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Image Points and image lines – two viewpoints

13/04/2011

In a previous post, I talked about the image points and image lines. In this post, I will talk about how to interpret the image geometry which is different from the geometry we learned before.

  • Point of view I: good, easy

Let an image point with coordinates (x,y). A line passing the point is

ax+by+c=0

Hence we denote m=[x,y,1]^T as an image point and l=[a,b,c]^T as an image line. Then we know:
a) a point is on a line iff l^Tm=0
b) the line passing two points is l=m_1\times m_2
c) the intersection point of two lines is m\sim l_1 \times l_2

Remark: This point of view is quite easy. No need to care what happens outside the image plane.

  • Point of view II: a 3D interseption

Image point: The formula of the normalized image plane is z=1. So the 3D coordinates of a point (x,y) on the image plane is [x,y,1]^T.
Image Line: A line in the image plane can be treated as the intersection line between a plane \pi passing through the origin and plane z=1. Note any plane \pi passing through the origin is uniquely determined by its norm vector l. Hence any vector

l\Rightarrow \pi \Rightarrow intersect with plane z=1\Rightarrow an image line

Any points m on the image line must have l^Tm=0 since l is orthogonal to all vectors in the plane \pi

Remark: here we interpret them in 3D camera reference frame.

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