Stéphanie Bernhardt

Compressed Sensing with Basis Mismatch: Performance Bounds and Sparse-Based Estimator

Compressed sensing (CS) is a promising emerging domain that outperforms the classical limit of the Shannon sampling theory if the measurement vector can be approximated as the linear combination of few basis vectors extracted from a redundant dictionary matrix. Unfortunately, in realistic scenario, the knowledge of this basis or equivalently of the entire dictionary is often uncertain, i.e., corrupted by a Basis Mismatch (BM) error. The consequence of the BM problem is that the estimation accuracy in terms of Bayesian Mean Square Error (BMSE) of popular sparse-based estimators collapses even if the support is perfectly estimated and in the high Signal to Noise Ratio (SNR) regime. This saturation effect considerably limits the effective viability of these estimation schemes. In the first part of this work, the Bayesian Cramér-Rao Bound (BCRB) is derived for CS model with unstructured BM. We show that the BCRB foresees the saturation effect of the estimation accuracy of standard sparse-based estimators as for instance the OMP, Cosamp or the BP. In addition, we provide an approximation of this BMSE threshold. In the second part and in the context of the structured BM model, a new estimation scheme called Bias-Correction Estimator (BiCE) is proposed and its statistical properties are studied. The BiCE acts as a post-processing estimation layer for any sparse-based estimator and mitigates considerably the BM degradation. Finally, the BiCE i) is a blind algorithm, i.e., is unaware of the uncorrupted dictionary matrix, ii) is generic since it can be associated to any sparse-based estimator, iii) is fast, i.e., the additional computational cost remains low, and iv) has good statistical properties. To illustrate our results and propositions, the BiCE is applied in the challenging context of the compressive sampling of non-bandlimited impulsive signals.

Sampling FRI signals with the SOS kernel: Bounds and optimal kernel

Recently it has been shown that using appropriate sampling kernel, finite rate of innovation signals can be perfectly recon structed even tough they are non-bandlimited. In the presence of noise, reconstruction is achieved by an estimation procedure of all the parameters of the incoming signal. In this paper we consider the estimation of a finite stream of pulses using the Sum of Sincs (SoS) kernel. We derive the Cramér Rao Bound (BCRB) relative to the estimated parameters. The SoS kernel is used since it is configurable by a vector of weights: we propose a family of kernels which maximizes the Bayesian Fisher Information (BIM) i.e. the total amount of information about each of the parameter in the measures. The advantage of the proposed family is that it can be user-adjusted to favor one specific parameter. The variety of the resulting kernel goes from a perfect sinusoid to the Dirichlet kernel.

Compressed Sensing with uncertainty - the Bayesian estimation perspective

The Compressed Sensing (CS) framework outperforms the sampling rate limits given by Shannons theory. This gap is possible since it is assumed that the signal of interest admits a linear decomposition of few vectors in a given sparsifying Basis (Fourier, Wavelet, ...). Unfortunately in realistic operating systems, uncertain knowledge of the CS model is inevitable and must be evaluated. Typically, this uncertainty drastically degrades the estimation performance of sparse-based estimators in the low noise variance regime. In this work, the Off-Grid (OG) and Basis Mismatch (BM) problems are compared in a Bayesian estimation perspective. At first glance, we are tempted to think that these two acronyms stand for the same problem. However, by comparing their Bayesian Cramèr-Rao Bounds (BCRB) for the estimation of a L-sparse amplitude vector based on N measurements, it is shown that the BM problem has a lower BCRB than the OG one in a general context. To go further into the analysis we provide for i.i.d. Gaussian amplitudes and in the low noise variance regime an interesting closed-form expression of a normalized 2-norm criterion of the difference of the two BCRB matrices. Based on the analysis of this closed-form expression, we obtain two conclusions. Firstly, the two uncertainty problems cannot be confused for a non-zero mismatch error variance and with finite N and L. Secondly, the two problems turn to be similar for any mismatch error variance in the large system regime, i.e., for N, L → ∞ with constant aspect ratio N/L → ρ.

Performance analysis for sparse based biased estimator: Application to line spectra analysis

Dictionary based sparse estimators are based on the matching of continuous parameters of interest to a discretized sampling grid. Generally, the parameters of interest do not lie on this grid and there exists an estimator bias even at high Signal to Noise Ratio (SNR). This is the off-grid problem. In this work, we propose and study analytical expressions of the Bayesian Mean Square Error (BMSE) of dictionary based biased estimators at high SNR. We also show that this class of estimators is efficient and thus reaches the Bayesian Cramér-Rao Bound (BCRB) at high SNR. The proposed results are illustrated in the context of line spectra analysis and several popular sparse estimators are compared to our closed-form expressions of the BMSE.

Sparse-based estimators improvement in case of Basis mismatch

Compressed sensing theory promises to sample sparse signals using a limited number of samples. It also resolves the problem of under-determined systems of linear equations when the unknown vector is sparse. Those promising applications induced a growing interest for this field in the past decade. In compressed sensing, the sparse signal estimation is performed using the knowledge of the dictionary used to sample the signal. However, dictionary mismatch often occurs in practical applications, in which case the estimation algorithm uses an uncertain dictionary knowledge. This mismatch introduces an estimation bias even when the noise is low and the support (i.e. location of non-zero amplitudes) is perfectly estimated. In this paper we consider that the dictionary suffers from a structured mismatch, this type of error being of particular interest in sparse estimation applications. We propose the Bias-Correction Estimator (BiCE) post-processing step which enhances the non-zero amplitude estimation of any sparse-based estimator in the presence of a structured dictionary mismatch. We give the theoretical Bayesian Mean Square Error of the proposed estimator and show its statistical efficiency in the low noise variance regime.