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Algebraic Reconstruction of Piecewise-Smooth Functions of Two Variables from Fourier Data

Michael Levinov, Yosef Yomdin, Dmitry Batenkov

TL;DR

This work tackles the problem of reconstructing a 2D piecewise smooth function $F$ on $T^2$ from bandlimited Fourier data, addressing Gibbs phenomena and discontinuity recovery. It extends a 1D algebraic reconstruction framework to 2D by processing slices: first recover $ ext{psi}_{ ext{omega_y}}(x)=\hat F_x(\omega_y)$ for $|\omega_y|\le N$, then rebuild each slice $F_x$ from these data. The authors provide explicit error bounds for the jump location $\xi(x)$, the jump magnitudes $A_l(x)$, and the pointwise slice values with rates $N^{-d-2}$, $N^{l-d-1}$, and $N^{-d-1}$ under the condition $N^2 \le M$ and a positive lower bound on $A_0(x)$. Numerical experiments on synthetic data corroborate the predicted convergence and demonstrate high-accuracy recovery of the jump curve, magnitudes, and slices, highlighting potential applications in medical imaging and spectral PDE solvers.

Abstract

We investigate the problem of reconstructing a 2D piecewise smooth function from its bandlimited Fourier measurements. This is a well known and well studied problem with many real world implications, in particular in medical imaging. While many techniques have been proposed over the years to solve the problem, very few consider the accurate reconstruction of the discontinuities themselves. In this work we develop an algebraic reconstruction technique for two-dimensional functions consisting of two continuity pieces with a smooth discontinuity curve. By extending our earlier one-dimensional method, we show that both the discontinuity curve and the function itself can be reconstructed with high accuracy from a finite number of Fourier measurements. The accuracy is commensurate with the smoothness of the pieces and the discontinuity curve. We also provide a numerical implementation of the method and demonstrate its performance on synthetic data.

Algebraic Reconstruction of Piecewise-Smooth Functions of Two Variables from Fourier Data

TL;DR

This work tackles the problem of reconstructing a 2D piecewise smooth function on from bandlimited Fourier data, addressing Gibbs phenomena and discontinuity recovery. It extends a 1D algebraic reconstruction framework to 2D by processing slices: first recover for , then rebuild each slice from these data. The authors provide explicit error bounds for the jump location , the jump magnitudes , and the pointwise slice values with rates , , and under the condition and a positive lower bound on . Numerical experiments on synthetic data corroborate the predicted convergence and demonstrate high-accuracy recovery of the jump curve, magnitudes, and slices, highlighting potential applications in medical imaging and spectral PDE solvers.

Abstract

We investigate the problem of reconstructing a 2D piecewise smooth function from its bandlimited Fourier measurements. This is a well known and well studied problem with many real world implications, in particular in medical imaging. While many techniques have been proposed over the years to solve the problem, very few consider the accurate reconstruction of the discontinuities themselves. In this work we develop an algebraic reconstruction technique for two-dimensional functions consisting of two continuity pieces with a smooth discontinuity curve. By extending our earlier one-dimensional method, we show that both the discontinuity curve and the function itself can be reconstructed with high accuracy from a finite number of Fourier measurements. The accuracy is commensurate with the smoothness of the pieces and the discontinuity curve. We also provide a numerical implementation of the method and demonstrate its performance on synthetic data.

Paper Structure

This paper contains 22 sections, 23 theorems, 242 equations, 3 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Problem setting
  3. Summary of our results
  4. Related work
  5. Code availability
  6. Acknowledgements
  7. Organization
  8. The algebraic reconstruction method
  9. The first stage
  10. Proof of \ref{['thm:main-result-for-psi']}
  11. The second stage
  12. Proof of \ref{['thm:general-theorem-for-fx']}
  13. Jump location
  14. Jump magnitudes
  15. Pointwise values
  16. ...and 7 more sections

Key Result

Lemma 1.1

Let $\xi$ as in eq:boundary-curve s.t. $\xi\in C_{(-\pi,\pi)}^{d_{\xi}}$ where $d_{\xi}\geq d$, then for each $\omega_{y}\in\mathbb{Z}$, $\psi_{\omega_{y}}$ satisfies the following properties:

Figures (3)

  • Figure 1: Our model of the piecewise-smooth 2D function $F: \IfEqCase{2}{ {1}{\mathbb{T}}} [\mathbb{T}^{2}]\to\mathbb{R}$ with two continuity pieces. \ref{['fig:graph_for_algebraic reconstruction method_02']} The graph of $F$. \ref{['fig:graph_for_algebraic reconstruction method_01']} The continuity domains $D_1,D_2$ and and the jump curve $\Sigma$.
  • Figure 2: Exact function with its discontinuity curve and approximated functions with its approximated discontinuity curve
  • Figure 3: Approximation errors computing by our algorithm for the function in \ref{['fig:approximate-F']}.

Theorems & Definitions (52)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Lemma 1.1
  • Theorem 1
  • Theorem 2
  • Proposition 6
  • proof
  • ...and 42 more