Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Numerical Analysis of the Non-uniform Sampling Problem

Thomas Strohmer

TL;DR

The paper tackles the problem of reconstructing band-limited signals from nonuniform samples by contrasting two finite-dimensional models. It shows that naïve truncated-frame approaches are ill-posed and unstable, while a trigonometric-polynomial model yields a well-posed, structure-preserving framework with fast, FFT-based algorithms. A multilevel reconstruction algorithm is proposed to estimate bandwidth adaptively from noisy data, avoiding a priori bandwidth knowledge and mitigating overfitting. Regularization via truncated SVD or CG is discussed, along with analysis of ill-conditioning and preconditioning limitations under nonuniform sampling geometries. Practical demonstrations in spectroscopy and geophysics highlight the method’s stability, efficiency, and applicability to real-world, irregularly sampled, noisy data.

Abstract

We give an overview of recent developments in the problem of reconstructing a band-limited signal from non-uniform sampling from a numerical analysis view point. It is shown that the appropriate design of the finite-dimensional model plays a key role in the numerical solution of the non-uniform sampling problem. In the one approach (often proposed in the literature) the finite-dimensional model leads to an ill-posed problem even in very simple situations. The other approach that we consider leads to a well-posed problem that preserves important structural properties of the original infinite-dimensional problem and gives rise to efficient numerical algorithms. Furthermore a fast multilevel algorithm is presented that can reconstruct signals of unknown bandwidth from noisy non-uniformly spaced samples. We also discuss the design of efficient regularization methods for ill-conditioned reconstruction problems. Numerical examples from spectroscopy and exploration geophysics demonstrate the performance of the proposed methods.

Numerical Analysis of the Non-uniform Sampling Problem

TL;DR

The paper tackles the problem of reconstructing band-limited signals from nonuniform samples by contrasting two finite-dimensional models. It shows that naïve truncated-frame approaches are ill-posed and unstable, while a trigonometric-polynomial model yields a well-posed, structure-preserving framework with fast, FFT-based algorithms. A multilevel reconstruction algorithm is proposed to estimate bandwidth adaptively from noisy data, avoiding a priori bandwidth knowledge and mitigating overfitting. Regularization via truncated SVD or CG is discussed, along with analysis of ill-conditioning and preconditioning limitations under nonuniform sampling geometries. Practical demonstrations in spectroscopy and geophysics highlight the method’s stability, efficiency, and applicability to real-world, irregularly sampled, noisy data.

Abstract

We give an overview of recent developments in the problem of reconstructing a band-limited signal from non-uniform sampling from a numerical analysis view point. It is shown that the appropriate design of the finite-dimensional model plays a key role in the numerical solution of the non-uniform sampling problem. In the one approach (often proposed in the literature) the finite-dimensional model leads to an ill-posed problem even in very simple situations. The other approach that we consider leads to a well-posed problem that preserves important structural properties of the original infinite-dimensional problem and gives rise to efficient numerical algorithms. Furthermore a fast multilevel algorithm is presented that can reconstruct signals of unknown bandwidth from noisy non-uniformly spaced samples. We also discuss the design of efficient regularization methods for ill-conditioned reconstruction problems. Numerical examples from spectroscopy and exploration geophysics demonstrate the performance of the proposed methods.

Paper Structure

This paper contains 13 sections, 3 theorems, 51 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Nonuniform sampling, frames, and numerical algorithms
  3. Truncated frames lead to ill-posed problems
  4. Estimation of regularization parameter
  5. CG and regularization of the truncated frame method
  6. Trigonometric polynomials and efficient signal reconstruction
  7. Multi-dimensional nonuniform sampling
  8. Bandwidth estimation and regularization
  9. A multilevel signal reconstruction algorithm
  10. Solution of ill-conditioned sampling problems
  11. Applications
  12. An example from spectroscopy
  13. Approximation of geophysical potential fields

Key Result

Theorem 1.1

If the set ${\{{\hbox{sinc}}(\cdot - t_j)\}_{j \in {\mathbb Z}}}$ is a frame for ${{\boldsymbol B}}$, then the function $f \in {{\boldsymbol B}}$ is uniquely defined by the sampling set $\{f(t_j)\}_{j \in {\mathbb Z}}$. In this case we can recover $f$ from its samples by or equivalently by with $R$ being the frame Gram matrix with entries $R_{j,l}= {\hbox{sinc}}(t_j - t_l)$ and $b=\{b_j\}=\{f(t_

Figures (2)

  • Figure 1: Example from spectroscopy -- comparison of reconstruction methods.
  • Figure 2: Approximation of synthetic gravity anomaly from 1000 non-uniformly spaced noisy samples by the multilevel algorithm of Section \ref{['ss:ml']}. The algorithm iteratively determines the optimal bandwidth (i.e. level) for the approximation.

Theorems & Definitions (4)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Theorem 3.1: and Algorithm