We investigate combinations of 3D-2D matches of points and lines for a single calibrated camera that imply a nontrivial constraint on the camera’s position. Specifically, we address the minimal cases for which n point matches and m line matches constrain the camera center to lie on an algebraic surface in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathbb {R}^3$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>3</mn> </msup> </math></EquationSource> </InlineEquation>. For two points <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$(m,n) = (0,2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, the constraint is well-known to be a self-intersecting torus. We complete the classification problem by deriving explicit, general equations for the cases <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$(m,n) \in \{ (2,0), (1, 1) \}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mo stretchy="false">{</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> for two lines and one point, one line, respectively. We also report preliminary experiments investigating the suitability of these constraints for pose estimation tasks.

Fundamental Constraints on Camera Centers Arising from 3D-2D Matches of Points and Lines

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2026年1月29日

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Jaired Collins,
Taci Ata Kucukpinar,
Timothy Duff,
Joshua Fraser,
Guna Seetharaman,
Kannappan Palaniappan

摘要

We investigate combinations of 3D-2D matches of points and lines for a single calibrated camera that imply a nontrivial constraint on the camera’s position. Specifically, we address the minimal cases for which n point matches and m line matches constrain the camera center to lie on an algebraic surface in $\mathbb {R}^3$ $R^{3}$ . For two points $(m,n) = (0,2)$ $(m, n) = (0, 2)$ , the constraint is well-known to be a self-intersecting torus. We complete the classification problem by deriving explicit, general equations for the cases $(m,n) \in \{ (2,0), (1, 1) \}$ $(m, n) \in {(2, 0), (1, 1)}$ for two lines and one point, one line, respectively. We also report preliminary experiments investigating the suitability of these constraints for pose estimation tasks.