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Image Points and image lines

  • Points

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


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:


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


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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