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Image Points and image lines

12/04/2011
  • Points

Suppose an image point on the normalized image plane has coordinates as (x,y) (in pixels or meters). Then its homogeneous coordinate is

m=[x,y,1]^T

Note the third coordinate is 1. No need to explain.

  • Lines

Given an image point m=[x,y,1]^T, what is the line passing through this point? Let’s consider the line equation in a 2D plane. For point (x,y), the line equation is:

ax+by+c=0

Write l=[a,b,c]^T, then the above equation is

l^Tm=0

So given an image point m with homogeneous coordinates, we can use l to denote the line.Then we can say l is a line on the image plane. Note
1. there are infinite number of lines pass through one point. This is obviously.
2. Once a,b,c is determined, the line is determine.
3. l and kl are the same line.

  • Is a point on a line?

Given an image point m and an image line l, it is clear that the point locates on the line if and only if

l^T m=0

  • Point determined by two lines

Let l_1 and l_2 be two image lines. What is the intersection points of these two lines? It is easy to derive. If the intersection point is m, then l_1^Tm=0 and l_2^Tm=0. So

m\sim l_1\times l_2

\sim denotes equality up to a scalar factor. Note the third element of l_1\times l_2 should be normalized to 1.

  • Line determined by two points

Given two image points m_1 and m_2, the line pass the two points is

l=m_1 \times m_2

Note here is = instead of ~ since we need not to normalize.

  • Distance between a line and a point

Given an image point m and an image line l, what is the distance between the line and point? I guess the distance is

d=\frac{l^T m}{l^Tl}

When the point is on the line, the above equation d=0; whether it is correct when point is not on the line. I have not proved yet. Edit: Wrong! Consider a line in 2D ax+by+c=0, then the distance between a point (x_0,y_0) and this line is d= \frac{|ax_0+by_0+c|}{\sqrt{a^2+b^2}}. Obviously this is not the same as the above equation.

Finally, it is worthwhile to note all the points and lines discussed here are limited to the normalized image plane!!!

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