[Paper Review] Weighted frames of exponentials and stable recovery of multidimensional functions from nonuniform Fourier samples

Ben Adcock, Milana Gatarić+1 more|arXiv (Cornell University)|May 13, 2014

Mathematical Analysis and Transform Methods43 references3 citations

TL;DR

This paper establishes that nonuniform Fourier sampling points with sufficient density—without requiring minimal separation—can stably recover multidimensional, compactly supported functions via weighted Fourier frames. It provides explicit, dimensionless frame bounds for functions supported on spheres and enables stable, quasi-optimal reconstruction using the NUGS algorithm, validated on practical sampling schemes like radial and spiral patterns.

ABSTRACT

In this paper, we consider the problem of recovering a compactly supported multivariate function from a collection of pointwise samples of its Fourier transform taken nonuniformly. We do this by using the concept of weighted Fourier frames. A seminal result of Beurling shows that sample points give rise to a classical Fourier frame provided they are relatively separated and of sufficient density. However, this result does not allow for arbitrary clustering of sample points, as is often the case in practice. Whilst keeping the density condition sharp and dimension independent, our first result removes the separation condition and shows that density alone suffices. However, this result does not lead to estimates for the frame bounds. A known result of Groechenig provides explicit estimates, but only subject to a density condition that deteriorates linearly with dimension. In our second result we improve these bounds by reducing the dimension dependence. In particular, we provide explicit frame bounds which are dimensionless for functions having compact support contained in a sphere. Next, we demonstrate how our two main results give new insight into a reconstruction algorithm---based on the existing generalized sampling framework---that allows for stable and quasi-optimal reconstruction in any particular basis from a finite collection of samples. Finally, we construct sufficiently dense sampling schemes that are often used in practice---jittered, radial and spiral sampling schemes---and provide several examples illustrating the effectiveness of our approach when tested on these schemes.

Motivation & Objective

• To remove the minimal separation requirement between nonuniform Fourier sampling points while preserving stable recovery of compactly supported multivariate functions.
• To provide explicit, dimension-independent frame bounds for weighted Fourier frames when functions are supported on a sphere.
• To extend the generalized sampling framework (NUGS) to the multivariate setting for stable and quasi-optimal reconstruction from finite, nonuniform samples.
• To analyze practical sampling schemes—jittered, radial, and spiral—under the new theoretical framework and validate their effectiveness numerically.
• To establish a theoretical foundation that guarantees stability and accuracy of reconstruction in any finite-dimensional function space using nonuniform samples.

Proposed method

• Introduce weighted Fourier frames to handle nonuniform sampling without requiring separation between sampling points.
• Leverage Beurling's density condition with a sharp, dimension-independent threshold: δ_E^∘ < 1/4 for compact, convex, symmetric supports.
• Derive explicit frame bounds that are dimensionless for functions supported on a sphere, improving upon Gröchenig’s bounds with linear dimension dependence.
• Apply the NUGS (Nonuniform Generalized Sampling) framework to reconstruct functions in any finite-dimensional space from nonuniform Fourier samples.
• Construct and analyze practical sampling schemes (jittered, radial, spiral) that satisfy the required density condition δ_E^∘ < 1/4.
• Use numerical experiments to demonstrate stable and quasi-optimal reconstruction on test functions using these sampling patterns.

Experimental results

Research questions

• RQ1 Can the minimal separation condition in nonuniform Fourier sampling be removed while still ensuring stable recovery of multidimensional functions?
• RQ2 What are explicit, dimension-independent frame bounds for weighted Fourier frames when the function support is a sphere?
• RQ3 How can the NUGS algorithm be extended and justified in the multivariate setting for stable reconstruction from nonuniform samples?
• RQ4 Do practical sampling schemes like radial and spiral trajectories satisfy the theoretical density condition δ_E^∘ < 1/4 and enable stable reconstruction?
• RQ5 What is the relationship between sampling bandwidth K and reconstruction accuracy in the NUGS framework for nonuniform samples?

Key findings

• Density alone, with δ_E^∘ < 1/4, is sufficient for a weighted Fourier frame to exist, even without separation between sampling points.
• Explicit frame bounds are derived that are dimensionless for functions supported on a sphere, significantly improving upon Gröchenig’s bounds with linear dimension dependence.
• The NUGS algorithm ensures stable and quasi-optimal reconstruction when the sampling density satisfies δ_E^∘ < 1/4 and the sampling bandwidth K is sufficiently large.
• Jittered, radial, and spiral sampling schemes all satisfy the required (K, δ_E^∘)-density condition with δ_E^∘ < 0.25, enabling stable recovery.
• Numerical experiments confirm that reconstructions from these schemes achieve high accuracy, even at low densities, when the theoretical conditions are met.
• The theoretical framework supports reconstruction in any finite-dimensional space, including wavelets, polynomials, and curvelets, with stable recovery guaranteed under the density condition.