Atomic Norm Denoising for Complex Exponentials With Unknown Waveform Modulations

Abstract

Non-stationary blind super-resolution is an extension of the traditional super-resolution problem, which deals with the problem of recovering fine details from coarse measurements. The non-stationary blind super-resolution problem appears in many applications including radar imaging, 3D single-molecule microscopy, computational photography, etc. There is a growing interest in solving non-stationary blind super-resolution task with convex methods due to their robustness to noise and strong theoretical guarantees. Motivated by the recent work on atomic norm minimization in blind inverse problems, we focus here on the signal denoising problem in non-stationary blind super-resolution. In particular, we use an atomic norm regularized least-squares problem to denoise a sum of complex exponentials with unknown waveform modulations. We quantify how the mean square error depends on the noise variance and the true signal parameters. Numerical experiments are also implemented to illustrate the theoretical result.Index terms-Atomic norm denoising, non-stationary blind deconvolution, line spectrum estimation, demodulation, mean square error.

Atomic Norm Denoising In Blind Two-Dimensional Super-Resolution

“…The significance of this work is that, to the best of our knowledge, it is the first work to address atomic norm denoising beyond the 1D blind super-resolution scheme. The work in this paper is closely related to that in [13], [20] and [19]. The focus of [13], [20] is on blind 2D super-resolution in the noise-free case, whereas in this paper, we consider the case in which the signal is contaminated by noise.…”

“…On the other hand, the work in [19] addresses the problem of blind super-resolution in 1D space. Extending the 1D framework to higher dimensions is non-trivial and comes with significant mathematical differences due to multiple reasons.…”

“…Extending the 1D framework to higher dimensions is non-trivial and comes with significant mathematical differences due to multiple reasons. First, the existence of the solution for the problem in [19] is shown by using a 1D polynomial that consists of multiple shifted versions of a single kernel. Such formulation fails in our problem as our 2D trigonometric vector polynomial has to satisfy different constraints.…”

“…Thus, we propose using multiple different kernels, as we will show in Section VI. Second, the MSE in [19] does not depend explicitly on the dimensionality of the problem and is entirely dependent on the formulation. Third, while [19] is based on using a 1D atomic norm, which is shown to be reformulated as an SDP, such reformulation does not exist for MD atomic norm and an alternative approach that is based on relaxing the problem is applied.…”

“…An atomic norm denoising algorithm for a mixture of sinusoidal from noisy samples is proposed in [18] in which the denoising error is derived and shown to be directly proportional to the noise level and inversely proportional to the number of samples. Finally, the authors in [19] study the problem of denoising a sum of complex exponentials with unknown waveform modulations using an atomic norm regularized least-squares problem. The theoretical MSE of the estimator is derived and shown to be proportional to the noise variance and the actual signal parameters.…”