Triangulation

Definition

Triangulation: given correspondences and known camera poses, find the 3D point. With perfect observations the viewing rays intersect exactly at $X$; with noise we look for the point with minimal distance to all rays.

Linear triangulation

$u\simeq PX\Rightarrow u\times PX=0$. With $P_k^T$ the rows of $P$, each camera gives two linear equations:

$$\left[\begin{array}{c}y{P}_3^T-P_2^T\\ P_1^T-x{P}_3^T\\ y^{\prime }{P}_3^{\prime T}-P_2^{\prime T}\\ P_1^{\prime T}-x^{\prime }{P}_3^{\prime T}\end{array}\right]X=0$$

Solved with SVD in the least squares sense.

The inverse problem is pose estimation (PnP): known 3D points and 2D-3D correspondences, unknown camera pose. In SfM, triangulation follows the motion estimation.

Appears in

• Lecture 3, Triangulation
• Lecture 3, Three components of SfM
• Lecture 1, History: stereo vision (70s-80s)

Related concepts

• Structure from Motion
• Epipolar Geometry
• Depth Ambiguity