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N-view triangulation: DLT homogeneous method 1 (1).

I’ve become interested in N-view triangulation methods recently, so I started with the most straightforward of them: the DLTs, of Direct Linear Transformation algorithms. It turns out there are three variants on the DLT algorithm for N-view triangulation.

The first DLT algorithm I’ll describe is a homogeneous DLT algorithm, from a 2004 CVPR paper by Richard Hartley and Frederick Schaffalitzky, HS2004. Don’t get too excited about DLTs for final solutions though; in R. Hartley’s papers, comments such as these are common:

Though this method of triangulation may seem attractive, the cost function that it is minimizing has no particular meaning, and the method is not reliable.

Let

Then the method mentioned in HS2004 is as follows.

The relationship of \(\mathbf{x} = \mathbf{PX}\) for the \(n\) correspondences is in terms of homogeneous coordinates, and can be written with the unknown scale factor \(k_i\),

\[k_i\mathbf{x} = \mathbf{PX}\]

Rewriting,

\[\mathbf{PX} - \mathbf{x}k_i = 0\]

Then, this relationship is linear in the unknowns \(\mathbf{X}\), \(k_i\)s,

\[\begin{bmatrix} \mathbf{P}_1 & \mathbf{x}_1 & & & \\ \mathbf{P}_2 & & \mathbf{x}_2 & &\\ \vdots & & & \ddots & \\ \mathbf{P}_n & & & & \mathbf{x}_n \end{bmatrix} \begin{bmatrix} \mathbf{X}\\ -k_1\\ -k_2\\ \vdots\\ -k_n\end{bmatrix} = \mathbf{0}\]

I use SVD to solve these type of \(\mathbf{AX} = 0\) problems. The solution with norm equal to 1 is the last column of the V matrix from SVD. The triangulated point is the first four rows of the last column of the V matrix, and remember to normalize by the fourth element such that

\[\mathbf{X} = \begin{bmatrix} X\\ Y\\ Z\\ 1 \end{bmatrix}\]

There’s two more versions, continue to Page 2, and a discussion of normalization relevant to this method on Page 3.

Listing of N-view Triangulation posts.

References

[HS2004] Richard Hartley and Frederick Schaffalitzky, “L\(_\infty\) minimization in geometric reconstruction problems,” CVPR 2004. link

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