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Discretization, sampling, and the Fourier ratio

A. Iosevich, E. Palsson, A. Yavicoli

TL;DR

The paper shows that smooth continuous signals, when discretized, exhibit spectral compressibility captured by the Fourier ratio $FR(f)$, enabling stable recovery from incomplete random samples without explicit sparsity. By deriving explicit FR bounds for grid-discretized functions on the unit square and for bandlimited spherical signals, it connects $C^2$ regularity to predictable sampling requirements. The main contributions include deterministic bounds $FR(g)\le r_N$ and $FR_L(f)\le r_L$, and concrete sample complexities of order $\frac{r^2}{\varepsilon^2}\log(r/\varepsilon)^2\log D$ (with polylog factors on the sphere) that guarantee recovery via standard $\ell^1$ minimization. This framework unifies continuous harmonic analysis with discrete compressed sensing, offering robust missing-data imputation guarantees for gridded imaging data and spherical data without assuming sparsity in the physical domain.

Abstract

We derive fundamental sampling bounds for smooth signals in continuous settings without sparsity assumptions. By introducing the Fourier ratio as a measure of spectral compressibility induced by smoothness, we obtain explicit, deterministic bounds linking signal regularity to recoverability from incomplete random samples. For functions in $C^{2}([0,1]^{2})$ sampled on an $N$ by $N$ grid, we show that a random subset of spatial samples of size $$ C\frac{r_{N}^{2}}{\eps^{2}}\log(r_{N}/\eps)^{2}\log(N^{2}) $$ suffices, with high probability, to recover the entire discretized signal via $\ell^{1}$ minimization with relative $L^{2}$ error $O(\eps)$. We develop a parallel theory for bandlimited functions on the unit sphere, obtaining analogous recovery guarantees with sample complexity scaling polylogarithmically in the bandwidth. Our results establish smoothness as a deterministic prior that enforces compressibility in the Fourier domain, bridging continuous harmonic analysis with discrete compressed sensing in a unified information-theoretic framework.

Discretization, sampling, and the Fourier ratio

TL;DR

The paper shows that smooth continuous signals, when discretized, exhibit spectral compressibility captured by the Fourier ratio , enabling stable recovery from incomplete random samples without explicit sparsity. By deriving explicit FR bounds for grid-discretized functions on the unit square and for bandlimited spherical signals, it connects regularity to predictable sampling requirements. The main contributions include deterministic bounds and , and concrete sample complexities of order (with polylog factors on the sphere) that guarantee recovery via standard minimization. This framework unifies continuous harmonic analysis with discrete compressed sensing, offering robust missing-data imputation guarantees for gridded imaging data and spherical data without assuming sparsity in the physical domain.

Abstract

We derive fundamental sampling bounds for smooth signals in continuous settings without sparsity assumptions. By introducing the Fourier ratio as a measure of spectral compressibility induced by smoothness, we obtain explicit, deterministic bounds linking signal regularity to recoverability from incomplete random samples. For functions in sampled on an by grid, we show that a random subset of spatial samples of size suffices, with high probability, to recover the entire discretized signal via minimization with relative error . We develop a parallel theory for bandlimited functions on the unit sphere, obtaining analogous recovery guarantees with sample complexity scaling polylogarithmically in the bandwidth. Our results establish smoothness as a deterministic prior that enforces compressibility in the Fourier domain, bridging continuous harmonic analysis with discrete compressed sensing in a unified information-theoretic framework.
Paper Structure (11 sections, 8 theorems, 125 equations, 2 figures)

This paper contains 11 sections, 8 theorems, 125 equations, 2 figures.

Table of Contents

  1. Introduction
  2. A compressed sensing narrative driven by smoothness
  3. Relation to frame-based erasure-resilient design
  4. Data distribution on the unit square
  5. Data distribution on the unit sphere
  6. Proofs of the main results
  7. Proof of Proposition \ref{['prop:discretization']}
  8. Proof of Theorem \ref{['thm:ZNd_recovery']}
  9. Proof of Theorem \ref{['thm:missing-values-unit-square']}
  10. Proof of Theorem \ref{['thm:sphere_missing_values']}
  11. Proof of Theorem \ref{['thm:sphere_121']}

Key Result

Theorem 1.2

Let $f$ be a real-valued function on $[0,1]^2$ which is 1-periodic in each variable and belongs to $C^2([0,1]^2)$. Fix an integer $N\ge 2$ and define a function $g:{\mathbb Z}_N^2\to{\mathbb R}$ by Assume that Define the discrete Fourier transform of $g$ by where here and throughout, $\chi(t)=e^{2\pi i t/N}$. Note that this matches the normalization in Theorem thm:ZNd_recovery when $d=2$, since

Figures (2)

  • Figure 1: Pipeline in inverse-problem form. A smooth continuous field is sampled on a grid, a Fourier ratio bound is obtained deterministically from regularity, and stable recovery from incomplete samples follows from $\ell^1$ minimization with an explicit error bound.
  • Figure 2: Reconstruction pipeline for spherical inverse problems. A bandlimited and smooth signal on the sphere is sampled at random locations, regularity yields an explicit spherical Fourier ratio bound, and stable recovery from incomplete point measurements follows from $\ell^1$ minimization of spherical harmonic coefficients.

Theorems & Definitions (30)

  • Remark 1.1: Normalization of empirical norms
  • Theorem 1.2
  • Remark 1.3: Sampling and imputation viewpoint
  • Remark 1.4: Algorithmic interpretation
  • Remark 1.5: Interpretation of the Fourier ratio bound
  • Proposition 1.6
  • Theorem 1.7: Recovery on ${\mathbb Z}_N^d$
  • Remark 1.8: Relation to sparse spherical harmonic compressed sensing
  • Remark 1.9: Spherical sampling viewpoint
  • Theorem 1.10
  • ...and 20 more