Triangulation of Points Constrained to a Plane

TL;DR

This paper derives a formula for the algebraic complexity of planar triangulation from multiple views and demonstrates that incorporating planar constraints improves reconstruction speed and accuracy.

Contribution

It provides a new formula for the number of critical points in planar triangulation and validates the benefits of planar constraints through experiments.

Findings

Derived a formula for the number of complex critical points in planar triangulation.

Incorporating planar constraints leads to faster and more accurate point reconstruction.

Validated theoretical results with experiments on synthetic and real data.

Abstract

We study the set of image tuples arising from fixed cameras observing varying planar 3-dimensional point configurations. We derive a formula for the number of complex critical points of the triangulation problem, which seeks to reconstruct such configurations from noisy image data. Valid for an arbitrary number of views, this formula quantifies the intrinsic algebraic complexity of planar triangulation. We validate our theoretical findings through numerical experiments on both synthetic and real data, demonstrating that incorporating the planar incidence constraints leads to faster point reconstruction and improved accuracy compared to unconstrained triangulation.