA and CC. In other words, therer is a λR\lambda \in \R such that X=C+λAX = C + \lambda A. Let xx be the image of XX, then x=PX=P(C+λA)=PC+λPA=λPAx = PX = P(C + \lambda A) = PC + \lambda PA = \lambda PA.
  • All the points on the line passing through points AA and CC are projected on the same image xx.
  • If AA moves, then the image xx is moved as well.
  • CC 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=[p1p2p3p4]P = [p_1 \vert p_2 \vert p_3 \vert p_4] where pip_i is a column vector.
  • In world coordinates, the points which are located on xx, yy, and zz-axis at infinity are (1,0,0,0)t(1, 0, 0, 0)^t, (0,1,0,0)t(0, 1, 0, 0)^t, and (0,0,1,0)t(0, 0, 1, 0)^t.
  • The images of these points are P(1,0,0,0)t=p1P(1, 0, 0, 0)^t = p_1, P(0,1,0,0)t=p2P(0, 1, 0, 0)^t = p_2, and P(0,0,1,0)t=p3P(0, 0, 1, 0)^t = p_3.
  • The image of the origin in world coordinates is P(0,0,0,1)t=p4P(0, 0, 0, 1)^t = p_4,
  • To sum up, p1p_1, p2p_2, and p3p_3 are the images of points on xx, yy, and zz-axis at infinity. p4p_4 is the image of the origin in world coordinates.

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

RowVector

  • Let P=[p1tp2tp3t]tP = [p^{1t} \vert p^{2t} \vert p^{3t}]^t where pip^i is a row vector.
  • For the point XX on the principle plane which is parallel with the image plane and passes through the camera center, PX=(x,y,0)tPX = (x, y, 0)^t form.
  • Since the line passing through points CC and XX is parallel with the image plane, this line intersects with the image plane at infinity whose image is also at infinity.
  • p3tX=0p^{3t}X = 0, which means that XX is on the plane p3p^3.
  • In the same way, the point YY such that PY=(0,y,w)tPY = (0, y, w)^t is on the plane p1p^1 and the point ZZ such that PZ=(x,0,w)tPZ = (x, 0, w)^t is on the plane p2p^2.

RowPlane

  • p1p^1 and p2p^2 are called axis planes, and p3p^3 is called the principle plane.

4. Principle point

  • Principle axis is the line passing through point CC and perpendicular to the principle plane. So p3p^3 is the normal vector of the principle plane.
  • The point qq in the normal direction of the principle plane at infinity is q=(p113t,p213t,p313t,0)tq = (p^{3t}_{11}, p^{3t}_{21}, p^{3t}_{31}, 0)^t where pij3tp^{3t}_{ij} is the (i,j)(i, j) element of p3p^3.
  • The image of qq is called principle point.

5. Backprojection

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