Finite Rate Of Innovation Signals Research Articles

ECT image reconstruction based on sensitive field expansion and optimization

Electrical capacitance tomography (ECT) is an efficient method for addressing the issue of two-phase flow monitoring. Most current methods result in low image reconstruction accuracy due to soft field issues. This paper propose an ECT image reconstruction method based on sensitive field expansion and optimization, which improve reconstruction efficiency and accuracy. Firstly, a sensitivity field optimization method based on flow pattern identification was proposed. The flow pattern recognition determines the flow pattern to which the input signal belongs and selects the sensitive field for the corresponding flow pattern. Secondly, a sensitive field expansion method based on feature extraction is proposed. The ECT image is modeled as a finite rate of innovation signal, sampled using an exponential reproducing kernel. Feature information is extracted from the input signal for data fusion, and the optimized sensitive field is expanded into a new sensitive field distribution matrix by zero-padding and random reorganization. Then, based on optimized and expanded sensitive fields, a sparse image reconstruction method is proposed. Image reconstruction by constructing comprehensive observation equation to obtain the original permittivity distribution vector of the ECT system using the sparsity of the sensitivity matrix and capacitance signal. Finally, the ECT hardware system and the upper computer measurement and imaging integration software are designed. The experimental results show that the method outperforms other existing algorithms in terms of imaging indexes, has better imaging results.

Sub-Nyquist Sampling and Measurement of FRI Signals With Additive Shot Noise

To mitigate the impact of shot noise on the parameter measurement process of Finite Rate of Innovation (FRI) signals, this paper proposes a sub-Nyquist modulation sampling system with two measurement channels. The noisy signal is sampled in the main channel after being filtered by a sinc kernel to obtain the aliasing Fourier coefficients, whereas the FRI signal is shifted out of the baseband with a cosine modulation function in the accessory channel to expose the shot noise information to the low frequency band. The burst time parameters of shot noise can be estimated using the annihilating filter method from the samples in the accessory channel, while the parameters of the FRI signal will be measured from the observations in the main channel after the shot noise information has been removed. If the Rate of Innovation of the FRI signal is ρ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>L</i></sub> and is affected by <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</i> impulses of shot noise, the entire measurement process only requires ρ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>L</i></sub> ∙ τ + 2 <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</i> points to be sampled per unit time τ. To enhance the robustness of the measurement process in actual circuit systems, a corrected annihilating filter method is proposed, and the parameter measurement error when employing this method is analyzed. Numerical and hardware experiments demonstrate that, compared to classical anti-noise algorithms, the proposed method not only measures the parameters of FRI signals contaminated by shot noise more accurately, but also significantly improves the robustness of the measurement process in non-ideal environments.

Image reconstruction based on frequency domain feature extraction for EMT

Image reconstruction of electromagnetic tomography (EMT) is often ill-posed due to the limited prior information about imaging features, leading to low reconstruction accuracy. In this paper, a novel image reconstruction method based on frequency domain feature extraction for EMT is proposed to improve the reconstruction accuracy. The two-dimensional EMT image is first modeled as a one-dimensional finite-length streams of Dirac signal, which is a typical signal with finite rate of innovation (FRI). Then the reconstruction results of the recent algorithms, such as the LBP, Landweber and TV algorithms, can be also modeled as FRI signals. The frequency domain feature of such signals can be extracted with a distributed FRI sampling system. Based on the obtained FRI samples, a new EMT measurement equation is established, and a joint image reconstruction algorithm is proposed, to improve the quality of the reconstructed image. Finally, simulation results have shown that the proposed method outperforms the related methods with better indicators such as image error and correlation coefficient.

Two-Dimensional Separable Gridless Direction-of-Arrival Estimation Based on Finite Rate of Innovation

In order to solve the problem that the gridless DOA estimation algorithms based on generalized finite rate of innovation (FRI) signal reconstruction model are not suitable for two-dimensional DOA estimation using planar array, a separable gridless DOA estimation algorithm exploiting bi-orthogonal sparse linear array (BSLA) structure is proposed in this article, which is called 2D-SGFRI. The 2D-SGFRI algorithm firstly recovers the covariance data of the virtual array formed by BSLA through the matrix completion method, so as to obtain the complete covariance data vectors about two independent parameters respectively. Next, since the covariance data vector satisfies the constraints of annihilation filter equations, the generalized FRI signal reconstruction model can be utilized to retrieve DOA from the covariance data vector. Compared with the existing DOA estimation algorithms based on generalized FRI signal reconstruction model, the 2D-SGFRI algorithm can be can be effectively applied to two-dimensional DOA estimation, and can obtain stable estimation results. At the same time, due to the reduction of the dimension of positive semidefinite matrix, the 2D-SGFRI algorithm can significantly reduce the computational complexity compared with the two-dimensional DOA estimation algorithms based on atomic norm minimization (ANM). A series of simulation experiments are shown to verify the effectiveness and superiority of 2D-SGFRI algorithm.

Stable Phaseless Sampling and Reconstruction of Real-Valued Signals with Finite Rate of Innovation

A spatial signal is defined by its evaluations on the whole domain. In this paper, we consider stable reconstruction of real-valued signals with finite rate of innovation (FRI), up to a sign, from their magnitude measurements on the whole domain or their phaseless samples on a discrete subset. FRI signals appear in many engineering applications such as magnetic resonance spectrum, ultra wide-band communication and electrocardiogram. For an FRI signal, we introduce an undirected graph to describe its topological structure, establish the equivalence between its graph connectivity and its phase retrievability by point evaluation measurements on the whole domain, apply the graph connected component decomposition to find its unique landscape decomposition and the set of FRI signals that have the same magnitude measurements. We construct discrete sets with finite density so that magnitude measurements of an FRI signal on the whole domain are determined by its phaseless samples taken on those discrete subsets, and we show that the corresponding phaseless sampling procedure has bi-Lipschitz property with respect to a new induced metric on the signal space and the standard $\ell ^{p}$ -metric on the sampling data set. In this paper, we also propose an algorithm with linear complexity to reconstruct an FRI signal from its (un)corrupted phaseless samples on the above sampling set without restriction on the noise level and apriori information whether the original FRI signal is phase retrievable. The algorithm is theoretically guaranteed to be stable, and numerically demonstrated to approximate the original FRI signal in magnitude measurements.

Reconstruction of finite rate of innovation signals in a noisy scenario: a robust, accurate estimation algorithm

The paradigmatic example of signals with finite rate of innovation (FRI) is a linear combination of a finite number of Diracs per time unit, a.k.a. spike sequence. Many researchers have investigated the problem of estimating the innovative part of a spike sequence, i.e., time instants tks and weights cks of Diracs and proposed various deterministic or stochastic algorithms, particularly while the samples were corrupted by digital noise. In the presence of noise, maximum likelihood estimation method proved to be a powerful tool for reconstructing FRI signals, which is inherently an optimization problem. Wein and Srinivasan presented an algorithm, namely IterML, for reconstruction of streams of Diracs in noisy situations, which achieved promising reconstruction error and runtime. However, IterML is prone to limited resolution of search grid for tk, so as to avoid a phenomenon known as the curse of dimensionality, that makes it an inappropriate algorithm for applications that require highly accurate reconstruction of time instants. In order to overcome this shortcoming, we introduce a novel modified local best particle swarm optimization (MLBPSO) algorithm aimed at maximizing likelihood estimation of innovative parameters of a sparse spike sequence given noisy low-pass filtered samples. We demonstrate via extensive simulations that MLBPSO algorithm outperforms the IterML in terms of robustness to noise and accuracy of estimated parameters while maintaining comparable computational cost.

Implementation of particle swarm optimization algorithm for estimating the innovative parameters of a spike sequence from noisy samples via maximum likelihood method

Since the introduction of finite rate of innovation (FRI) signals in 2002, various deterministic and stochastic techniques have been proposed to estimate the innovative parameters of FRI signal, e.g. time instants tk and weights ck belonging to a linear combination of finite number of Diracs, from its noisy samples regardless of sampling kernel's type. Having analyzed the Bayesian methods dedicated to the retrieval of signal innovations, particularly the IterML algorithm introduced by Wein & Srinivasan, we discover some limitations which still leave room for improvement. In this article, we present a novel stochastic hybrid algorithm utilizing both maximum–likelihood-estimation (MLE) and Modified Particle Swarm Optimization (MPSO) in order to improve upon IterML in terms of robustness to noise and accuracy of estimated parameters. Relying on extensive simulations, our proposed algorithm provably achieves better performance than IterML while maintaining comparable computational cost. Due to high dependency of this problem on the trade-off between the level of noise and the number of samples, we also investigate this compromise in order to achieve better reconstruction error metrics than IterML.

Exponential-Reproducing-Kernel-Based Sparse Sampling Method for Finite Rate of Innovation Signal with Arbitrary Pulse Echo Position

An exponential reproducing kernel (ERK) has been applied to data sparse sampling for finite rate of innovation (FRI) signals with the characteristic of flexible parameter setting in contrast to other sampling kernels. However, the signal reconstruction process may fail if a pulse echo is present in certain positions. To solve this problem, a novel ERK sparse sampling method was developed for arbitrary pulse echo positions. A constraint relationship between the pulse echo position and reproduced-exponent area was deduced, revealing the cause of invalid signal reconstruction from sparse sampling data. A new sampling time interval calculation algorithm is presented in this paper. Through the proposed method, FRI signals can be reconstructed accurately with a pulse echo present at arbitrary positions without increasing the quantity of sparse sampling data. The signal reconstruction effectiveness and accuracy were verified through simulation experiments. The proposed technique can be used to improve the flexibility of the ERK sparse sampling method in actual applications.

Finite-Rate-of-Innovation-Sampling-Based Super-Resolution Radar Imaging

We address the problem of estimating target locations that are sparse in a pulse-Doppler radar's unambiguous region by sampling the received signal at sub-Nyquist rates. The received signal is modeled as a finite-rate-of-innovation (FRI) signal, and the problem of estimating the delays in a single transceiver radar is formulated as one of recovery of sparse common-support (SCS) FRI signals, which arises in the context of channel estimation in multiple input, multiple output communication systems. The delays are estimated by the SCS-FRI reconstruction method. We present a new method termed delay focusing to estimate the Doppler shifts. To obtain overall performance gains, we also present an extended method called dual focusing , which combines both delay and Doppler focusing schemes, and has the capability to superresolve targets in the delay-Doppler plane. The performance of the recovery methods in the presence of noise is also analyzed. We demonstrate that the proposed estimation methods are robust to noise. Monte Carlo performance analysis in the presence of noise shows that the dual focusing method accurately resolves closely spaced targets and yields a significant decrease in normalized mean-square error of up to 10–20 dB for the estimated Doppler shifts. We also simulate the scenario where multiple targets are in a formation , that is, when they are closely spaced along both delay and Doppler axes, and show that the dual focusing method achieves a hit rate of nearly 100% at a much lower signal-to-noise ratios than that required for Doppler focusing.

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

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 Cramer–Rao’s lower bound (CRLB) to validate these contributions.

IterML: A Fast, Robust Algorithm for Estimating Signals With Finite Rate of Innovation

Recently, various methods have emerged for sub- Nyquist sampling and reconstruction of signals with finite rate of innovation (FRI). These methods seek to sample parametric signals at close to their information rate and later reconstruct the parameters of interest. Some proposed reconstruction algorithms are based on annihilating filters and root-finding. Stochastic methods based on Gibbs sampling were subsequently proposed with the intent of improving robustness to noise, but these may run too slowly for some real-time applications. We present a fast maximum-likelihood-based deterministic greedy algorithm, IterML, for reconstructing FRI signals from noisy samples. We show in simulation that it achieves comparable or better performance than previous algorithms at a much lower computational cost. We also uncover a fundamental flaw in the application of MMSE (minimum mean squared error) estimation, a technique employed by some existing methods, to the problem in question.

A finite rate of innovation algorithm for fast and accurate spike detection from two-photon calcium imaging

Objective. Inferring the times of sequences of action potentials (APs) (spike trains) from neurophysiological data is a key problem in computational neuroscience. The detection of APs from two-photon imaging of calcium signals offers certain advantages over traditional electrophysiological approaches, as up to thousands of spatially and immunohistochemically defined neurons can be recorded simultaneously. However, due to noise, dye buffering and the limited sampling rates in common microscopy configurations, accurate detection of APs from calcium time series has proved to be a difficult problem. Approach. Here we introduce a novel approach to the problem making use of finite rate of innovation (FRI) theory (Vetterli et al 2002 IEEE Trans. Signal Process. 50 1417–28). For calcium transients well fit by a single exponential, the problem is reduced to reconstructing a stream of decaying exponentials. Signals made of a combination of exponentially decaying functions with different onset times are a subclass of FRI signals, for which much theory has recently been developed by the signal processing community. Main results. We demonstrate for the first time the use of FRI theory to retrieve the timing of APs from calcium transient time series. The final algorithm is fast, non-iterative and parallelizable. Spike inference can be performed in real-time for a population of neurons and does not require any training phase or learning to initialize parameters. Significance. The algorithm has been tested with both real data (obtained by simultaneous electrophysiology and multiphoton imaging of calcium signals in cerebellar Purkinje cell dendrites), and surrogate data, and outperforms several recently proposed methods for spike train inference from calcium imaging data.

2D Finite Rate of Innovation Reconstruction Method for Step Edge and Polygon Signals in the Presence of Noise

The finite rate of innovation (FRI) principle is developed for sampling a class of non-bandlimited signals that have a finite number of degrees of freedom per unit of time, i.e., signals with FRI. This sampling scheme is later extended to three classes of sampling kernels with compact support and applied to the step edge reconstruction problem by treating the image row by row. In this paper, we regard step edges as 2D FRI signals and reconstruct them block by block. The step edge parameters are obtained from the 2D moments of a given image block. Experimentally, our technique can reconstruct the edge more precisely and track the Cramér-Rao bounds (CRBs) closely with a signal-to-noise ratio (SNR) larger than 4 dB on synthetic step edge images. Experiments on real images show that our proposed method can reconstruct the step edges under practical conditions, i.e., in the presence of various types of noise and using a real sampling kernel. The results on locating the corners of data matrix barcodes using our method also outperform some state-of-the-art barcode decoders.

Multichannel Sampling of Multidimensional Parametric Signals

The contribution of this paper is two-fold; first, we present a novel approach for sampling and reconstructing two dimensional signals with parametric structure or namely signals with Finite Rate of Innovation (FRI). The considered signals are bi-level polygons and set of 2-D Diracs and it will be shown that with the use of ACMP method, projection-slice theorem, Radon projections and exponential splines as sampling kernels, such signals can be perfectly reconstructed. Then, we present a possible extension of the theory of sampling multidimensional FRI signals, discussed above, to the case of multichannel acquisition systems. The essential issue of our considered multichannel system is that each channel receives the input signal with an unknown geometric transformation which needs to be estimated. We pose both the synchronization stage and the signal reconstruction stage as a parametric estimation problem and demonstrate that a simultaneous exact synchronization of the channels and reconstruction of the FRI signal is possible.

Performance Bounds and Design Criteria for Estimating Finite Rate of Innovation Signals

In this paper, we consider the problem of estimating finite rate of innovation (FRI) signals from noisy measurements, and specifically analyze the interaction between FRI techniques and the underlying sampling methods. We first obtain a fundamental limit on the estimation accuracy attainable regardless of the sampling method. Next, we provide a bound on the performance achievable using any specific sampling approach. Essential differences between the noisy and noise-free cases arise from this analysis. In particular, we identify settings in which noise-free recovery techniques deteriorate substantially under slight noise levels, thus quantifying the numerical instability inherent in such methods. This instability, which is only present in some families of FRI signals, is shown to be related to a specific type of structure, which can be characterized by viewing the signal model as a union of subspaces. Finally, we develop a methodology for choosing the optimal sampling kernels based on a generalization of the Karhunen--Lo\`eve transform. The results are illustrated for several types of time-delay estimation problems.

Multichannel Sampling of Signals With Finite Rate of Innovation

In this letter, we present a possible extension of the theory of sampling signals with finite rate of innovation (FRI) to the case of multichannel acquisition systems. The essential issue of a multichannel system is that each channel introduces different unknown delays and gains that need to be estimated for the calibration of the channels. We pose both the synchronization stage and the signal reconstruction stage as a parametric estimation problem and demonstrate that a simultaneous exact synchronization of the channels and reconstruction of the FRI signal is possible. We also consider the case of noisy measurements and evaluate the Cramer-Rao bounds (CRB) of the proposed system. Numerical results as well as the CRB show clearly that multichannel systems are more resilient to noise than the single-channel ones.

Compressive Sampling of EEG Signals with Finite Rate of Innovation

Analyses of electroencephalographic signals and subsequent diagnoses can only be done effectively on long term recordings that preserve the signals' morphologies. Currently, electroencephalographic signals are obtained at Nyquist rate or higher, thus introducing redundancies. Existing compression methods remove these redundancies, thereby achieving compression. We propose an alternative compression scheme based on a sampling theory developed for signals with a finite rate of innovation (FRI) which compresses electroencephalographic signals during acquisition. We model the signals as FRI signals and then sample them at their rate of innovation. The signals are thus effectively represented by a small set of Fourier coefficients corresponding to the signals' rate of innovation. Using the FRI theory, original signals can be reconstructed using this set of coefficients. Seventy-two hours of electroencephalographic recording are tested and results based on metrices used in compression literature and morphological similarities of electroencephalographic signals are presented. The proposed method achieves results comparable to that of wavelet compression methods, achieving low reconstruction errors while preserving the morphologiies of the signals. More importantly, it introduces a new framework to acquire electroencephalographic signals at their rate of innovation, thus entailing a less costly low-rate sampling device that does not waste precious computational resources.

Sampling Schemes for Multidimensional Signals With Finite Rate of Innovation

In this paper, we consider the problem of sampling signals that are nonband-limited but have finite number of degrees of freedom per unit of time and call this number the rate of innovation. Streams of Diracs and piecewise polynomials are the examples of such signals, and thus are known as signals with finite rate of innovation (FRI). We know that the classical (band-limited sine) sampling theory does not enable perfect reconstruction of such signals from their samples since they are not band-limited. However, the recent results on FRI sampling suggest that it is possible to sample and perfectly reconstruct such nonband-limited signals using a rich class of kernels. In this paper, we extend those results in higher dimensions using compactly supported kernels that reproduce polynomials (satisfy Strang-Fix conditions). In fact, the polynomial reproduction property of the kernel makes it possible to obtain the continuous moments of the signal from its samples. Using these moments and the annihilating filter method (Prony's method), the innovative part of the signal, and therefore, the signal itself is perfectly reconstructed. In particular, we present local (directional-derivatives-based) and global (complex-moments-based, Radon-transform-based) sampling schemes for classes of FRI signals such as sets of Diracs, bilevel, and planar polygons, quadrature domains (e.g., circles, ellipses, and cardioids), 2D polynomials with polygonal boundaries, and n-dimensional Diracs and convex polytopes. This work has been explored in a promising way in super-resolution algorithms and distributed compression, and might find its applications in photogrammetry, computer graphics, and machine vision.