Direct linear transformation: Difference between revisions

Revision as of 19:33, 5 October 2020

Revision as of 19:35, 5 October 2020

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:<math> \mathbf{0} = \mathbf{B} \, \mathbf{a} </math>

where <math> \mathbf{B} </math> is a <math> N \times 6 </math> matrix which holds the known vectors <math> \mathbf{b}_{k} </math> in its rows. The unknown <math> \mathbf{a} </math> can be determined, for example, by a [[singular value decomposition]] of <math> \mathbf{B} </math>; <math> \mathbf{a} </math> is a right singular vector of <math> \mathbf{B} </math> corresponding to a singular value that equals zero. Once <math> \mathbf{a} </math> has been determined, the elements of matrix <math> \mathbf{A} </math> can be found by a simple rearrangementrearranged from a 6-dimensional vector to a <math> 2 \times 3 mathbf{a}</math> matrix. Notice that the scaling of <math> \mathbf{a} </math> or <math> \mathbf{A} </math> is not important (except that it must be non-zero) since the defining equations already allow for unknown scaling.

In practice the vectors <math> \mathbf{x}_{k} </math> and <math> \mathbf{y}_{k} </math> may contain noise which means that the similarity equations are only approximately valid. As a consequence, there may not be a vector <math> \mathbf{a} </math> which solves the homogeneous equation <math> \mathbf{0} = \mathbf{B} \, \mathbf{a} </math> exactly. In these cases, a [[total least squares]] solution can be used by choosing <math> \mathbf{a} </math> as a right singular vector corresponding to the smallest singular value of <math> \mathbf{B}. </math>