Direct linear transformation

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

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

where ${\displaystyle \mathbf {x} _{k}}$ and ${\displaystyle \mathbf {y} _{k}}$ are known vectors, ${\displaystyle \,\propto }$ denotes equality up to an unknown scalar multiplication, and ${\displaystyle \mathbf {A} }$ is a matrix (or linear transformation) which contains the unknowns to be solved.

This type of relation appears frequently in projective geometry. Practical examples include the relation between 3D points in a scene and their projection onto the image plane of a pinhole camera,^[1] and homographies.