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A hybrid reconstruction of piece-wise smooth functions from non-uniform Fourier data

Guohui Song, Congzhi Xia

TL;DR

The paper addresses reconstructing piece-wise smooth functions from non-uniform Fourier data by extending the standard filter method to non-uniform samples via admissible frames. A stable extrapolation step, following Demanet 2018, is added to recover near-edge values, enabling a hybrid reconstruction that achieves uniform exponential accuracy across the entire domain. The authors provide explicit parameter choices and error bounds, and validate the approach with numerical experiments on jittered and log-sampled data, including cases with multiple jumps. The work offers a robust, implementable framework for high-accuracy non-uniform Fourier reconstructions in applications where Gibbs phenomena hinder traditional methods.

Abstract

In this paper, we consider the problem of reconstructing piece-wise smooth functions from their non-uniform Fourier data. We first extend the filter method for uniform Fourier data to the non-uniform setting by using the techniques of admissible frames. We show that the proposed non-uniform filter method converges exponentially away from the jump discontinuities. However, the convergence rate is significantly slower near the jump discontinuities due to the Gibbs phenomenon. To overcome this issue, we combine the non-uniform filter method with a stable extrapolation method to recover the function values near the jump discontinuities. We show that the proposed hybrid method could achieve exponential accuracy uniformly on the entire domain. Numerical experiments are provided to demonstrate the performance of the proposed method.

A hybrid reconstruction of piece-wise smooth functions from non-uniform Fourier data

TL;DR

The paper addresses reconstructing piece-wise smooth functions from non-uniform Fourier data by extending the standard filter method to non-uniform samples via admissible frames. A stable extrapolation step, following Demanet 2018, is added to recover near-edge values, enabling a hybrid reconstruction that achieves uniform exponential accuracy across the entire domain. The authors provide explicit parameter choices and error bounds, and validate the approach with numerical experiments on jittered and log-sampled data, including cases with multiple jumps. The work offers a robust, implementable framework for high-accuracy non-uniform Fourier reconstructions in applications where Gibbs phenomena hinder traditional methods.

Abstract

In this paper, we consider the problem of reconstructing piece-wise smooth functions from their non-uniform Fourier data. We first extend the filter method for uniform Fourier data to the non-uniform setting by using the techniques of admissible frames. We show that the proposed non-uniform filter method converges exponentially away from the jump discontinuities. However, the convergence rate is significantly slower near the jump discontinuities due to the Gibbs phenomenon. To overcome this issue, we combine the non-uniform filter method with a stable extrapolation method to recover the function values near the jump discontinuities. We show that the proposed hybrid method could achieve exponential accuracy uniformly on the entire domain. Numerical experiments are provided to demonstrate the performance of the proposed method.
Paper Structure (6 sections, 5 theorems, 56 equations, 8 figures)

This paper contains 6 sections, 5 theorems, 56 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Non-uniform Fourier reconstruction of piece-wise smooth functions
  3. Filter method for non-uniform Fourier reconstruction
  4. Hybrid filter-extrapolation method
  5. Numerical experiments
  6. Conclusions

Key Result

Lemma 1

For any $x\in [0,1]$, if $\frac{n^{2}\gamma_{x}^{2}}{2m^{2}} \geq p_x$, then we have and

Figures (8)

  • Figure 1: Filtered reconstruction $f_{m}^{\mathop{\mathrm{filter}}\nolimits}$ (top row) and hybrid reconstruction $f_{m, M, \delta}^{\mathop{\mathrm{hybrid}}\nolimits}$ (bottom row) for the jittered sampling with $m=128, 256, 512$.
  • Figure 2: Pointwise approximation errors of the filter reconstruction $f_{m}^{\mathop{\mathrm{filter}}\nolimits}$ and the hybrid reconstruction $f_{m, M, \delta}^{\mathop{\mathrm{hybrid}}\nolimits}$ for the jittered sampling with $m=128, 256, 512$.
  • Figure 3: Filtered reconstruction $f_{m}^{\mathop{\mathrm{filter}}\nolimits}$ (top row) and hybrid reconstruction $f_{m, M, \delta}^{\mathop{\mathrm{hybrid}}\nolimits}$ (bottom row) for the log sampling with $m=128, 256, 512$.
  • Figure 4: Pointwise approximation errors of the filter reconstruction $f_{m}^{\mathop{\mathrm{filter}}\nolimits}$ and the hybrid reconstruction $f_{m, M, \delta}^{\mathop{\mathrm{hybrid}}\nolimits}$ for the log sampling with $m=128, 256, 512$.
  • Figure 5: Filtered reconstruction $f_{m}^{\mathop{\mathrm{filter}}\nolimits}$ (top row) and hybrid reconstruction $f_{m, M, \delta}^{\mathop{\mathrm{hybrid}}\nolimits}$ (bottom row) for the jittered sampling with $m=128, 256, 512$.
  • ...and 3 more figures

Theorems & Definitions (12)

  • Lemma 1
  • proof
  • Proposition 1
  • proof
  • Theorem 3.1
  • proof
  • Proposition 2
  • proof
  • Theorem 4.1
  • proof
  • ...and 2 more