Super-resolution via superset selection and pruning

Abstract: We present a pursuit-like algorithm that we call the "superset method" for recovery of sparse vectors from consecutive Fourier measurements in the super-resolution regime. The algorithm has a subspace identification step that hinges on the translation invariance of the Fourier transform, followed by a removal step to estimate the solution's support. The superset method is always successful in the noiseless regime (unlike L1-minimization) and generalizes to higher dimensions (unlike the matrix pencil method). Relative robustness to noise is demonstrated numerically.

Super-resolution via Superset Selection and Pruning

The paper, authored by Laurent Demanet, Deanna Needell, and Nam Nguyen, introduces a novel pursuit-like algorithm, termed the "superset method," for the recovery of sparse vectors from consecutive Fourier measurements in the super-resolution regime. This method emphasizes two distinct computational steps—subspace identification and removal—to improve sparse recovery performance, offering a compelling alternative to existing techniques such as $\ell_1$ minimization and the matrix pencil method.

Overview

The superset method leverages the translation invariance of the Fourier transform to identify and refine the support set of sparse vectors. The procedure begins with a subspace identification phase, analogous to the matrix pencil method, but circumvents intricate eigenvalue computations. This initial phase relies on constructing a Hankel matrix from the Fourier measurements, facilitating the determination of the subspace spanned by the signal components. The removal step, reminiscent of certain greedy pursuit algorithms, further refines the set of potential solution components identified in the first step, achieving a more precise estimation of the signal's support.

Numerical Results and Claims

The algorithm exhibits clear advantages in the noiseless regime by consistently recovering the true sparse signal, a feat not reliably accomplished by $\ell_1$ minimization, especially when signals have closely spaced nonzero components. Numerical robustness to noise is demonstrated, asserting the method's efficacy in practical applications. Specifically, experimental results highlight the superset method's capability to accurately reconstruct signals in scenarios with high coherence—a challenge for traditional methods like matrix pencil and $\ell_1$-minimization.

In coherence-intense setups, the superset method performs admirably, recovering signals with minimal measurements even when noise is present. This positions the algorithm as a valuable tool in handling scenarios where the spacing between components in sparse vectors is considerably narrower than the classical Rayleigh limit.

Theoretical Implications

The superset method’s reliance on the inherent structure of Fourier measurements underscores its distinction from generic compressed sensing frameworks. By exploiting translation invariance, researchers can redefine recovery bounds and push the horizontal limits on the applicability of super-resolution techniques across various dimensions. This departure from conventional approaches invites a reconsideration of theoretical foundations regarding sparse recovery and super-resolution.

Future Prospects

The findings invite further exploration into algorithmic enhancement tailored to multidimensional data frameworks and potential cross-application in signal processing fields beyond super-resolution. The simplicity and scalability of the superset method suggest promising extensions into related areas and novel applications requiring precise and efficient sparse recovery amidst noise and coherence challenges.

Conclusion

The paper delivers an insightful advancement in super-resolution, proposing a meticulous method with strong numerical evidence of its effectiveness in the difficult scenarios of sparse recovery. Its unique approach and potential adaptability may catalyze future comprehensive theoretical analyses and inspire enhancements in computational techniques across the field of Fourier-based signal processing.