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A Data-Driven Interpolation Method on Smooth Manifolds via Diffusion Processes and Voronoi Tessellations

Alvaro Almeida Gomez

Abstract

We propose a data-driven interpolation method for approximating real-valued functions on smooth manifolds, based on the Laplace--Beltrami operator and Voronoi tessellations. Given pointwise evaluations of a function, the method constructs a continuous extension over the manifold by exploiting diffusion processes and the intrinsic geometry of the data. The proposed approach is entirely data-driven and requires neither a training phase nor any preprocessing prior to inference. Furthermore, the computational complexity of the inference step scales linearly in the number of sample points, thereby providing substantial improvements in scalability and computational efficiency compared to classical data driven interpolation methods, including neural networks, radial basis function networks, and Gaussian process regression. We further show that the interpolant has vanishing gradient at the interpolation points and, with high probability as the number of samples increases, attenuates high-frequency components of the signal. Moreover, the proposed method minimizes a total variation-type energy, thereby yielding a closed-form analytical approximation to the compressed sensing problem in the case where the forward operator is the identity. Finally, we present applications to sparse computational tomography reconstruction. Numerical experiments demonstrate that the proposed method achieves competitive reconstruction quality while significantly reducing computational time compared to classical total variation-based reconstruction methods.

A Data-Driven Interpolation Method on Smooth Manifolds via Diffusion Processes and Voronoi Tessellations

Abstract

We propose a data-driven interpolation method for approximating real-valued functions on smooth manifolds, based on the Laplace--Beltrami operator and Voronoi tessellations. Given pointwise evaluations of a function, the method constructs a continuous extension over the manifold by exploiting diffusion processes and the intrinsic geometry of the data. The proposed approach is entirely data-driven and requires neither a training phase nor any preprocessing prior to inference. Furthermore, the computational complexity of the inference step scales linearly in the number of sample points, thereby providing substantial improvements in scalability and computational efficiency compared to classical data driven interpolation methods, including neural networks, radial basis function networks, and Gaussian process regression. We further show that the interpolant has vanishing gradient at the interpolation points and, with high probability as the number of samples increases, attenuates high-frequency components of the signal. Moreover, the proposed method minimizes a total variation-type energy, thereby yielding a closed-form analytical approximation to the compressed sensing problem in the case where the forward operator is the identity. Finally, we present applications to sparse computational tomography reconstruction. Numerical experiments demonstrate that the proposed method achieves competitive reconstruction quality while significantly reducing computational time compared to classical total variation-based reconstruction methods.

Paper Structure

This paper contains 18 sections, 8 theorems, 130 equations, 16 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Proposed Methodology
  3. Gaussian approximations
  4. Infinitesimal scale epsilon, Voronoi cells, and numerical stability
  5. Computational Complexity
  6. Smoothness and Decay Properties
  7. High-Frequency Attenuation via Gradient Smoothing
  8. Applications
  9. Data-driven numerical interpolation
  10. Angular Regularization for Sparse-Angle Tomographic Projections
  11. Relation to compressed sensing.
  12. Numerical Experiments
  13. Synthetic Experiment on the Square [0,1] x [0,1]
  14. Sparse-View Tomographic Reconstruction
  15. Conclusion and Future Work
  16. ...and 3 more sections

Key Result

Theorem 2.1

Let $g \in C^{\infty}(\mathcal{M})$. Then its Gaussian approximation $g_{\varepsilon}$ satisfies the infinitesimal expansion where $\Delta$ denotes the Laplace--Beltrami operator. In particular, $g_{\varepsilon}$ converges pointwise to $g$ as $\varepsilon \to 0$: $\blacktriangleleft$$\blacktriangleleft$

Figures (16)

  • Figure 1: The domain $\mathcal{M} = [0,1]^2$ and the function $f(x,y) = \sin(10\pi x)\cos(10\pi y)$.
  • Figure 2: Interpolation results over the domain $[0,1]^2$ using different data-driven methods. Each block contains five subplots. The top block corresponds to training with 100 samples, and the bottom block to 500 samples. In both cases, $10^4$ out-of-sample points are used for evaluation. The panel labeled $f(x)$ shows the training data, while the others display the predictions of the proposed method, a feedforward neural network, Gaussian process regression, and a radial basis function network.
  • Figure 3: Interpolation results over the domain $[0,1]^2$ using different data-driven methods. Each block contains five subplots. The top block corresponds to training with 1000 samples, and the bottom block to 2000 samples. In both cases, $10^4$ out-of-sample points are used for evaluation. The panel labeled $f(x)$ shows the training data, while the others display the predictions of the proposed method, a feedforward neural network, Gaussian process regression, and a radial basis function network.
  • Figure 4: From left to right: Shepp-Logan phantom, human head, and human abdomen. The human images were taken from imagesrad.
  • Figure 5: Sinograms of tomographic projections of the Shepp--Logan phantom reconstructed using different interpolation methods. The first two rows correspond to $100$ angular samples, while the last two rows correspond to $300$ angular samples. From left to right, the panels represent: Noisy sinogram (noisy sparse projections from the training set), Proposed method, Cubic spline interpolation, Gaussian process regression (GPR), Feed-forward neural network (FNN), and Radial basis function (RBF) network.
  • ...and 11 more figures

Theorems & Definitions (8)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Lemma C.1
  • Lemma C.2
  • Lemma C.3
  • Lemma C.4