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How to triangulate two 3d lines

  1. Feb 21, 2016 #1
    Hi guys,
    I am working on an computer vision project.
    the project uses two cameras to triangulate an object in front of the cameras.


    1. The problem statement, all variables and given/known data
    Express the object's location in 3d coordinates relative to the cameras.

    2. Relevant equations
    From the software i can get two line equations.
    The line equations are for lines going through the centers of the lens and the center of the object itself
    1: a1x1+b1y1+c1=0
    2: a2x2+b2y2+c2=0
    Known parameters are the a1,a2,b1,b2,c1,c2
    Also know is the relation between the two cameras (The Fundamental matrix 3X3)
    Also, the object coordinates in 2d on a projection plane from each camera is known (i.e. x1,y1 and x2,y2 )

    Sadly, the lines do not meet, so its also needed to calculate the shortest line that connects both lines and treat its middle as the object's center.

    3. The attempt at a solution
    how to calculate a vector that originates from the middle of the two cameras, to the object.
     
  2. jcsd
  3. Feb 21, 2016 #2
    Let's say that your original coordinate system is ##(O,\vec i,\vec j,\vec k) ##
    You want to set a camera in ##O'## and ##O''##, and set a coordinate system in ##O'## and ##O''##.

    If ##M## has coordinates ##(x,y,z)## in the original coordinate system, its coordinates ##(x',y',z')## in ##(O',\vec I,\vec J,\vec K) ## satisfy

    ## \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \vec{OO'} + P . \begin{pmatrix} x' \\ y' \\ z' \end{pmatrix} ##

    where ##P## is the change of basis matrix from ##(\vec i,\vec j,\vec k)## to ##(\vec I,\vec J,\vec K) ##. Similarly for the other camera.
     
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