Linear Triangulation

Triangulation reconstructs a 3D point from its projections in two or more calibrated views. It is the inverse of the projection operation: given pixel observations and camera poses, recover the world point.

Problem Statement

Given: $n\ge 2$ camera projection matrices $\{P_i\}$ (each $3\times 4$) and corresponding pixel observations $\{{\mathbf{p}}_i=(u_i,v_i)\}$.

Find: 3D point $\mathbf{X}=[X,Y,Z{]}^T$ such that ${\mathbf{p}}_i\sim P_i[\mathbf{X},1{]}^T$.

Assumptions:

• Camera poses and intrinsics are known
• Correspondences are correct
• The point is visible from all views (not occluded)

Derivation (DLT)

For each view $i$, the projection constraint ${\mathbf{p}}_i\sim P_i{\mathbf{X}}_h$ gives (by cross-multiplication):

$$u_i({\mathbf{p}}_3^{(i)T}{\mathbf{X}}_h)-{\mathbf{p}}_1^{(i)T}{\mathbf{X}}_h=0$$ $$v_i({\mathbf{p}}_3^{(i)T}{\mathbf{X}}_h)-{\mathbf{p}}_2^{(i)T}{\mathbf{X}}_h=0$$

where ${\mathbf{p}}_k^{(i)T}$ is the $k$-th row of $P_i$ and ${\mathbf{X}}_h=[X,Y,Z,1{]}^T$.

Stacking all views gives a $2n\times 4$ system:

$$A{\mathbf{X}}_h=0$$

Solve via SVD: ${\mathbf{X}}_h$ is the right singular vector corresponding to the smallest singular value. Dehomogenize: $\mathbf{X}=[X_h/w,Y_h/w,Z_h/w{]}^T$.

Limitations

• No uncertainty modeling: The DLT minimizes algebraic error, not geometric (reprojection) error
• Baseline sensitivity: For small baselines (nearly parallel views), the triangulation is poorly conditioned — small pixel errors lead to large depth errors
• Outlier sensitivity: A single incorrect correspondence corrupts the result; no robustness mechanism

For high-accuracy 3D reconstruction, triangulation should be followed by bundle adjustment that jointly optimizes points and cameras to minimize reprojection error.

API

#![allow(unused)]
fn main() {
let X = triangulate_point_linear(&projection_matrices, &pixel_points)?;
}

Where each projection matrix $P_i=K_i[R_i|{\mathbf{t}}_i]$ is precomputed from the camera intrinsics and pose.