Reconstruction of Sparse Signals under Gaussian Noise and Saturation (original) (raw)
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2021
Most compressed sensing algorithms do not account for the effect of saturation in noisy compressed measurements, though saturation is an important consequence of the limited dynamic range of existing sensors. The few algorithms that handle saturation effects either simply discard saturated measurements, or impose additional constraints to ensure consistency of the estimated signal with the saturated measurements (based on a known saturation threshold) given uniform-bounded noise. In this paper, we instead propose a new data fidelity function which is directly based on ensuring a certain form of consistency between the signal and the saturated measurements, and can be expressed as the negative logarithm of a certain carefully designed likelihood function. Our estimator works even in the case of Gaussian noise (which is unbounded) in the measurements. We prove that our data fidelity function is convex. We moreover, show that it satisfies the condition of Restricted Strong Convexity an...
Non-Convex Compressed Sensing from Noisy Measurements
This paper proposes solution to the following non-convex optimization problem: min || x ||p subject to || y .. Ax ||q Such an optimization problem arises in a rapidly advancing branch of signal processing called ‘Compressed Sensing’ (CS). The problem of CS is to reconstruct a k-sparse vector xnX1, from noisy measurements y = Ax +.. , where AmXn (m<n) is the measurement matrix and ..mX1 is additive noise. In general the optimization methods developed for CS minimizes a sparsity promoting l1-norm (p=1) for Gaussian noise (q=2). This is restrictive for two reasons: i) theoretically it has been shown that, with positive fractional norms (0<p<1), the sparse vector x can be reconstructed by fewer measurements than required by l1-norm; and ii) Noises other than Gaussian require the norm of the misfit (q) to be something other than 2. To address these two issues an Iterative Reweighted Least Squares based algorithm is proposed here to solve the aforesaid optimization problem.
Detection and estimation with compressive measurements
Dept. of ECE, Rice University, …, 2006
The recently introduced theory of compressed sensing enables the reconstruction of sparse or compressible signals from a small set of nonadaptive, linear measurements. If properly chosen, the number of measurements can be much smaller than the number of Nyquist rate samples. Interestingly, it has been shown that random projections are a satisfactory measurement scheme. This has inspired the design of physical systems that directly implement similar measurement schemes. However, despite the intense focus on the reconstruction of signals, many (if not most) signal processing problems do not require a full reconstruction of the signal—we are often interested only in solving some sort of detection problem or in the estimation of some function of the data. In this report, we show that the compressed sensing framework is useful for a wide range of statistical inference tasks. In particular, we demonstrate how to solve a variety of signal detection and estimation problems given the measurements without ever reconstructing the signals themselves. We provide theoretical bounds along with experimental results.
A short note on non-convex compressed sensing
In this note, we summarize the results we recently proved in [14] on the theoretical performance guarantees of the decoders ∆p. These decoders rely on ℓp minimization with p ∈ (0, 1) to recover estimates of sparse and compressible signals from incomplete and inaccurate measurements. Our guarantees generalize the results of [2] and [16] about decoding by ℓp minimization with p = 1, to the setting where p ∈ (0, 1) and are obtained under weaker sufficient conditions. We also present novel extensions of our results in [14] that follow from the recent work of De-Vore et al. in [8]. Finally, we show some insightful numerical experiments displaying the trade-off in the choice of p ∈ (0, 1] depending on certain properties of the input signal.
Convex Feasibility Methods for Compressed Sensing
We present a computationally-efficient method for recovering sparse signals from a series of noisy observations, known as the problem of compressed sensing (CS). CS theory requires solving a convex constrained minimization problem. We propose to transform this optimization problem into a convex feasibility problem (CFP), and solve it using subgradient projection methods, which are iterative, fast, robust and convergent schemes for solving CFPs. As opposed to some of the recentlyintroduced CS algorithms, such as Bayesian CS and gradient projections for sparse reconstruction, which become prohibitively inefficient as the problem dimension and sparseness degree increase, the newly-proposed methods exhibit a marked robustness with respect to these factors. This renders the subgradient projection methods highly viable for large-scale compressible scenarios.
Compressed Sensing with Non-Gaussian Noise and Partial Support Information
IEEE Signal Processing Letters, 2015
We study the problem of recovering sparse and compressible signals using a weighted minimization with from noisy compressed sensing measurements when part of the support is known a priori. To better model different types of non-Gaussian (bounded) noise, the minimization program is subject to a data-fidelity constraint expressed as the norm of the residual error. We show theoretically that the reconstruction error of this optimization is bounded (stable) if the sensing matrix satisfies an extended restricted isometry property. Numerical results show that the proposed method, which extends the range of and comparing with previous works, outperforms other noise-aware basis pursuit programs. For , since the optimization is not convex, we use a variant of an iterative reweighted algorithm for computing a local minimum.
Robust sampling and reconstruction methods for compressed sensing
2009 IEEE International Conference on Acoustics, Speech and Signal Processing, 2009
Recent results in compressed sensing show that a sparse or compressible signal can be reconstructed from a few incoherent measurements. Since noise is always present in practical data acquisition systems, sensing, and reconstruction methods are developed assuming a Gaussian (light-tailed) model for the corrupting noise. However, when the underlying signal and/or the measurements are corrupted by impulsive noise, commonly employed linear sampling operators, coupled with current reconstruction algorithms, fail to recover a close approximation of the signal. In this paper, we propose robust methods for sampling and reconstructing sparse signals in the presence of impulsive noise. To solve the problem of impulsive noise embedded in the underlying signal prior the measurement process, we propose a robust nonlinear measurement operator based on the weighed myriad estimator. In addition, we introduce a geometric optimization problem based on 1 minimization employing a Lorentzian norm constraint on the residual error to recover sparse signals from noisy measurements. Analysis of the proposed methods show that in impulsive environments when the noise posses infinite variance we have a finite reconstruction error and furthermore these methods yield successful reconstruction of the desired signal. Simulations demonstrate that the proposed methods significantly outperform commonly employed compressed sensing sampling and reconstruction techniques in impulsive environments, while providing comparable performance in less demanding, light-tailed environments.
Performance Analysis of Compressed Sensing Given Insufficient Random Measurements
Etri Journal, 2013
Most of the literature on compressed sensing has not paid enough attention to scenarios in which the number of acquired measurements is insufficient to satisfy minimal exact reconstruction requirements. In practice, encountering such scenarios is highly likely, either intentionally or unintentionally, that is, due to high sensing cost or to the lack of knowledge of signal properties. We analyze signal reconstruction performance in this setting. The main result is an expression of the reconstruction error as a function of the number of acquired measurements.
Support recovery in compressed sensing: An estimation theoretic approach
2009 IEEE International Symposium on Information Theory, 2009
Compressed sensing (CS) deals with the reconstruction of sparse signals from a small number of linear measurements. One of the main challenges in CS is to find the support of a sparse signal from a set of noisy observations. In the CS literature, several information-theoretic bounds on the scaling law of the required number of measurements for exact support recovery have been derived, where the focus is mainly on random measurement matrices.
IJERT-Compressive Sensing Reconstruction for Sparse Signals with Convex Optimization
International Journal of Engineering Research and Technology (IJERT), 2014
https://www.ijert.org/compressive-sensing-reconstruction-for-sparse-signals-with-convex-optimization https://www.ijert.org/research/compressive-sensing-reconstruction-for-sparse-signals-with-convex-optimization-IJERTV3IS090011.pdf The theory of compressive sampling (CS), also known as compressed sensing. It is a modern sensing scheme that goes against the common theory in data acquisition. The CS theory claims that one can recover images or signals from fewer samples or measurements than the traditional methods use. To achieve this recovery, CS theory depends on two basic principles: the first is the sparsity of signal, which relates to the signals of interest, and the incoherence, which relates to the sensing method. In this paper we will give a simple review on the CS theory and the analog to information (AIC) system will be discussed briefly supported with two examples of signal reconstruction from undersampled signals. Simulation results show the powerful of the CS reconstruction for both sparse in time and spars in frequency signals.