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Object Pose from 2-D to 3-D Point and Line Correspondences

Thai Quynh Phong, Radu Horaud, Adnan Yassine, Pham Dinh Tao

arXiv:2610.19734v1cs.CV

TL;DR

Estimating an object’s rotation and translation relative to a camera from point or line correspondences remains a challenging pose problem. The paper represents pose with dual number quaternions and minimizes quadratic constraints using trust-region optimization. Its method is reported as more stable than a linear method as correspondence counts vary, and it degrades more gently as mismatches increase.

  • Problem

    Estimating an object’s rotation and translation relative to a camera from 3-D points and their image projections is a nonlinear pose problem.

    Source: §1 · p. 1, §1.2 · p. 2

  • Method

    The method represents pose with dual number quaternions and minimizes the sum of squared quadratic correspondence constraints using trust-region optimization.

    Source: §1.2 · p. 3, §2 · p. 4

  • Results

    The non-linear method is more stable than the linear method as correspondence counts vary, and its results degrade gently as mismatches increase while the linear method performs badly.

    Source: §6 · pp. 11–12

  • Takeaways & Limitations

    For this pose problem, trust-region optimization can attain the global solution independently of initialization in a very small number of iterations.

    Source: §5 · p. 9

    For large-residual problems, the trust-region method requires replacing Hebden’s algorithm with a more time-consuming safeguarding algorithm.

    Source: §4.2 · p. 9

Abstract

from arXiv · show

In this paper we present a method for optimally estimating the rotation and translation between a camera and a 3-D object from point and/or line correspondences. First we devise an error function and second we show how to minimize this error function. The quadratic nature of this function is made possible by representing rotation and translation with a dual number quaternion. We provide a detailed account of the computational aspects of a trust-region optimization method. This method compares favourably with Newton's method which has extensively been used to solve the problem at hand, with Faugeras-Toscani linear method [6] for calibrating a camera, and with the Levenberg-Marquardt non-linear optimization method. Finally we present some experimental results which demonstrate the robustness of our method with respect to image noise and matching errors.

1 Introduction

The paper addresses estimating an object’s rotation and translation relative to a camera from 3-D points and their image projections, a problem with applications including camera calibration and robotics. It proposes quadratic constraints and trust-region optimization to estimate pose from point and line correspondences.

1 Introduction

  • The pose problem is to recover the rigid transformation between object- and camera-centered frames from 3-D points, their image projections, and a projection model.The paper also calls this the perspective n-point problem, exterior camera calibration, or camera location and orientation problem.Source: §1 · p. 1

1.1 Background

  • Earlier methods include closed-form solutions for limited correspondence sets, while larger sets generally require iterative numerical solutions.The paper notes closed-form cases for 3 points, 4 coplanar or general-position points, and 3 lines; with more than 6 points, iterative numerical solutions are generally required.Source: §1.1 · pp. 1–2
  • The paper argues that representation, error-function choice, and optimization method are crucial because single-view object pose is nonlinear.Prior linear methods can be noise-sensitive when rotation-matrix orthogonality constraints are not enforced, while Newton-based methods can have local-minimum or initialization and stability issues,,.Source: §1.1 · p. 2
  • Prior numerical approaches face trade-offs: linearized methods can need good initialization, and some robust outlier-handling methods are time-consuming.Liu et al. report good performance only when Euler angles are below 300; Kumar and Hanson describe a combinatorial outlier method that considers all triplets of line correspondences.Source: §1.1 · p. 2

1.2 Approach

  • Dual number quaternions represent the pose so that line-correspondence coplanarity constraints become quadratic, with two constraints per line.The paper identifies its use for 2-D/3-D exterior camera calibration as novel relative to earlier computer-vision uses for 3-D/3-D pose estimation.Source: §1.2 · pp. 2–3
  • Point correspondences also yield two quadratic constraints each, but unlike line correspondences they do not allow rotation and translation to be separated.The resulting point-based error function has the same structure as the line-based one.Source: §1.2 · p. 3
  • The method minimizes the sum of squared quadratic constraints using a trust-region method that dynamically adjusts a local quadratic approximation.The paper compares its algorithm with Newton and Levenberg–Marquardt methods.Source: §1.2 · p. 3
  • The experiments compare pose estimates from the proposed method and Faugeras–Toscani’s camera-calibration method on the same data, examining correspondence count, image noise, and matching errors.The Faugeras–Toscani method estimates both intrinsic and extrinsic parameters; its intrinsic estimates provide the camera model for the comparison.Source: §1.2 · p. 3

2 Object pose from line correspondences

Line correspondences are converted into two coplanarity constraints per match, and pose is estimated by minimizing their quadratic residuals using a dual-number-quaternion representation.

2 Object pose from line correspondences

  • Each line correspondence yields two quadratic constraints: its transformed direction and point must satisfy coplanarity with the camera-center/image-line plane.The plane contains the camera center and image line, with its normal defining the coplanarity tests.Source: §2 · pp. 3–4, §2.2 · p. 5

2.1 Rotation, translation, and dual number quaternions

  • A dual number quaternion represents rigid motion as a screw, enabling quaternion algebra to express pose constraints quadratically.A screw combines rotation about an offset axis with translation along that axis.Source: §2.1 · pp. 4–5, §2.2 · p. 5

2.2 The error function

  • The pose objective sums squared constraint residuals while penalizing r^T r=1 and r^T s=0; λ=50 worked well, but too-large λ slows convergence.The authors report good results with λ=50 and warn that an overly large value can make convergence inefficient.Source: §2.2 · p. 5
  • For line correspondences, rotation can alternatively be estimated first, after which translation reduces to linear least squares.This separates the rotation and translation estimates rather than optimizing them together.Source: §2.2 · pp. 5–6

3 Object pose from point correspondences

Point correspondences yield quadratic constraints from camera-ray collinearity, giving the same sum-of-squares error structure used for line correspondences.

3 Object pose from point correspondences

  • Each point correspondence supplies two constraints expressing collinearity among the object point, its image projection, and the camera center.The two constraints are derived from the point’s projection geometry.Source: §3 · p. 6
  • Representing pose with a dual number quaternion turns the point constraints into quadratic forms, matching the line-based error structure.The resulting error function has the same sum-of-squares-of-quadratic-constraints structure as for line correspondences.Source: §3 · p. 6
  • The point-correspondence objective minimizes two quadratic constraints per match.The constraints are the two collinearity conditions for each point correspondence.Source: §3 · p. 6

4 The trust-region optimization method

The trust-region method minimizes the pose error by repeatedly optimizing a local quadratic approximation inside a restricted neighborhood. Its practical variant uses a Gauss–Newton Hessian approximation, while its comparison with Levenberg–Marquardt identifies a large-residual limitation shared by both methods’ Hessian approximation assumption.

4 The trust-region optimization method

  • The method approximates the least-squares error locally with a quadratic form based on its gradient and Hessian, then minimizes that form within a spherical trust region.The trust radius changes dynamically according to the quality of the local approximation.Source: §4 · pp. 6–8
  • Unlike Newton’s method, which can increase the objective, the trust-region method restricts each local quadratic search and accepts a step only when the approximation is satisfactory.The trust radius is adjusted according to the ratio of actual to predicted error reduction.Source: §4 · pp. 7–8

4.1 The practical trust-region algorithm

  • The practical algorithm uses the Gauss–Newton approximation H(x) = J(x)^T J(x), modified by αI, and uses Hebden’s algorithm when the unconstrained step exceeds the trust radius.The paper gives α = 0.001 as an example and sets r = 0.25 and s = 0.75 for its application.Source: §4 · p. 8, §4.1 · p. 9

4.2 Comparison with Levenberg-Marquardt

  • Levenberg–Marquardt and the trust-region method both assume J(x_k)^T J(x_k) approximates the Hessian, an assumption the paper says is unjustified for large-residual problems.For large residuals, the exact Hessian is not positive semidefinite, so Levenberg–Marquardt cannot be used; the trust-region method can instead use Moré–Sorensen safeguarding, at greater computational cost than Hebden’s algorithm.Source: §4.2 · p. 9

5 Experimental results

Experiments compare pose estimates against a ground-truth solution across correspondence counts and image noise, using transformation-space and image-space errors. The nonlinear method degrades gently as matching errors increase, while the linear method gives erroneous results.

5 Experimental results

  • The trust-region algorithm can attain the global solution independently of initialization in a very small number of iterations, which the authors relate to its quadratic constraints.The experiments include point and line implementations, including rotation-then-translation and simultaneous rotation-and-translation variants for line correspondences.Source: §5 · pp. 9–10
  • The evaluations compare solutions with ground truth in transformation space and measure image-space reprojection error over all 460 calibrating points.The reported image-space summaries include the largest, smallest, and average errors.Source: §5 · pp. 9–10

5.1 Matching errors

  • As point-matching errors increase, the nonlinear method’s rotation and translation errors degrade gently, while the linear method gives erroneous results.Figure 9 plots rotation-axis, rotation-angle, and translation errors against the number of mismatched point pairs.Source: §5.1 · p. 10

5 Experimental results

  • Table 1 reports that the linear method’s rotation is relatively stable as point correspondences decrease, while translation degrades gently and intrinsic parameters vary substantially.The 460-point solution is used as ground truth.Source: Table 1 · p. 11
  • With 100 point correspondences, the linear method’s rotation resists image noise, but translation is unreliable above a maximum noise amplitude of half a pixel.The table’s first column gives the maximum noise amplitude around nominal image locations.Source: Table 2 · p. 11

6 Discussion

The nonlinear method is reported to be robust to image noise and matching errors, and more stable than the linear method as correspondence counts vary. Its rotation-then-translation implementation is faster, but the paper notes a trade-off between efficiency and accuracy.

6 Discussion

  • The final result is independent of initialization, a stated improvement over other approaches using Newton’s method,, and.Source: §6 · p. 11
  • The nonlinear method resists image noise, remains more stable as correspondence counts vary, and degrades gently under mismatches, unlike the linear method.The noise comparisons include Tables 4, 6, and 8; matching-error behavior is illustrated in Figure 9.Source: §6 · pp. 11–12, Table 6 · p. 12, Table 8 · p. 13
  • Rotation-then-translation is more efficient than jointly estimating rotation and translation because the trust-region optimization uses 4 rather than 8 parameters.The paper describes an efficiency–accuracy compromise and reports 0.03 seconds for convergence with 30 line correspondences on a Sun/Sparc10.Source: §6 · p. 12
  • Table 7 reports that the rotation-and-translation implementation with point correspondences seems to provide the most accurate results and the most stable results as correspondence counts vary.The caption qualifies this as an apparent result rather than an unqualified guarantee.Source: Table 7 · p. 13
  • The authors consider the pose method suitable when robustness and accuracy are needed, and suggest the trust-region method for other computer-vision minimization problems.Examples include structure-from-motion, reconstruction from multiple views, and fundamental-matrix estimation [5].Source: §6 · pp. 13–14

A Minimization of the local quadratic form

The section characterizes how to minimize a local quadratic form subject to a spherical constraint. It derives a multiplier-based solution and outlines Hebden’s iterative procedure for finding it.

A Minimization of the local quadratic form

  • The local problem minimizes a quadratic form over a compact sphere, and its solution is characterized by a multiplier μ* satisfying positive semidefiniteness, stationarity, feasibility, and complementary slackness.The multiplier is unique.Source: A Minimization of the local quadratic… · p. 14
  • When μ* > 0, the solution lies on the constraint boundary, reducing the problem to finding a multiplier that satisfies the boundary condition.Source: A Minimization of the local quadratic… · p. 14
  • The function ϕ can be expressed through the eigensystem of the symmetric matrix H, and its stated properties support finding the multiplier through a scalar root search.For μ* > 0, the passage says there is a unique solution d* = d(μ*) to the boundary equation.Source: A Minimization of the local quadratic… · p. 14
  • The function ϕ is positive, strictly decreasing on its stated interval, and tends to zero as μ tends to infinity.When the Gauss–Newton Hessian approximation is used, the text states that μ* > 0 and Hebden’s method can determine the unique solution efficiently.Source: A Minimization of the local quadratic… · p. 14
  • Hebden’s algorithm iterates on μ using Cholesky factorization and triangular solves, avoiding explicit knowledge of H’s eigensystem.The section says the algorithm converges on average after 4 iterations.Source: A Minimization of the local quadratic… · p. 15
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