Fitting instead of annihilation: Improved recovery of noisy FRI signals

Gilliam, Christopher; Blu, Thierry

doi:10.1109/icassp.2014.6853556

Cited by 17 publications

(5 citation statements)

References 22 publications

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“…3) Stopping Criteria: Because of the non-convexity of (P1), we should not expect the algorithm to always find the global optimal solution in general. In fact, it is not the optimal solution of (P1) that we should seek but rather a solution that (i) satisfies the annihilation constraint and (ii) has a fitting error a − Gb 2 2 below the noise level [41]. After all, our goal is to use the constrained minimization as a tool to find a valid solution of (P2) -any solution that meets the two criteria is a valid one for the FRI reconstruction.…”

Section: B Essential Ingredientsmentioning

confidence: 99%

Towards Generalized FRI Sampling With an Application to Source Resolution in Radioastronomy

Pan

Blu

Vetterli

2017

IEEE Trans. Signal Process.

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100

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It is a classic problem to estimate continuous-time sparse signals, like point sources in a direction-of-arrival problem, or pulses in a time-of-flight measurement. The earliest occurrence is the estimation of sinusoids in time series using Prony's method. This is at the root of a substantial line of work on high resolution spectral estimation. The estimation of continuous-time sparse signals from discrete-time samples is the goal of the sampling theory for finite rate of innovation (FRI) signals. Both spectral estimation and FRI sampling usually assume uniform sampling. But not all measurements are obtained uniformly, as exemplified by a concrete radioastronomy problem we set out to solve. Thus, we develop the theory and algorithm to reconstruct sparse signals, typically sum of sinusoids, from nonuniform samples. We achieve this by identifying a linear transformation that relates the unknown uniform samples of sinusoids to the given measurements. These uniform samples are known to satisfy the annihilation equations. A valid solution is then obtained by solving a constrained minimization such that the reconstructed signal is consistent with the given measurements and satisfies the annihilation constraint. Thanks to this new approach, we unify a variety of FRI-based methods. We demonstrate the versatility and robustness of the proposed approach with five FRI reconstruction problems, namely Dirac reconstructions with irregular time or Fourier domain samples, FRI curve reconstructions, Dirac reconstructions on the sphere, and point source reconstructions in radioastronomy. The proposed algorithm improves substantially over state-of-the-art methods and is able to reconstruct point sources accurately from irregularly sampled Fourier measurements under severe noise conditions.

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Section: B Essential Ingredientsmentioning

confidence: 99%

Towards Generalized FRI Sampling With an Application to Source Resolution in Radioastronomy

Pan

Blu

Vetterli

2017

IEEE Trans. Signal Process.

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100

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“…The Fourier transform, over-zero detection, and other conventional methods have the problem of low estimation accuracy [3]. The FRI algorithm [4][5][6][7][8] achieves the estimation of multiple signals based on the finite rate of innovation model through the zeroed filter with the polynomial ratio method, which has gained applications in DOA and frequency estimation.…”

Section: Introductionmentioning

confidence: 99%

Frequency estimation of underwater acoustic targets based on improved FRI method

Wang,

Guo,

Sun

2024

J. Phys.: Conf. Ser.

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The line spectrum information of a ship’s radiated noise signal is often regarded as an important feature used to distinguish between targets. In this paper, an improved FRI (finite rate of innovation) method is proposed to estimate the low-frequency line spectra of the radiated noise from underwater acoustic passive targets under the condition of finite sampling points. First, a preliminary rough estimation of the frequency is performed using the Fourier transform. Then, the multichannel signal data are iteratively fitted based on the polynomial-ratio form model. The optimal estimation results are finally determined based on the residuals between the data and the model in the fitting process. Simulation experiments prove that compared with the original FRI algorithm, the method proposed in this paper can estimate line spectral frequency more efficiently under the condition of low signal-to-noise ratio (SNR), and can achieve fast convergence with shorter operation time. The experimental data processing results prove that the method can realize high-precision extraction of line spectrum information of ship noise.

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“…Finally, another alternative method has been proposed based on fitting a signal model to the noisy data [25], [28]. Yet, robust estimation of the nonlinear signal innovations remains a challenging problem for practical applications.…”

Section: Reconstruction Of Finite Rate Of Innovationmentioning

confidence: 99%

“…Alternatively, stochastic recovery algorithms have been proposed in FRI setting based on Gibbs sampling [25]. Briefly, the idea is to draw samples from a multivariate posterior probability distribution to infer the parameters of the FRI signal by calculating some statistic measures from the drawn samples.…”

Section: Reconstruction Of Finite Rate Of Innovationmentioning

confidence: 99%

“…Proof: We start by replacing the original problem (20) with the following unconstrained minimization problem (25) where is the Lagrangian to be minimized, is a vector Lagrange multiplier for the constraints and is a scalar Lagrange multiplier for the constraint . The solution is then obtained by setting all the partial derivatives of with respect to to 0…”

Section: Fri Model Fittingmentioning

confidence: 99%

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Reconstruction of Finite Rate of Innovation Signals with Model-Fitting Approach

Dogan

Gilliam

Blu

et al. 2015

IEEE Trans. Signal Process.

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Abstract-Finite rate of innovation (FRI) is a recent framework for sampling and reconstruction of a large class of parametric signals that are characterized by finite number of innovations (parameters) per unit interval. In the absence of noise, exact recovery of FRI signals has been demonstrated. In the noisy scenario, there exist techniques to deal with non-ideal measurements. Yet, the accuracy and resiliency to noise and model mismatch are still challenging problems for real-world applications. We address the reconstruction of FRI signals, specifically a stream of Diracs, from few signal samples degraded by noise and we propose a new FRI reconstruction method that is based on a model-fitting approach related to the structured-TLS problem. The model-fitting method is based on minimizing the training error, that is, the error between the computed and the recovered moments (i.e., the FRI-samples of the signal), subject to an annihilation system. We present our framework for three different constraints of the annihilation system. Moreover, we propose a model order selection framework to determine the innovation rate of the signal; i.e., the number of Diracs by estimating the noise level through the training error curve. We compare the performance of the model-fitting approach with known FRI reconstruction algorithms and Cramér-Rao's lower bound (CRLB) to validate these contributions.Index Terms-Annihilating Filter, Cadzow, Cramér-Rao's lower bound (CRLB), finite-rate-of-innovation, iterative quadratic maximum likelihood (IQML), Kumaresan-Tufts, matrix pencil, model fitting, noise, reconstruction, sampling, structured total least squares (STLS), total least squares (TLS).

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Fitting instead of annihilation: Improved recovery of noisy FRI signals

Cited by 17 publications

References 22 publications

Towards Generalized FRI Sampling With an Application to Source Resolution in Radioastronomy

Towards Generalized FRI Sampling With an Application to Source Resolution in Radioastronomy

Frequency estimation of underwater acoustic targets based on improved FRI method

Reconstruction of Finite Rate of Innovation Signals with Model-Fitting Approach

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