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

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

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.

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

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.

Paper Structure

This paper contains 23 sections, 4 theorems, 42 equations, 9 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Works
  3. Key Contributions
  4. Surface Learning
  5. Surface Learning with Noiseless Data
  6. Challenges Imposed by Noisy Data in Surface Learning
  7. RAIN-FIT Algorithm
  8. Motivation
  9. Conditions on Feature Functions
  10. Noise Compensation with Ordered Feature Sets
  11. Remarks on Arbitrary Analytic Feature Sets
  12. Computational Aspects of the Proposed Algorithm
  13. Algorithm Convergence Analysis
  14. Smooth representation of data
  15. Numerical results
  16. ...and 8 more sections

Key Result

Lemma 5

$\Hat{\mathbf{M}}_{\mathcal{D}_{L}}$ generated through RAIN-FIT exhibits the following entry-wise convergence property: $\blacktriangleleft$$\blacktriangleleft$

Figures (9)

  • Figure 1: Elliptic cone data points corrupted with 10% noise level while the normal vectors are calculated.
  • Figure 2: Poisson Reconstruction for the Elliptic cone data points corrupted by 10% noise level.
  • Figure 3: RAIN-Fit for the Elliptic cone data points corrupted by 10% noise level.
  • Figure 4: Elliptic-cone results.
  • Figure 5: Clebsch cube results.
  • ...and 4 more figures

Theorems & Definitions (11)

  • Example 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Example 2
  • Example 3
  • Remark 4
  • Lemma 5
  • Proposition 6
  • Proposition 7
  • ...and 1 more