Direct linear transformation

Direct linear transformation (DLT) is an algorithm which solves a set of variables from a set of similarity relations: Direct linear transformation (DLT) is an algorithm which solves a set of variables from a set of similarity relations: 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 projective geometry. Practical examples include the relation between 3D points in a scene and their projection onto the image plane of a pinhole camera, and homographies. An ordinary system of linear equations can be solved, for example, by rewriting it as a matrix equation X = A Y {displaystyle mathbf {X} =mathbf {A} ,mathbf {Y} } where matrices X {displaystyle mathbf {X} } and Y {displaystyle mathbf {Y} } contain the vectors x k {displaystyle mathbf {x} _{k}} and y k {displaystyle mathbf {y} _{k}} in their respective columns. Given that there exists a unique solution, it is given by