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

    for 

where  and  are known vectors,  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, and homographies. Introduction 

An ordinary system of linear equations

    for 

can be solved, for example, by rewriting it as a matrix equation  where matrices  and  contain the vectors  and  in their respective columns. Given that there exists a unique solution, it is given by

 

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 which is dependent on k.  As a consequence,  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 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. Example 

Suppose that . Let  and  be two known vectors, and we want to find the  matrix  such that

 

where  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  from the left

Since  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  and matrix :

 ,   ,   and   

and the above homogeneous equation becomes

    for 

This can also be written in the matrix form:

   for 

where  and  both are 6-dimensional vectors defined as

    and   

So far, we have 1 equation and 6 unknowns. A set of homogeneous equations can be written in the matrix form

where  is a  matrix which holds the known vectors  in its rows. The unknown  can be determined, for example, by a singular value decomposition of ;  is a right singular vector of  corresponding to a singular value that equals zero. Once  has been determined, the elements of matrix  can rearranged from vector . Notice that the scaling of  or  is not important (except that it must be non-zero) since the defining equations already allow for unknown scaling. In practice the vectors  and  may contain noise which means that the similarity equations are only approximately valid. As a consequence, there may not be a vector  which solves the homogeneous equation  exactly. In these cases, a total least squares solution can be used by choosing  as a right singular vector corresponding to the smallest singular value of

More general cases 

The above example has  and , but the general strategy for rewriting the similarity relations into homogeneous linear equations can be generalized to arbitrary dimensions for both  and 

If  and  the previous expressions can still lead to an equation

    for   

where  now is   Each k provides one equation in the  unknown elements of  and together these equations can be written  for the known  matrix  and unknown 2q-dimensional vector   This vector can be found in a similar way as before. In the most general case  and . The main difference compared to previously is that the matrix  now is  and anti-symmetric. When  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

    for      and for 

where  is a M-dimensional basis of the space of  anti-symmetric matrices. Example p = 3 

In the case that p = 3 the following three matrices  can be chosen

 ,   ,   

In this particular case, the homogeneous linear equations can be written as

    for   

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 . However, since  has rank = 2, at most two equations are linearly independent. In practice, therefore, it is common to only use two of the three matrices , for example, for m=1, 2. However, the linear dependency between the equations is dependent on , which means that in unlucky cases it would have been better to choose, for example, m=2,3. As a consequence, if the number of equations is not a concern, it may be better to use all three equations when the matrix  is constructed. 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  or by allowing  to become larger than necessary for determining

References

External links 
 Homography Estimation by Elan Dubrofsky (§2.1 sketches the "Basic DLT Algorithm")

 A DLT Solver based on MATLAB by Hsiang-Jen (Johnny) Chien

Geometry in computer vision
Projective geometry