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arxiv: 2605.06572 · v1 · submitted 2026-05-07 · 💻 cs.CV · cs.NA· math.NA

Solving Minimal Problems Without Matrix Inversion Using FFT-Based Interpolation

Haidong Wu , Snehal Bhayani , Janne Heikkil\"a This is my paper

Pith reviewed 2026-05-08 12:20 UTC · model grok-4.3

classification 💻 cs.CV cs.NAmath.NA
keywords minimal problemscamera geometryhidden variable resultantsFFT interpolationpolynomial solversnumerical stability
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The pith

Minimal camera geometry problems can be solved by reconstructing determinant polynomials via inverse FFT interpolation from samples, avoiding matrix inversion entirely.

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

The paper presents a sampling-based solver for minimal problems in camera geometry estimation, which are systems of multivariate polynomial equations. Existing Gröbner-basis and resultant methods typically require matrix inversion in the online phase, which can introduce numerical issues. The proposed approach builds sparse hidden-variable resultants and reconstructs the determinant polynomial in one hidden variable by evaluating it at sample points and applying inverse fast Fourier transform interpolation. This yields the hidden variable roots, after which the remaining unknowns are found by locating rank-1 deficient submatrices and applying Cramer's rule, with a GCD criterion to handle noise. Experiments indicate competitive speed and strong stability for small-scale instances compared with traditional solvers.

Core claim

The determinant polynomial in the hidden variable can be efficiently reconstructed from sampled evaluations via inverse fast Fourier transform interpolation without symbolic expansion, after which the hidden variable is solved and the remaining unknowns recovered by identifying rank-1 deficient submatrices through a greatest common divisor criterion.

What carries the argument

Sparse hidden-variable resultants whose determinant polynomial in the hidden variable is recovered by inverse fast Fourier transform interpolation from sampled point evaluations.

If this is right

  • Matrix inversion is eliminated from the online solver, removing one source of numerical instability.
  • The method achieves competitive runtime and strong stability on small-scale minimal problems.
  • The GCD-based selection provides robustness to noise when identifying deficient submatrices.
  • It offers a practical alternative to Gröbner-basis and traditional resultant solvers for camera geometry tasks.

Where Pith is reading between the lines

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

  • The sampling-plus-interpolation strategy might apply to polynomial systems in other domains such as robotics or algebraic vision tasks.
  • Hybrid implementations combining this interpolation with selective symbolic preprocessing could further improve scalability.
  • The approach suggests that many resultant-based solvers could be reframed as evaluation-and-reconstruction pipelines.

Load-bearing premise

The determinant polynomial can be accurately reconstructed from a modest number of sampled evaluations via IFFT without symbolic work, and the GCD criterion reliably identifies the correct rank-1 deficient submatrix under realistic noise.

What would settle it

Applying the method to a known exact minimal problem and obtaining either incorrect roots from the reconstructed polynomial or failure of the GCD to select the proper submatrix would show the approach does not hold.

Figures

Figures reproduced from arXiv: 2605.06572 by Haidong Wu, Janne Heikkil\"a, Snehal Bhayani.

Figure 1
Figure 1. Figure 1: High-level overview of the proposed solver pipeline, illustrating the offline construction and online numerical stages. view at source ↗
Figure 2
Figure 2. Figure 2: Histograms of Log10 of normalized equation residual error for three selected minimal problems. view at source ↗
read the original abstract

Estimating camera geometry typically involves solving minimal problems formulated as systems of multivariate polynomial equations, which often pose computational challenges when using existing Gr\"obner-basis or resultant-based methods due to matrix inversion needed in the online solver. Here we propose a sampling-based, matrix inversion-free method that constructs the solvers using sparse hidden-variable resultants. The determinant polynomial in the hidden variable is efficiently reconstructed via inverse fast Fourier transform interpolation from sampled evaluations, avoiding symbolic expansion. Solving this polynomial yields the hidden variable, and the remaining unknowns are recovered by identifying rank-1 deficient submatrices and applying Cramer's rule. A greatest common divisor-based criterion ensures robust submatrix identification under noise. Experiments on diverse minimal problems demonstrate that the proposed solver achieves strong numerical stability and competitive runtime, particularly for small-scale problems, providing a practical alternative to traditional Gr\"obner-basis and resultant-based solvers.

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 paper proposes a sampling-based solver for minimal problems in camera geometry that constructs sparse hidden-variable resultants, reconstructs the determinant polynomial in the hidden variable via IFFT interpolation from floating-point determinant evaluations at FFT grid points (avoiding symbolic expansion and online matrix inversion), solves the resulting univariate polynomial, and recovers the remaining unknowns by applying Cramer's rule to rank-1 deficient submatrices identified via a GCD criterion. Experiments on diverse minimal problems are reported to show strong numerical stability and competitive runtime, especially for small-scale instances, positioning the method as a practical alternative to Gröbner-basis and classical resultant solvers.

Significance. If the central claims hold, the work provides a concrete, inversion-free pipeline for algebraic minimal solvers that could improve numerical robustness in structure-from-motion and multi-view geometry pipelines. The use of FFT-based interpolation and GCD-based submatrix selection is a novel practical twist on resultant techniques, and the empirical demonstration of stability on real minimal problems would be a useful addition to the computer-vision solver literature.

major comments (2)
  1. [Method description (reconstruction and GCD criterion)] The stability claim rests on the assumption that det(M(x_i)) evaluations at the FFT sample points remain sufficiently accurate for stable IFFT reconstruction even when the resultant matrix M(x) is near-singular. No a-priori bound on the condition number of M at these points is supplied, and the manuscript does not analyze how ill-conditioning propagates through the IFFT step or the subsequent GCD test.
  2. [Experiments] The experimental section reports strong numerical stability and competitive runtimes, yet the provided abstract and supporting description lack the exact sampling density, noise model, quantitative error tables, and comparison baselines (e.g., Gröbner-basis solvers with and without inversion). Without these, it is impossible to verify whether the observed stability genuinely supports the claim that the IFFT+GCD pipeline outperforms traditional methods under realistic noise.
minor comments (2)
  1. [Preliminaries] Notation for the hidden-variable resultant matrix and the precise definition of the GCD criterion should be introduced earlier and used consistently throughout.
  2. [Experiments] Figure captions and table headers would benefit from explicit statements of the noise level and number of Monte-Carlo trials used in each experiment.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the constructive feedback on our work. We address the major comments point by point below, indicating where revisions will be made to the manuscript.

read point-by-point responses

  1. Referee: [Method description (reconstruction and GCD criterion)] The stability claim rests on the assumption that det(M(x_i)) evaluations at the FFT sample points remain sufficiently accurate for stable IFFT reconstruction even when the resultant matrix M(x) is near-singular. No a-priori bound on the condition number of M at these points is supplied, and the manuscript does not analyze how ill-conditioning propagates through the IFFT step or the subsequent GCD test.

    Authors: We acknowledge the absence of an a priori bound in the current manuscript. The proposed method prioritizes practical applicability and numerical performance over theoretical bounds, which can be difficult to obtain for the diverse set of minimal problems in computer vision. In the revised version, we will include an empirical analysis of the condition numbers at the sampled points, along with a discussion of error propagation in the IFFT reconstruction and the GCD-based submatrix selection. Additional experiments will be added to quantify the impact of ill-conditioning. This will provide readers with a clearer understanding of the method's robustness. revision: partial

  2. Referee: [Experiments] The experimental section reports strong numerical stability and competitive runtimes, yet the provided abstract and supporting description lack the exact sampling density, noise model, quantitative error tables, and comparison baselines (e.g., Gröbner-basis solvers with and without inversion). Without these, it is impossible to verify whether the observed stability genuinely supports the claim that the IFFT+GCD pipeline outperforms traditional methods under realistic noise.

    Authors: We will make the requested details explicit in the revised manuscript. We will update the abstract and experimental section to specify the sampling density used for the IFFT (twice the degree of the univariate polynomial), describe the noise model (additive Gaussian noise on the input measurements), provide quantitative tables with error statistics, and include comparisons against Gröbner basis solvers implemented with and without matrix inversion. These changes will enable independent verification of the stability and runtime claims. revision: yes

standing simulated objections not resolved
  • Deriving a general a-priori bound on the condition numbers of the resultant matrices at FFT grid points across all minimal problems.

Circularity Check

0 steps flagged

No circularity: derivation uses standard algebraic and numerical primitives without self-reduction

full rationale

The paper constructs solvers via sparse hidden-variable resultants, reconstructs the determinant polynomial in the hidden variable by IFFT interpolation of numerical determinant samples at FFT grid points, solves the resulting univariate polynomial, and recovers remaining variables via rank-1 deficient submatrix selection (using a GCD criterion) followed by Cramer's rule. None of these steps reduce by construction to fitted inputs, self-definitions, or load-bearing self-citations; they follow directly from established resultant theory and FFT properties. Stability claims rest on empirical tests rather than definitional equivalence. The method is therefore self-contained against external algebraic benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Abstract-only review reveals no free parameters or invented entities; the approach rests on standard mathematical properties of the FFT and the existence of sparse hidden-variable resultants for the target problem class.

axioms (2)
  • standard math The univariate determinant polynomial can be exactly recovered from a sufficient number of point evaluations via inverse FFT interpolation
    Invoked for the reconstruction step that replaces symbolic expansion.
  • domain assumption Sparse hidden-variable resultants exist and produce a univariate polynomial whose roots correspond to valid solutions of the minimal problem
    Foundation for setting up the sampled evaluations.

pith-pipeline@v0.9.0 · 5455 in / 1360 out tokens · 33403 ms · 2026-05-08T12:20:20.104440+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

40 extracted references · 2 canonical work pages

  1. [1]

    A sparse resultant based method for efficient minimal solvers

    Snehal Bhayani, Zuzana Kukelova, and Janne Heikkila. A sparse resultant based method for efficient minimal solvers. InProceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), 2020. 1, 2, 3, 4, 6, 7, 8

    2020

  2. [2]

    Computing stable resultant-based minimal solvers by hiding a variable

    Snehal Bhayani, Zuzana Kukelova, and Janne Heikkil ¨a. Computing stable resultant-based minimal solvers by hiding a variable. In2020 25th International Conference on Pattern Recognition (ICPR), pages 6104–6111. IEEE, 2021. 2, 3

    2021

  3. [3]

    Sparse resultant-based minimal solvers in computer vision and their connection with the action matrix.Journal of Math- ematical Imaging and Vision, 66(3):335–360, 2024

    Snehal Bhayani, Janne Heikkil ¨a, and Zuzana Kukelova. Sparse resultant-based minimal solvers in computer vision and their connection with the action matrix.Journal of Math- ematical Imaging and Vision, 66(3):335–360, 2024. 2

    2024

  4. [4]

    Improv- ing numerical accuracy of gr ¨obner basis polynomial equa- tion solvers

    Martin Byrod, Klas Josephson, and Kalle Astrom. Improv- ing numerical accuracy of gr ¨obner basis polynomial equa- tion solvers. In2007 IEEE 11th International Conference on Computer Vision, pages 1–8. IEEE, 2007. 1

    2007

  5. [5]

    A column-pivoting based strategy for monomial ordering in numerical gr ¨obner basis calculations

    Martin Byr ¨od, Klas Josephson, and Kalle ˚Astr¨om. A column-pivoting based strategy for monomial ordering in numerical gr ¨obner basis calculations. InComputer Vision– ECCV 2008: 10th European Conference on Computer Vi- sion, Marseille, France, October 12-18, 2008, Proceedings, Part IV 10, pages 130–143. Springer, 2008. 1

    2008

  6. [6]

    Fast and stable polynomial equation solving and its application to computer vision.International Journal of Computer Vision, 84(3):237–256, 2009

    Martin Byr ¨od, Klas Josephson, and Kalle ˚Astr¨om. Fast and stable polynomial equation solving and its application to computer vision.International Journal of Computer Vision, 84(3):237–256, 2009. 1

    2009

  7. [7]

    A subdivision-based al- gorithm for the sparse resultant.Journal of the ACM (JACM), 47(3):417–451, 2000

    John F Canny and Ioannis Z Emiris. A subdivision-based al- gorithm for the sparse resultant.Journal of the ACM (JACM), 47(3):417–451, 2000. 3

    2000

  8. [8]

    Locally opti- mized ransac

    Ond ˇrej Chum, Ji ˇr´ı Matas, and Josef Kittler. Locally opti- mized ransac. InJoint pattern recognition symposium, pages 236–243. Springer, 2003. 1

    2003

  9. [9]

    Springer, 1998

    David A Cox, John Little, and Donal O’shea.Using alge- braic geometry. Springer, 1998. 1, 2, 3

    1998

  10. [10]

    Revisiting the p3p problem

    Yaqing Ding, Jian Yang, Viktor Larsson, Carl Olsson, and Kalle ˚Astr¨om. Revisiting the p3p problem. InProceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pages 4872–4880, 2023. 1

    2023

  11. [11]

    A practical method for the sparse resultant

    Ioannis Emiris and John Canny. A practical method for the sparse resultant. InProceedings of the 1993 international symposium on Symbolic and algebraic computation, pages 183–192, 1993. 3

    1993

  12. [12]

    A general solver based on sparse resul- tants.arXiv preprint arXiv:1201.5810, 2012

    Ioannis Z Emiris. A general solver based on sparse resul- tants.arXiv preprint arXiv:1201.5810, 2012. 3

    work page arXiv 2012

  13. [13]

    Efficient incremental algorithms for the sparse resultant and the mixed volume

    Ioannis Z Emiris and John F Canny. Efficient incremental algorithms for the sparse resultant and the mixed volume. Journal of Symbolic Computation, 20(2):117–149, 1995. 1

    1995

  14. [14]

    Symbolic and numeric methods for exploiting structure in constructing resultant matrices.Journal of Symbolic Computation, 33(4):393–413,

    Ioannis Z Emiris and Victor Y Pan. Symbolic and numeric methods for exploiting structure in constructing resultant matrices.Journal of Symbolic Computation, 33(4):393–413,

  15. [15]

    Random sample consensus: a paradigm for model fitting with applications to image analy- sis and automated cartography.Commun

    MA FISCHLER AND. Random sample consensus: a paradigm for model fitting with applications to image analy- sis and automated cartography.Commun. ACM, 24(6):381– 395, 1981. 1

    1981

  16. [16]

    Analysis and solutions of the three point per- spective pose estimation problem

    Robert M Haralick, Chung-nan Lee, Kars Ottenburg, and Michael N¨olle. Analysis and solutions of the three point per- spective pose estimation problem. InCVPR, pages 592–598,

  17. [17]

    An efficient hidden vari- able approach to minimal-case camera motion estimation

    Richard Hartley and Hongdong Li. An efficient hidden vari- able approach to minimal-case camera motion estimation. IEEE transactions on pattern analysis and machine intelli- gence, 34(12):2303–2314, 2012. 3

    2012

  18. [18]

    Using sparse elimination for solving min- imal problems in computer vision

    Janne Heikkila. Using sparse elimination for solving min- imal problems in computer vision. InProceedings of the IEEE International Conference on Computer Vision (ICCV),

  19. [19]

    Reconstructing the world in six days

    Heinly Jared, Johannes L Schonberger, Enrique Dunn, and Jan-Michael Frahm. Reconstructing the world in six days. InProceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), 2015. 1

    2015

  20. [20]

    A minimal solution to relative pose with unknown focal length and radial distortion

    Fangyuan Jiang, Yubin Kuang, Jan Erik Solem, and Kalle ˚Astr¨om. A minimal solution to relative pose with unknown focal length and radial distortion. InAsian Conference on Computer Vision, pages 443–456. Springer, 2014. 1

    2014

  21. [21]

    Resultant based incremental recovery of camera pose from pairwise matches

    Yoni Kasten, Meirav Galun, and Ronen Basri. Resultant based incremental recovery of camera pose from pairwise matches. In2019 IEEE Winter Conference on Applications of Computer Vision (WACV), pages 1080–1088. IEEE, 2019. 3

    2019

  22. [22]

    PhD thesis, Czech Technical University, 2013

    Zuzana Kukelova.Algebraic methods in computer vision. PhD thesis, Czech Technical University, 2013. 1, 3

    2013

  23. [23]

    A minimal solution to the autocalibration of radial distortion

    Zuzana Kukelova and Tomas Pajdla. A minimal solution to the autocalibration of radial distortion. In2007 IEEE Con- ference on Computer Vision and Pattern Recognition, pages 1–7. IEEE, 2007. 1

    2007

  24. [24]

    Auto- matic generator of minimal problem solvers

    Zuzana Kukelova, Martin Bujnak, and Tomas Pajdla. Auto- matic generator of minimal problem solvers. InComputer Vision–ECCV 2008: 10th European Conference on Com- puter Vision, Marseille, France, October 12-18, 2008, Pro- ceedings, Part III 10, pages 302–315. Springer, 2008. 1

    2008

  25. [25]

    Poly- nomial eigenvalue solutions to minimal problems in com- puter vision.IEEE Transactions on Pattern Analysis and Machine Intelligence, 34(7):1381–1393, 2011

    Zuzana Kukelova, Martin Bujnak, and Tomas Pajdla. Poly- nomial eigenvalue solutions to minimal problems in com- puter vision.IEEE Transactions on Pattern Analysis and Machine Intelligence, 34(7):1381–1393, 2011. 2, 3

    2011

  26. [26]

    Effi- cient solvers for minimal problems by syzygy-based reduc- tion

    Viktor Larsson, Kalle Astrom, and Magnus Oskarsson. Effi- cient solvers for minimal problems by syzygy-based reduc- tion. InProceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 820–829, 2017. 1

    2017

  27. [27]

    Poly- nomial solvers for saturated ideals

    Viktor Larsson, Kalle Astrom, and Magnus Oskarsson. Poly- nomial solvers for saturated ideals. InProceedings of the IEEE International Conference on Computer Vision, pages 2288–2297, 2017. 1

    2017

  28. [28]

    Beyond grob- ner bases: Basis selection for minimal solvers

    Viktor Larsson, Magnus Oskarsson, Kalle Astrom, Alge Wallis, Zuzana Kukelova, and Tomas Pajdla. Beyond grob- ner bases: Basis selection for minimal solvers. InProceed- ings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 3945–3954, 2018. 1, 3

    2018

  29. [29]

    Gaps: Generator for automatic polynomial solvers.arXiv preprint arXiv:2004.11765, 2020

    Bo Li and Viktor Larsson. Gaps: Generator for automatic polynomial solvers.arXiv preprint arXiv:2004.11765, 2020. 1, 3, 6, 7, 8

    work page arXiv 2004

  30. [30]

    An efficient solution to the five-point relative pose problem.IEEE transactions on pattern analysis and machine intelligence, 26(6):756–770, 2004

    David Nist ´er. An efficient solution to the five-point relative pose problem.IEEE transactions on pattern analysis and machine intelligence, 26(6):756–770, 2004. 1

    2004

  31. [31]

    Usac: A universal framework for random sample consensus.IEEE transactions on pattern analysis and machine intelligence, 35(8):2022–2038, 2012

    Rahul Raguram, Ondrej Chum, Marc Pollefeys, Jiri Matas, and Jan-Michael Frahm. Usac: A universal framework for random sample consensus.IEEE transactions on pattern analysis and machine intelligence, 35(8):2022–2038, 2012. 1

    2022

  32. [32]

    Efficient & effective prioritized matching for large-scale image-based localization.IEEE transactions on pattern analysis and ma- chine intelligence, 39(9):1744–1756, 2016

    Torsten Sattler, Bastian Leibe, and Leif Kobbelt. Efficient & effective prioritized matching for large-scale image-based localization.IEEE transactions on pattern analysis and ma- chine intelligence, 39(9):1744–1756, 2016. 1

    2016

  33. [33]

    Visual odom- etry [tutorial].IEEE robotics & automation magazine, 18(4): 80–92, 2011

    Davide Scaramuzza and Friedrich Fraundorfer. Visual odom- etry [tutorial].IEEE robotics & automation magazine, 18(4): 80–92, 2011. 1

    2011

  34. [34]

    Model- ing the world from internet photo collections.International journal of computer vision, 80:189–210, 2008

    Noah Snavely, Steven M Seitz, and Richard Szeliski. Model- ing the world from internet photo collections.International journal of computer vision, 80:189–210, 2008. 1

    2008

  35. [35]

    Citeseer, 2005

    Henrik Stew ´enius.Gr ¨obner basis methods for minimal prob- lems in computer vision. Citeseer, 2005. 1

    2005

  36. [36]

    How hard is 3-view triangulation really? InTenth IEEE In- ternational Conference on Computer Vision (ICCV’05) Vol- ume 1, pages 686–693

    Henrik Stew ´enius, Frederik Schaffalitzky, and David Nist´er. How hard is 3-view triangulation really? InTenth IEEE In- ternational Conference on Computer Vision (ICCV’05) Vol- ume 1, pages 686–693. IEEE, 2005. 1

    2005

  37. [37]

    Recent developments on direct relative orientation.ISPRS Journal of Photogrammetry and Remote Sensing, 60(4):284– 294, 2006

    Henrik Stewenius, Christopher Engels, and David Nist ´er. Recent developments on direct relative orientation.ISPRS Journal of Photogrammetry and Remote Sensing, 60(4):284– 294, 2006. 1

    2006

  38. [38]

    A minimal solution for relative pose with un- known focal length.Image and Vision Computing, 26(7): 871–877, 2008

    Henrik Stew ´enius, David Nist´er, Fredrik Kahl, and Frederik Schaffalitzky. A minimal solution for relative pose with un- known focal length.Image and Vision Computing, 26(7): 871–877, 2008. 1

    2008

  39. [39]

    Cambridge university press, 2003

    Joachim V on Zur Gathen and J¨urgen Gerhard.Modern com- puter algebra. Cambridge university press, 2003. 5

    2003

  40. [40]

    A conic transformation approach for solving the perspective-three- point problem

    Haidong Wu, Snehal Bhayani, and Janne Heikkil ¨a. A conic transformation approach for solving the perspective-three- point problem. In2025 IEEE/CVF Winter Conference on Applications of Computer Vision (WACV), pages 3237–3245. IEEE, 2025. 1

    2025