# Why does direct linear transform (DLT) yield a unique solution?

I try to triangulate point correspondences from 2 images in order to reconstruct the 3D positions of those points. I found the DLT method as an easy way to achieve that. The system which needs to be solved is shown here on page 23

whereas $$\mathit{x}, \mathit{y}, \mathit{x'}, \mathit{y'}$$ are the projected image points,$$\pmb{p_i^T}$$ are the rows of the 3x4 camera matrix P. Having 2 point correspondences from two views leads to the matrix $$A$$. So the matrix $$A$$ is a 4x4 matrix. What we want to find is the 3D point $$X$$ which is in homogeneous coordinates. According to the literature a unique solution for this can be found using SVD. But I don't understand why a unique solution for $$X$$ can be found at all. Since this is a homogeneous linear system, it should either have the trivial solution $$X=\overrightarrow{0}$$ or infinitely many. What am I missing here?

• I did not read the book, but homogeneous coordinates notoriously represent a point by a line. Which is logical, because we add a dimension (3D $\rightarrow$ 4D). So finding a dim 1 solution space is not a surprise. Commented Aug 24, 2023 at 15:17
• Well but then A should possess a 1-dimensional null-space and X can only be determined up to a non-zero scale factor (infinitely many solutions). So no unique solution.
– NMO
Commented Aug 25, 2023 at 9:36
• Yes, precisely. When representing a point in $n$ dimensions by homogeneous coordinates in $n+1$ dimensions, those homogeneous coordinates are up to a non-zero scale factor. A point is represented by a line. Cf. this Wikipedia page : en.wikipedia.org/wiki/Homogeneous_coordinates : "If homogeneous coordinates of a point are multiplied by a non-zero scalar then the resulting coordinates represent the same point". Commented Aug 25, 2023 at 11:46
• I understand. So the 4th component of X is a scaling factor right?
– NMO
Commented Aug 30, 2023 at 8:52
• Yes, that's it. Note that this representation has the additional benefit that you can represent points at infinity: these are points where this 4th component is zero. In this geometry, all conics (ellipse, parabola, hyperbola) are the same closed curve. Commented Aug 30, 2023 at 9:43

When representing a point in $$n$$ dimensions by homogeneous coordinates in $$n+1$$ dimensions, those homogeneous coordinates are up to a non-zero scale factor. A point is represented by a line. Quoting Wikipedia page on homogeneous coordinates: "If homogeneous coordinates of a point are multiplied by a non-zero scalar then the resulting coordinates represent the same point".