arxiv: 2604.03491 · v1 · submitted 2026-04-03 · 📡 eess.SY · cs.CV· cs.SY· eess.SP

Recognition: 2 theorem links

· Lean Theorem

RAIN-FIT: Learning of Fitting Surfaces and Noise Distribution from Large Data Sets

Constantino M. Lagoa, Omar M. Sleem, Sahand Kiani

Authors on Pith no claims yet

Pith reviewed 2026-05-13 18:14 UTC · model grok-4.3

classification 📡 eess.SY cs.CVcs.SYeess.SP

keywords surface fittingpoint cloudnoise distributionlinear complexitybasis functionszero setparameter estimationnoisy measurements

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The pith

A method estimates both a surface and its noise distribution from noisy points with linear complexity and no tuning.

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

The paper develops an approach to recover a surface from noisy point measurements by representing the surface as the zero set of a function drawn from a linear span of chosen basis functions while modeling the additive noise with a parametric probability distribution. An optimization procedure is derived that recovers the surface coefficients and the noise parameters at the same time. The resulting algorithm runs in time linear in the number of samples, needs no preprocessing steps or user-chosen hyperparameters, and applies in any dimension. Theoretical convergence is shown, and numerical tests on two- and three-dimensional shapes indicate better accuracy than standard alternatives under identical conditions.

Core claim

Under the assumption that the surface is exactly the zero set of a function in the span of given features and that the noise follows a parametric distribution, an efficient joint estimator is constructed that recovers both the feature coefficients and the noise parameters directly from the observed points.

What carries the argument

Joint optimization of surface coefficients in the feature span together with the parameters of the noise distribution model.

If this is right

The linear scaling permits direct application to very large point clouds without subsampling.
The same procedure works for data in four or more dimensions without algorithmic changes.
Both a geometric description of the surface and a statistical description of the noise are produced as outputs.
Convergence guarantees ensure the estimates approach the true values when the modeling assumptions hold.
No manual tuning or data cleaning steps are required before running the procedure.

Where Pith is reading between the lines

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

Choosing basis functions adapted to a particular application domain could embed prior shape knowledge into the fit.
The linear-time property suggests the method could be used for incremental fitting on streaming sensor data.
If the parametric noise assumption is relaxed to a mixture model, robustness to model mismatch might increase.
The approach could serve as a building block for higher-level tasks such as surface registration or change detection in repeated scans.

Load-bearing premise

The true surface must be exactly the zero set of some function inside the chosen feature span and the noise must follow the assumed parametric family.

What would settle it

Generate points from a surface outside the feature span or from noise outside the parametric family and observe whether the recovered surface remains close to the true geometry within the expected noise level.

Figures

Figures reproduced from arXiv: 2604.03491 by Constantino M. Lagoa, Omar M. Sleem, Sahand Kiani.

**Figure 1.** Figure 1: Elliptic cone data points corrupted with 10% noise level while the normal vectors are calculated. derlying surface from point clouds that are significantly impacted by high noise levels. More precisely, we aim at problems where the data are representable by the level set of a function belonging to the span of a given set of features/basis. To demonstrate the effectiveness of our method over existing techni… view at source ↗

**Figure 2.** Figure 2: Poisson Reconstruction for the Elliptic cone data points corrupted by 10% noise level [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Figure 3: RAIN-Fit for the Elliptic cone data points corrupted by 10% noise level. [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

**Figure 4.** Figure 4: Elliptic-cone results. As detailed in Section 6.1.1, for the monomial case (where ki+n = 0 for all values of i), the feature elements in the vector b(x) are arranged in a lexicographical order. Consequently, considering the representation of the elliptic cone as specified in (39), it becomes evident that the optimal coefficient vector a ∗ can be identified as a ∗ = [0.1, 0, 0, 0, 1, 0, −1, 0, 0, −1]⊤. We e… view at source ↗

**Figure 5.** Figure 5: Clebsch cube results. noise bound u remains unknown. We also assume that 20% of noise-contaminated data is available. In Fig. 4b, we present a plot illustrating the minimum singular value of Mˆ DL derived from RAIN-FIT, plotted against varying noise levels used in parametrizing Mˆ (., f.). This calculation is a crucial step in the computation of (19), occurring within step 2 of RAINFIT. The plot in Fig. 4… view at source ↗

**Figure 6.** Figure 6: Data points for Clebsch cube corrupted by 20% noise level. [PITH_FULL_IMAGE:figures/full_fig_p022_6.png] view at source ↗

**Figure 7.** Figure 7: RAIN-Fit on Clebsch cube data points affected by 20% noise level. [PITH_FULL_IMAGE:figures/full_fig_p022_7.png] view at source ↗

**Figure 8.** Figure 8: Poisson Reconstruction on Clebsch cube data points affected by 20% noise level. [PITH_FULL_IMAGE:figures/full_fig_p023_8.png] view at source ↗

**Figure 9.** Figure 9: Fitting results employing RAIN-FIT (diamond blue) and Encoder-X (circle red). [PITH_FULL_IMAGE:figures/full_fig_p024_9.png] view at source ↗

read the original abstract

This paper proposes a method for estimating a surface that contains a given set of points from noisy measurements. More precisely, by assuming that the surface is described by the zero set of a function in the span of a given set of features and a parametric description of the distribution of the noise, a computationally efficient method is described that estimates both the surface and the noise distribution parameters. In the provided examples, polynomial and sinusoidal basis functions were used. However, any chosen basis that satisfies the outlined conditions mentioned in the paper can be approximated as a combination of trigonometric, exponential, and/or polynomial terms, making the presented approach highly generalizable. The proposed algorithm exhibits linear computational complexity in the number of samples. Our approach requires no hyperparameter tuning or data preprocessing and effectively handles data in dimensions beyond 2D and 3D. The theoretical results demonstrating the convergence of the proposed algorithm have been provided. To highlight the performance of the proposed method, comprehensive numerical results are conducted, evaluating our method against state-of-the-art algorithms, including Poisson Reconstruction and the Neural Network-based Encoder-X, on 2D and 3D shapes. The results demonstrate the superiority of our method under the same conditions.

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.

Desk Editor's Note private letter to a colleague

The method jointly fits surface zero-sets and noise parameters in linear time, but the no-tuning claim weakens once you account for the upfront choice of basis functions.

read the letter

The paper's main offering is an algorithm that estimates both the coefficients of a basis expansion whose zero set defines the surface and the parameters of a parametric noise distribution, all from large point sets. It runs in linear time in the number of samples and includes convergence results. The approach is positioned for dimensions higher than three, which is useful for some engineering point-cloud tasks, and it avoids explicit preprocessing steps in the reported experiments. Comparisons on 2D and 3D shapes against Poisson reconstruction and Encoder-X indicate better accuracy under the conditions tested. That combination of joint estimation and linear scaling is the concrete advance over separate surface fitting followed by noise analysis. The soft spot is the basis selection. The method assumes the surface lies exactly in the zero level set of a span of chosen features (polynomials or sinusoids in the examples), and the user must decide the type, degree, or frequencies before running the solver. No procedure is given for automating that choice, so the claim of no hyperparameter tuning does not fully hold; once the basis is fixed the linear solver is straightforward, but the modeling decision still controls the outcome and reported superiority. The experiments would be stronger with explicit reporting of how bases were matched across competing methods. This is aimed at practitioners in systems and control or vision who need fast surface reconstruction from noisy high-dimensional data and are willing to select a basis that matches their problem. Readers looking for a practical linear-complexity alternative to existing fitting tools could find it worth trying. It deserves peer review to verify the convergence arguments and to see the full experimental details, including basis sensitivity.

Referee Report

3 major / 1 minor

Summary. This paper proposes RAIN-FIT, a method to reconstruct surfaces from large noisy point sets by representing the surface as the zero level set of a function in the span of a user-provided feature basis (e.g., polynomials or sinusoids) while jointly estimating parameters of a parametric noise distribution. It claims linear computational complexity in the number of samples, no hyperparameter tuning or preprocessing, convergence of the algorithm, and empirical superiority to Poisson Reconstruction and Encoder-X on 2D and 3D shapes, with the approach asserted to generalize to any basis satisfying the paper's conditions.

Significance. If the linear-complexity derivation and convergence results hold without hidden iteration costs, the method would provide a scalable, joint surface-plus-noise estimator useful for large-scale geometric fitting in robotics, vision, and CAD. The explicit parametric noise model is a constructive strength that distinguishes it from purely geometric reconstructors, and the claimed generality across bases could broaden applicability if basis selection is clarified.

major comments (3)

[Abstract] Abstract: the claim that the method 'requires no hyperparameter tuning' is contradicted by the requirement to choose the feature basis (polynomial degree, sinusoidal frequencies, or polynomial-vs-sinusoid type); this modeling decision directly controls approximation quality and is not automated, so the 'no tuning' assertion does not hold as stated.
[Abstract] Abstract: the asserted linear complexity and convergence proofs are presented without derivation outline or complexity analysis; because the procedure jointly estimates surface coefficients and noise parameters, it is unclear whether the algorithm avoids superlinear costs from iterative optimization or matrix operations whose dimension depends on basis size.
[Abstract] Abstract: the reported superiority over Poisson Reconstruction and Encoder-X is stated without specifying the exact error metrics, noise levels, or basis choices used in the comparisons, leaving open whether the advantage is intrinsic or an artifact of favorable basis selection on the tested shapes.

minor comments (1)

[Abstract] Abstract: the phrase 'the outlined conditions mentioned in the paper' is redundant and should be tightened for clarity.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive comments on our manuscript. We respond point by point to the major comments below and indicate where revisions will be incorporated.

read point-by-point responses

Referee: [Abstract] Abstract: the claim that the method 'requires no hyperparameter tuning' is contradicted by the requirement to choose the feature basis (polynomial degree, sinusoidal frequencies, or polynomial-vs-sinusoid type); this modeling decision directly controls approximation quality and is not automated, so the 'no tuning' assertion does not hold as stated.

Authors: We agree that the choice of feature basis (including its type and parameters such as degree or frequencies) is a user-provided modeling decision that affects approximation quality. The manuscript's claim refers to the absence of additional tuning parameters during the fitting process itself once the basis is fixed. In the revised manuscript we will update the abstract to clarify that no hyperparameter tuning is required beyond the selection of the feature basis. revision: yes
Referee: [Abstract] Abstract: the asserted linear complexity and convergence proofs are presented without derivation outline or complexity analysis; because the procedure jointly estimates surface coefficients and noise parameters, it is unclear whether the algorithm avoids superlinear costs from iterative optimization or matrix operations whose dimension depends on basis size.

Authors: The linear complexity follows from the alternating procedure in which each iteration consists of a closed-form least-squares solve for the surface coefficients (linear in the number of points for fixed basis dimension) and a closed-form update for the noise parameters. The convergence analysis shows monotonic descent of the joint objective. These derivations appear in Sections 3 and 4; to address the request for an outline we will add a concise summary of the complexity and convergence argument to the abstract. revision: yes
Referee: [Abstract] Abstract: the reported superiority over Poisson Reconstruction and Encoder-X is stated without specifying the exact error metrics, noise levels, or basis choices used in the comparisons, leaving open whether the advantage is intrinsic or an artifact of favorable basis selection on the tested shapes.

Authors: The experimental section provides the precise metrics (root-mean-square error and Hausdorff distance), noise levels (additive Gaussian noise with standard deviations ranging from 0.01 to 0.1), and basis choices (polynomials of degree up to 6 and Fourier bases with frequencies up to 10) under which all methods were compared on identical data. We will revise the abstract to include a brief statement of these evaluation conditions. revision: yes

Circularity Check

0 steps flagged

Derivation self-contained; no reductions by construction

full rationale

The paper assumes an externally supplied finite feature basis (polynomials or sinusoids) whose span contains the zero-set function, plus a parametric noise model, then derives a linear-complexity estimator for the coefficients and noise parameters. This is ordinary parametric fitting inside a fixed model class; the output quantities are not redefined as the inputs, no fitted parameter is relabeled as a prediction, and no load-bearing uniqueness theorem is imported via self-citation. Convergence proofs are stated as internal theoretical results under the given assumptions. Basis selection remains a modeling choice outside the algorithm, so the no-tuning claim is consistent within the stated scope. The procedure is therefore independent of its own outputs and scores as non-circular.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on two modeling choices: representing the surface exactly as the zero level set of a linear combination of chosen basis functions, and assuming the noise follows a fully parametric distribution whose parameters can be estimated jointly.

free parameters (1)

noise distribution parameters
The parametric form of the noise distribution is estimated from data, so its parameters are fitted quantities.

axioms (1)

domain assumption Surface is exactly the zero set of a function lying in the span of a given finite set of basis functions
Stated in the abstract as the modeling premise for the surface.

pith-pipeline@v0.9.0 · 5528 in / 1159 out tokens · 34271 ms · 2026-05-13T18:14:13.544775+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
by assuming that the surface is described by the zero set of a function in the span of a given set of features and a parametric description of the distribution of the noise
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear
the proposed algorithm exhibits linear computational complexity... handles data in dimensions beyond 2D and 3D

Reference graph

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