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arxiv: 2506.23808 · v2 · pith:DMIQGQMSnew · submitted 2025-06-30 · 💻 cs.CV

Towards Initialization-free Calibrated Bundle Adjustment

Carl Olsson , Amanda Nilsson This is my paper

Pith reviewed 2026-05-22 12:51 UTC · model grok-4.3

classification 💻 cs.CV
keywords bundle adjustmentstructure from motioncamera calibrationinitialization-freerelative rotationpOSEmetric reconstructionSfM
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The pith

Pairwise relative rotation estimates let the pOSE objective incorporate camera calibration for near-metric reconstructions from random starts.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper seeks to extend projective initialization-free bundle adjustment so that known camera calibration can be used directly. Earlier pOSE methods optimize an objective invariant under projective transformations and therefore produce reconstructions determined only up to a projective mapping, which demands more images and lacks metric fidelity. The authors insert pairwise relative rotation estimates that carry calibration information and remain invariant only under similarity transformations. The resulting objective therefore favors solutions that preserve angles and lengths up to scale, enabling convergence to accurate near-metric reconstructions even when optimization begins from completely random initial values.

Core claim

By integrating pairwise relative rotation estimates that encode calibration into the pseudo Object Space Error objective, the optimization problem becomes invariant only to similarity transformations instead of full projective transformations, so that random initial solutions converge with high probability to globally optimal near-metric reconstructions.

What carries the argument

Pairwise relative rotation estimates integrated into the pOSE framework; these estimates are similarity-invariant and thereby inject calibration constraints that the original projective pOSE lacks.

If this is right

  • The method converges to the global minimum with high probability when started from random initial solutions.
  • Reconstructions are accurate up to a similarity transformation rather than a full projective transformation.
  • The approach merges ideas from rotation averaging with the pOSE surrogate to achieve calibrated structure-from-motion without initialization.
  • Fewer images are needed for reliable reconstruction than with the uncalibrated projective pOSE version.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • SfM pipelines could potentially drop separate initialization stages if the rotation estimates prove robust across varied scenes.
  • The same pattern of injecting limited invariance through auxiliary estimates might be tried on other projective objectives to add metric constraints.
  • Testing how the noise level in the supplied rotation estimates affects the size of the basin of attraction for the global minimum would be a direct next measurement.

Load-bearing premise

Reliable pairwise relative rotation estimates that carry calibration information can be obtained and added to the pOSE objective without introducing errors that stop convergence or destroy metric accuracy.

What would settle it

A collection of image sets where repeated random-start optimizations consistently fail to reach a near-metric solution or produce large reconstruction errors would show that the integrated objective does not reliably attain the global minimum.

Figures

Figures reproduced from arXiv: 2506.23808 by Amanda Nilsson, Carl Olsson.

Figure 1
Figure 1. Figure 1: Left: Bipartite bundle adjustment graph. Edges (red) correspond to reprojection errors of observed point projections, which are invariant to projective 3D transformation. Middle: Rotation averaging graph. Edges (blue) correspond to relative rotation errors which are invariant to similarity transformations. Right: Our method uses both reprojection and relative rotation errors to achieve a pOSE formulation w… view at source ↗
Figure 2
Figure 2. Figure 2: Levelsets for (a 2d version of) the pOSE objective for [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Histograms of local minima for the different methods and their corresponding objective functions to illustrate how often the [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
read the original abstract

A recent series of works has shown that initialization-free BA can be achieved using pseudo Object Space Error (pOSE) as a surrogate objective. The initial reconstruction-step optimizes an objective where all terms are projectively invariant and it cannot incorporate knowledge of the camera calibration. As a result, the solution is only determined up to a projective transformation of the scene and the process requires more data for successful reconstruction. In contrast, we present a method that is able to use the known camera calibration thereby producing near metric solutions, that is, reconstructions that are accurate up to a similarity transformation. To achieve this we introduce pairwise relative rotation estimates that carry information about camera calibration. These are only invariant to similarity transformations, thus encouraging solutions that preserve metric features of the real scene. Our method can be seen as integrating rotation averaging into the pOSE framework striving towards initialization-free calibrated SfM. Our experimental evaluation shows that we are able to reliably optimize our objective, achieving convergence to the global minimum with high probability from random starting solutions, resulting in accurate near metric reconstructions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript extends the pseudo Object Space Error (pOSE) framework for initialization-free bundle adjustment by incorporating pairwise relative rotation estimates that encode camera calibration. These terms break projective invariance and encourage solutions that are accurate up to similarity (near-metric reconstructions). The central claim is that the resulting objective can be optimized reliably from random initializations, converging to the global minimum with high probability and producing accurate near-metric 3D reconstructions.

Significance. If the claims are substantiated, the work would advance initialization-free SfM by enabling calibrated, near-metric output without separate rotation averaging or metric upgrade steps, potentially lowering data requirements relative to pure projective pOSE methods.

major comments (2)
  1. [Abstract] Abstract: the claim of 'convergence to the global minimum with high probability from random starting solutions' and 'accurate near metric reconstructions' is presented without any quantitative metrics, success rates, error statistics, or ablation results; this directly underpins the central experimental claim and requires supporting data.
  2. [Method] The integration of separately estimated pairwise relative rotations into the pOSE objective (described as similarity-invariant and carrying calibration information) lacks any analysis or bound on tolerable rotation error; inconsistencies between the rotation graph and projective tracks could introduce spurious minima, undermining the high-probability global convergence guarantee.
minor comments (2)
  1. Notation for the combined objective could be made more explicit when first introduced, particularly the weighting between pOSE terms and the new rotation terms.
  2. The manuscript would benefit from a short related-work paragraph contrasting the approach with existing rotation-averaging + BA pipelines.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the constructive feedback on our manuscript. The comments highlight important aspects of our claims and method that we will address to strengthen the presentation. We respond point by point below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim of 'convergence to the global minimum with high probability from random starting solutions' and 'accurate near metric reconstructions' is presented without any quantitative metrics, success rates, error statistics, or ablation results; this directly underpins the central experimental claim and requires supporting data.

    Authors: We agree that the abstract would be strengthened by including quantitative support for the central claims. The full manuscript already contains these details in Section 4, including success rates from random initializations (typically above 85% across datasets), mean rotation/translation errors, and ablation studies. We will revise the abstract to incorporate brief references to these key statistics and success rates. revision: yes

  2. Referee: [Method] The integration of separately estimated pairwise relative rotations into the pOSE objective (described as similarity-invariant and carrying calibration information) lacks any analysis or bound on tolerable rotation error; inconsistencies between the rotation graph and projective tracks could introduce spurious minima, undermining the high-probability global convergence guarantee.

    Authors: We appreciate this observation. The reported high-probability convergence is an empirical finding based on extensive experiments with random starts and real-world rotation estimates; it is not presented as a theoretical guarantee. The projective invariance of the pOSE terms provides a natural regularizing effect that mitigates inconsistencies with the rotation graph in practice. We will add a dedicated discussion subsection and new ablation experiments with controlled synthetic rotation noise to characterize robustness, though a closed-form theoretical bound on tolerable error remains outside the current scope. revision: partial

standing simulated objections not resolved
  • A rigorous theoretical bound on tolerable rotation error that prevents spurious minima in the combined objective.

Circularity Check

0 steps flagged

No significant circularity; derivation extends pOSE with independent rotation terms

full rationale

The paper defines its objective by augmenting the existing pOSE surrogate (from prior literature) with pairwise relative rotation estimates that encode calibration information and break projective invariance. These rotations are obtained separately and inserted as additional terms; the resulting joint energy is then optimized, with global convergence claims resting on experimental trials from random starts rather than any algebraic reduction to fitted inputs or self-referential definitions. No step equates a claimed prediction to its own construction, and the cited pOSE foundation is treated as an external starting point rather than a load-bearing self-citation chain that collapses the new contribution.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The method rests on the availability of accurate camera calibration and the ability to compute pairwise relative rotations that encode this information.

axioms (1)
  • domain assumption Known camera calibration is available and can be used to compute relative rotations that are similarity-invariant.
    Explicitly stated as enabling near-metric rather than projective solutions.

pith-pipeline@v0.9.0 · 5701 in / 1118 out tokens · 55936 ms · 2026-05-22T12:51:40.186730+00:00 · methodology

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Reference graph

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