Direct linear transformation (DLT) is an algorithm which solves a set of variables from a set of similarity relations: where
denotes equality up to an unknown scalar multiplication, and
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.
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 which is dependent on k. As a consequence,
Instead, the similarity relations are rewritten as proper linear homogeneous equations which then can be solved by a standard method.
The combination of rewriting the similarity equations as homogeneous linear equations and solving them by standard methods is referred to as a direct linear transformation algorithm or DLT algorithm.
DLT is attributed to Ivan Sutherland.
is the unknown scalar factor related to equation k. To get rid of the unknown scalars and obtain homogeneous equations, define the anti-symmetric matrix and multiply both sides of the equation with
the following homogeneous equations, which no longer contain the unknown scalars, are at hand In order to solve
from this set of equations, consider the elements of the vectors
: and the above homogeneous equation becomes This can also be written in the matrix form: where
both are 6-dimensional vectors defined as So far, we have 1 equation and 6 unknowns.
A set of homogeneous equations can be written in the matrix form where
can be determined, for example, by a singular value decomposition of
is not important (except that it must be non-zero) since the defining equations already allow for unknown scaling.
may contain noise which means that the similarity equations are only approximately valid.
In these cases, a total least squares solution can be used by choosing
, but the general strategy for rewriting the similarity relations into homogeneous linear equations can be generalized to arbitrary dimensions for both
the previous expressions can still lead to an equation where
The main difference compared to previously is that the matrix
the space of such matrices is no longer one-dimensional, it is of dimension This means that each value of k provides M homogeneous equations of the type where
can be chosen In this particular case, the homogeneous linear equations can be written as where
is the matrix representation of the vector cross product.
Notice that this last equation is vector valued; the left hand side is the zero element in
Each value of k provides three homogeneous linear equations in the unknown elements of
has rank = 2, at most two equations are linearly independent.
, which means that in unlucky cases it would have been better to choose, for example, m=2,3.
The linear dependence between the resulting homogeneous linear equations is a general concern for the case p > 2 and has to be dealt with either by reducing the set of anti-symmetric matrices