Direct linear transformation

Direct linear transformation (DLT) is an algorithm which solves a set of variables from a set of similarity relations:

x
          
          
            k
          
        
        ∝
        
          A
        
        
        
          
            y
          
          
            k
          
        
      
    
    {\displaystyle \mathbf {x} _{k}\propto \mathbf {A} \,\mathbf {y} _{k}}
  
   for 
  
    
      
        
        k
        =
        1
        ,
        …
        ,
        N
      
    
    {\displaystyle \,k=1,\ldots ,N}

where 
 
 
 
 
 
 x
 
 
 k
 
 
 
 
 {\displaystyle \mathbf {x} _{k}}
 
 and 
 
 
 
 
 
 y
 
 
 k
 
 
 
 
 {\displaystyle \mathbf {y} _{k}}
 
 are known vectors, 
 
 
 
 
 ∝
 
 
 {\displaystyle \,\propto }
 
 denotes equality up to an unknown scalar multiplication, and 
 
 
 
 
 A
 
 
 
 {\displaystyle \mathbf {A} }
 
 is a matrix (or linear transformation) which contains the unknowns to be solved.
This type of relation appears frequently in <a href="/facts/Projective_geometry/rHKSaEbx">projective geometry</a>. Practical examples include the relation between 3D points in a scene and their projection onto the image plane of a <a href="/facts/Pinhole_camera_model/Y0Jn9P9s">pinhole camera</a>, and <a href="/facts/Homography_(computer_vision)/hE8Pc5Dx">homographies</a>.

Direct linear transformation open-in-new

Direct linear transformation