We address two problems. First, reconstructing a sphere of a prescribed radius from a single calibrated view of its occluding contour. Second, reconstructing simultaneously a sphere of a prescribed radius and the camera focal length from a single view of the sphere’s occluding contour. A sphere’s occluding contour generally appears as an ellipse and existing reconstruction methods use ellipse fitting, thus requiring <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\ge 5$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>≥</mo> <mn>5</mn> </mrow> </math></EquationSource> </InlineEquation> contour points. The calibrated minimal solution requires 3 points, and a few methods can deal with it. The minimal solution with an unknown focal length requires 4 points, and there exists no method to deal with it. All existing methods share two shortcomings: (i) they fail for non-elliptic occluding contours, including parabola and hyperbola, and (ii) they use the point-to-ellipse distance, whose computation is not closed-form. On the first problem, we make the observation that the spherically-normalised contour points form a circle in space, which we reconstruct by plane fitting. This handles minimal 3-point and redundant <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$>3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>></mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation> point fitting, copes with elliptic and non-elliptic contours, and benefits from the simple point-to-plane distance. The reconstructed circle then leads to a one-parameter sphere family from which the actual sphere of prescribed radius is uniquely retrieved. We name our method <Emphasis FontCategory="NonProportional">SpherO</Emphasis>, where letter ‘O’ depicts a circle. We robustify <Emphasis FontCategory="NonProportional">SpherO</Emphasis> using random sampling at the 3-point plane fitting stage. Experimental comparisons show that <Emphasis FontCategory="NonProportional">SpherO</Emphasis> outperforms the current-best 3-point method. On the second problem, we make the observation that the spherically-normalised contour points generally form a non-circular spatial elliptic curve for wrong camera parameters. The calibration constraint is thus that the spherically-normalised points must be cocircular, which implies coplanarity. The coplanarity constraint allows us to solve the minimal 4-point case. We solve redundant <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$>4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>></mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation> point case by fitting planes. This simultaneously reconstructs a circle and the camera focal length from a non-circular spatial elliptic curve. Finally, the reconstructed circle and the camera focal length allow us to retrieve the sphere of prescribed radius. We name our method <Emphasis FontCategory="NonProportional">SpherOf</Emphasis>, where letter ‘<Emphasis FontCategory="NonProportional">f</Emphasis>’ is for the focal length. We robustify <Emphasis FontCategory="NonProportional">SpherOf</Emphasis> using random sampling at the 4-point coplanarity constraint formation and 3-point plane fitting stages. Experiments show that <Emphasis FontCategory="NonProportional">SpherOf</Emphasis> has comparable performance to <Emphasis FontCategory="NonProportional">SpherO</Emphasis>.

Reconstructing a Sphere and the Camera Focal Length from a Single View by Fitting Planes

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发表日期：

2026年3月6日

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Erol Ozgur,
Mohammad Alkhatib,
Youcef Mezouar,
Adrien Bartoli

摘要

We address two problems. First, reconstructing a sphere of a prescribed radius from a single calibrated view of its occluding contour. Second, reconstructing simultaneously a sphere of a prescribed radius and the camera focal length from a single view of the sphere’s occluding contour. A sphere’s occluding contour generally appears as an ellipse and existing reconstruction methods use ellipse fitting, thus requiring $\ge 5$ $\geq 5$ contour points. The calibrated minimal solution requires 3 points, and a few methods can deal with it. The minimal solution with an unknown focal length requires 4 points, and there exists no method to deal with it. All existing methods share two shortcomings: (i) they fail for non-elliptic occluding contours, including parabola and hyperbola, and (ii) they use the point-to-ellipse distance, whose computation is not closed-form. On the first problem, we make the observation that the spherically-normalised contour points form a circle in space, which we reconstruct by plane fitting. This handles minimal 3-point and redundant $>3$ $> 3$ point fitting, copes with elliptic and non-elliptic contours, and benefits from the simple point-to-plane distance. The reconstructed circle then leads to a one-parameter sphere family from which the actual sphere of prescribed radius is uniquely retrieved. We name our method SpherO, where letter ‘O’ depicts a circle. We robustify SpherO using random sampling at the 3-point plane fitting stage. Experimental comparisons show that SpherO outperforms the current-best 3-point method. On the second problem, we make the observation that the spherically-normalised contour points generally form a non-circular spatial elliptic curve for wrong camera parameters. The calibration constraint is thus that the spherically-normalised points must be cocircular, which implies coplanarity. The coplanarity constraint allows us to solve the minimal 4-point case. We solve redundant $>4$ $> 4$ point case by fitting planes. This simultaneously reconstructs a circle and the camera focal length from a non-circular spatial elliptic curve. Finally, the reconstructed circle and the camera focal length allow us to retrieve the sphere of prescribed radius. We name our method SpherOf, where letter ‘f’ is for the focal length. We robustify SpherOf using random sampling at the 4-point coplanarity constraint formation and 3-point plane fitting stages. Experiments show that SpherOf has comparable performance to SpherO.