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Wikipedia / Direct linear transformation

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

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

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 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

such that

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

from the left

the following homogeneous equations, which no longer contain the unknown scalars, are at hand

:

  1. , , and

and the above homogeneous equation becomes

This can also be written in the matrix form:

both are 6-dimensional vectors defined as

  1. and

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

can be determined, for example, by a singular value decomposition is not important (except that it must be non-zero) since the defining equations already allow for unknown scaling.

exactly. In these cases, a total least squares

More general cases

the previous expressions can still lead to an equation

  1. for

Each k and unknown 2q This vector can be found in a similar way as before.

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

  1. for

is a M anti-symmetric matrices.

Example p = 3

In the case that p can be chosen

  1. , ,

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

  1. for

is the matrix representation of the vector cross product.

Each value of k, for example, for m, which means that in unlucky cases it would have been better to choose, for example, m is constructed.

The linear dependence between the resulting homogeneous linear equations is a general concern for the case p

References