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Triangulation

Triangulation recovers a 3D point from its projections in two or more calibrated views. Theia implements several algebraic and multi-view refinements on top of classic DLT and midpoint closures. In pyTheia these live under pytheia.sfm, alongside RANSAC-wrapped robust triangulation for noisy correspondences.

For Python entry points and NumPy types, see Python API overview. C++ sources: src/theia/sfm/triangulation/.

Concepts

  • Projection matrices \(P_i \in \mathbb{R}^{3 \times 4}\) map homogeneous 3D points \(X\) to image pixels (or normalized coordinates, depending on how you built \(P\)). Two-view helpers assume consistent conventions with the rest of Theia (see fundamental_matrix_utils if you only have \(E\) or \(F\)).
  • Output is a homogeneous 4-vector \([x, y, z, w]^\top\); Euclidean 3D is \((x/w,\, y/w,\, z/w)\) when \(w \neq 0\).
  • Cheirality: a candidate should lie in front of the cameras that observe it. Use the library’s cheirality / angle checks before accepting a point.

Closed-form and multi-view methods

API (Python module pt.sfm) Role
Triangulate Two-view triangulation (“Triangulation Made Easy”, Lindstrom CVPR 2010).
TriangulateDLT Two-view DLT (Hartley & Zisserman).
TriangulateMidpoint Fast midpoint of rays; unit directions from each camera center.
TriangulateNViewSVD \(N\)-view SVD triangulation; approximately minimizes algebraic error.
TriangulateNView \(N\)-view L2 algebraic minimization; scalable and often better reprojection error than SVD alone.
EstimateTriangulation RANSAC over two-view samples; inputs: RansacParameters, list of Camera, list of 2D points (Vector2d / NumPy); returns: success, triangulated point, RansacSummary.

C++ declarations: triangulation.h, robust wrapper estimators_wrapper.h.

Quality checks

  • IsTriangulatedPointInFrontOfCameras — Given a FeatureCorrespondence (two-image match) and a relative rotation / translation between the two views, tests whether the implied structure is in front of both cameras.
  • SufficientTriangulationAngle — Given unit bearing directions from several views, returns whether the maximum angular baseline between any pair of rays is large enough (parameter: minimum angle in degrees). Use this to reject ill-conditioned points before or after bundle adjustment.

TriangulationMethodType (MIDPOINT / SVD / L2_MINIMIZATION) appears in reconstruction / track logic where multiple backends are selectable.

Typical use in a pipeline

  1. Estimate relative or absolute poses and build Camera objects with correct intrinsics.
  2. For each track, collect 2D observations and corresponding projection matrices (or camera centers + calibrated rays for TriangulateMidpoint).
  3. Triangulate (or EstimateTriangulation if matches are noisy).
  4. Filter with cheirality and triangulation angle, then refine with Bundle adjustment.

See also

  • Pose — two-view geometry and projection utilities
  • SfM — Reconstruction, Track, View
  • Bundle adjustment — joint refinement of structure and motion