Direct linear transformation

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

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

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

This type of relation appear frequently in projective geometry. Practical examples are the relation between 3D points in a scene and their projection onto the image plane of a pinhole camera and homographies.

Introduction

An ordinary linear equation

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

can be solved, for example, by rewriting it as a matrix equation $\mathbf {X} =\mathbf {A} \,\mathbf {Y}$ where matrices $mathbf{X}$ and $mathbf{Y}$ contain the vectors $\mathbf {x} _{k}$ and $\mathbf {y} _{k}$ in their respective columns. Given that there exists a unique solution, it is given by

$\mathbf {A} =\mathbf {X} \,\mathbf {Y} ^{T}\,(\mathbf {Y} \,\mathbf {Y} ^{T})^{-1}$

Solutions can also be described in the case that the equations are over or under determined.

What makes the direct linear transformation problem distinct from the above standard case is the fact that the left and right sides of the defining equation can differ by an unknown multiplicative factor. As a consequence, $\mathbf {A}$ cannot be computed as in the standard case. Instead, the similarity relations are rewritten as proper linear homogeneous equations which then can be solved by standard method. The combination of rewriting the similarity equations as homogeneous equations and solving them by standard methods is referred to as a direct linear transformation algorithm or DLT algorithm.