Multichannel Blind Deconvolution Research Articles

Rolling Bearing Composite Fault Diagnosis Method Based on Enhanced Harmonic Vector Analysis.

Composite fault diagnosis of rolling bearings is very challenging work, especially when the characteristic frequency ranges of different fault types overlap. To solve this problem, an enhanced harmonic vector analysis (EHVA) method was proposed. Firstly, the wavelet threshold (WT) denoising method is used to denoise the collected vibration signals to reduce the influence of noise. Next, harmonic vector analysis (HVA) is used to remove the convolution effect of the signal transmission path, and blind separation of fault signals is carried out. The cepstrum threshold is used in HVA to enhance the harmonic structure of the signal, and a Wiener-like mask will be constructed to make the separated signals more independent in each iteration. Then, the backward projection technique is used to align the frequency scale of the separated signals, and each fault signal can be obtained from composite fault diagnosis signals. Finally, to make the fault characteristics more prominent, a kurtogram was used to find the resonant frequency band of the separated signals by calculating its spectral kurtosis. Semi-physical simulation experiments are conducted using the rolling bearing fault experiment data to verify the effectiveness of the proposed method. The results show that the proposed method, EHVA, can effectively extract the composite faults of rolling bearings. Compared to fast independent component analysis (FICA) and traditional HVA, EHVA improves separation accuracy, enhances fault characteristics, and has higher accuracy and efficiency compared to fast multichannel blind deconvolution (FMBD).

Unrolled Compressed Blind-Deconvolution

The problem of sparse multichannel blind deconvolution (S-MBD) arises frequently in many engineering applications such as radar/sonar/ultrasound imaging. To reduce its computational and implementation cost, we propose a compression method that enables blind recovery from much fewer measurements with respect to the full received signal in time. The proposed compression measures the signal through a filter followed by a subsampling, allowing for a significant reduction in implementation cost. We derive theoretical guarantees for the identifiability and recovery of a sparse filter from compressed measurements. Our results allow for the design of a wide class of compression filters. We, then, propose a data-driven unrolled learning framework to learn the compression filter and solve the S-MBD problem. The encoder is a recurrent inference network that maps compressed measurements into an estimate of sparse filters. We demonstrate that our unrolled learning method is more robust to choices of source shapes and has better recovery performance compared to optimization-based methods. Finally, in data-limited applications (fewshot learning), we highlight the superior generalization capability of unrolled learning compared to conventional deep learning.

Algorithms for Sparse Multichannel Blind Deconvolution

In this paper we present two algorithms for sparse multichannel blind deconvolution. The first algorithm is based on a cascade of a forward and a backward prediction error filter (C-PEF). The second consists in an alternating minimization algorithm for estimating both the reflectivity series and the seismic wavelet (AM-SMBD). We also compare the algorithms with other state-of-the-art sparse blind deconvolution algorithms. Simulation results with synthetic data for different SNR levels showed that the AM-SMBD was outperformed (in terms of the Pearson Correlation Coefficient and the Gini Correlation Coefficient) other estimation methods, like the reduced SMBD, the TS-sparseTLS (Toeplitz-structured sparse total least square) and the SMBD-SPG (Sparse multichannel blind deconvolution via spectral projected-gradient). For the same data, the C-PEF was able to provide better results (in terms of the Gini Correlation Coefficient, visual inspection and frequency gain) when compared with the F-SMBD (Fast sparse multichannel blind deconvolution). In a simulation considering reflectivities with different levels of sparsity, the C-PEF seems to be more robust for less sparse data when compared with AM-SMBD and SMBD-SPG (up to a certain degree of sparsity). Finally, simulations considering a real land acquisition show that both algorithms were able to greatly improve the resolution of the seismic data.

Multichannel blind deconvolution for full Stokes image recovery

Random, time-varying blur due to atmospheric turbulence is known to significantly limit image quality. In division-of-time polarimeters, turbulence also randomly distorts the images that are used to infer the polarization content of the scene, leading to substantial Stokes image estimation errors. This research proposes a method to jointly estimate the full polarization properties of a scene along with the point spread functions (PSFs) for each imaging channel in the presence of isoplanatic turbulence. In particular, this research significantly expands on an existing algorithm, to include circular polarization in the framework of a generalized expectation maximization approach. The effectiveness of the approach is demonstrated on laboratory data using a surrogate phase screen placed near the entrance pupil of an imaging system.

Research on Co-channel Interference Suppression Method for Passive Radar Based on the Jiont Processing of Primary and Reference Channels

The illuminators of passive radar based civil communication signals are densely distributed. As a result, the co-channel illuminator always interferes with the primary and reference channels, resulting in poor detection performance. To solve the aforementioned problem, an improved signal processing flow with co-channel interference suppression is proposed in this paper. First, signals from all channels were processed jointly. The direct-path wave of each illuminator was estimated using the multi-channel blind deconvolution algorithm. Then, the direct-path wave of the primary illuminator was identified as the reference signal by applying the difference in the proportion of the primary illuminator signal energy among channels. Then, the clutter of each illuminator in the primary channel was suppressed by utilizing each of the above estimated signals. Finally, the residual signal, after cancellation, was used to compute the cross-ambiguity functions with the identified direct-path wave of the primary illuminator for target detection. The improved flow can promote the cancellation ratio and reduce the bottom noise of the cross-ambiguity function and missed alarm. Co-channel interference can be effectively suppressed using the improved processing flow without changing the radar system’s hardware. The validity of the proposed method were confirmed by the results of the simulation and experiment.

Identifiability Conditions for Compressive Multichannel Blind Deconvolution

In applications such as multi-receiver radars and ultrasound array systems, the observed signals are often modeled as the convolution of the transmit pulse signal and a set of sparse filters representing the sparse target scenes. A sparse multichannel blind deconvolution (MBD) problem simultaneously identifies the unknown signal and sparse filters, which is in general ill-posed. In this paper, we consider the identifiability problem of sparse-MBD and show that, similar to compressive sensing, it is possible to identify the sparse filters from compressive measurements of the output sequences. Specifically, we consider compressible measurements in the Fourier domain and derive deterministic identifiability conditions. Our main results demonstrate that $L$ -sparse filters can be identified from $\text{2}L^2$ Fourier measurements from two or more coprime channels. We also show that $\text{2}L$ measurements per channel are necessary. The sufficient condition sharpens as the number of channels increases and is asymptotically optimal, i.e., it suffices to acquire on the order of $L$ Fourier samples per channel. We also propose a kernel-based sampling scheme that acquires Fourier measurements from a commensurate number of time-domain samples. The gap between the sufficient and necessary conditions is illustrated through numerical experiments including comparing practical reconstruction algorithms. The proposed compressive MBD results require fewer measurements and fewer channels for identifiability compared to previous results, which aids in building cost-effective receivers.

Exact Recovery of Multichannel Sparse Blind Deconvolution via Gradient Descent

We study the multichannel sparse blind deconvolution (MCS-BD) problem, whose task is to simultaneously recover a kernel $a$ and multiple sparse inputs $\{x_i\}_{i=1}^p$ from their circulant convolu...

Blind Deconvolution Using Modulated Inputs

This paper considers the blind deconvolution of multiple modulated signals/filters, and an arbitrary filter/signal. Multiple inputs $\boldsymbol{s}_1, \boldsymbol{s}_2, \ldots, \boldsymbol{s}_N =: [\boldsymbol{s}_n]$ are modulated (pointwise multiplied) with random sign sequences $\boldsymbol{r}_1, \boldsymbol{r}_2, \ldots, \boldsymbol{r}_N =: [\boldsymbol{r}_n]$ , respectively, and the resultant inputs $(\boldsymbol{s}_n \odot \boldsymbol{r}_n) \in \mathbb {C}^Q, n \in [N]$ are convolved against an arbitrary input $\boldsymbol{h} \in \mathbb {C}^M$ to yield the measurements $\boldsymbol{y}_n = (\boldsymbol{s}_n\odot \boldsymbol{r}_n)\circledast \boldsymbol{h}, n \in [N] := 1,2,\ldots,N,$ where $\odot$ and $\circledast$ denote pointwise multiplication, and circular convolution. Given $[\boldsymbol{y}_n]$ , we want to recover the unknowns $[\boldsymbol{s}_n]$ and $\boldsymbol{h}$ . We make a structural assumption that unknowns $[\boldsymbol{s}_n]$ are members of a known $K$ -dimensional (not necessarily random) subspace, and prove that the unknowns can be recovered from sufficiently many observations using a regularized gradient descent algorithm whenever the modulated inputs $\boldsymbol{s}_n \odot \boldsymbol{r}_n$ are long enough, i.e, $Q \gtrsim KN+M$ (to within logarithmic factors, and signal dispersion/coherence parameters). Under the bilinear model, this is the first result on multichannel ( $N\geq 1$ ) blind deconvolution with provable recovery guarantees under near optimal (in the $N=1$ case) sample complexity estimates, and comparatively lenient structural assumptions on the convolved inputs. A neat conclusion of this result is that modulation of a bandlimited signal protects it against an unknown convolutive distortion. We discuss the applications of this result in passive imaging, wireless communication in unknown environment, and image deblurring. A thorough numerical investigation of the theoretical results is also presented using phase transitions, image deblurring experiments, and noise stability plots.

Multichannel Sparse Blind Deconvolution on the Sphere

Multichannel blind deconvolution is the problem of recovering an unknown signal $f$ and multiple unknown channels $x_{i}$ from their circular convolution $y_{i}=x_{i} \circledast f$ ( $i=1,2, {\dots },N$ ). We consider the case where the $x_{i}$ ’s are sparse, and convolution with $f$ is invertible. Our nonconvex optimization formulation solves for a filter $h$ on the unit sphere that produces sparse output $y_{i}\circledast h$ . Under some technical assumptions, we show that all local minima of the objective function correspond to the inverse filter of $f$ up to an inherent sign and shift ambiguity, and all saddle points have strictly negative curvatures. This geometric structure allows successful recovery of $f$ and $x_{i}$ using a simple manifold gradient descent (MGD) algorithm. The same approach is also applicable to blind gain and phase calibration with a Fourier sensing matrix. Our algorithm and analysis require fewer assumptions than previous algorithms for the same problem. Our theoretical findings are complemented by numerical experiments, which demonstrate superior performance of the proposed approach over the previous methods. Empirically, our algorithm has low computation cost (converging in a small number of iterations) and low memory footprint (solving only for the inverse filter of $f$ ).

Blind deconvolution estimation of an arterial input function for small animal DCE-MRI

PurposeOne of the main obstacles for reliable quantitative dynamic contrast-enhanced (DCE) MRI is the need for accurate knowledge of the arterial input function (AIF). This is a special challenge for preclinical small animal applications where it is very difficult to measure the AIF without partial volume and flow artifacts. Furthermore, using advanced pharmacokinetic models (allowing estimation of blood flow and permeability-surface area product in addition to the classical perfusion parameters) poses stricter requirements on the accuracy and precision of AIF estimation. This paper addresses small animal DCE-MRI with advanced pharmacokinetic models and presents a method for estimation of the AIF based on blind deconvolution. MethodsA parametric AIF model designed for small animal physiology and use of advanced pharmacokinetic models is proposed. The parameters of the AIF are estimated using multichannel blind deconvolution. ResultsEvaluation on simulated data show that for realistic signal to noise ratios blind deconvolution AIF estimation leads to comparable results as the use of the true AIF.Evaluation on real data based on DCE-MRI with two contrast agents of different molecular weights showed a consistence with the known effects of the molecular weight. ConclusionMulti-channel blind deconvolution using the proposed AIF model specific for small animal DCE-MRI provides reliable perfusion parameter estimates under realistic signal to noise conditions.

ENLIVE: An Efficient Nonlinear Method for Calibrationless and Robust Parallel Imaging

Robustness against data inconsistencies, imaging artifacts and acquisition speed are crucial factors limiting the possible range of applications for magnetic resonance imaging (MRI). Therefore, we report a novel calibrationless parallel imaging technique which simultaneously estimates coil profiles and image content in a relaxed forward model. Our method is robust against a wide class of data inconsistencies, minimizes imaging artifacts and is comparably fast, combining important advantages of many conceptually different state-of-the-art parallel imaging approaches. Depending on the experimental setting, data can be undersampled well below the Nyquist limit. Here, even high acceleration factors yield excellent imaging results while being robust to noise and the occurrence of phase singularities in the image domain, as we show on different data. Moreover, our method successfully reconstructs acquisitions with insufficient field-of-view. We further compare our approach to ESPIRiT and SAKE using spin-echo and gradient echo MRI data from the human head and knee. In addition, we show its applicability to non-Cartesian imaging on radial FLASH cardiac MRI data. Using theoretical considerations, we show that ENLIVE can be related to a low-rank formulation of blind multi-channel deconvolution, explaining why it inherently promotes low-rank solutions.

Robust, blind multichannel image identification and restoration using stack decoder

In this study, the authors introduce new solutions and improvements to the multi-channel blind image deconvolution problem. More precisely, authors’ contributions are threefold: (i) At first, a simplified version of the existing cross-relation method for blind system identification is proposed; but most importantly, the authors incorporate into the channel estimation cost function a sparsity constraint to deal with the challenging issue of channel order overestimation errors; (ii) then, once the channel identification is achieved, a new image restoration method based on the stack decoding algorithm is introduced; and (iii) finally, a refining approach using an ‘all-at-once’ optimisation technique with an improved mixed norm regularisation is considered. The performance of the proposed approach was evaluated using several numerical simulations. Blind system identification and image restoration tasks were evaluated with respect to several criteria: numerical complexity, robustness to noise effects and channel order estimation. The results obtained are promising and highlight the effectiveness of the proposed approach.

Multichannel blind deconvolution via maximum likelihood estimator: application in neural recordings

In many multidimensional data such as radar recordings and astrophotography images, the receiver sensors record a linear mixture of signals propagated by a few activities, and the signal of each activity is repetition of a specific waveform at different times and with different amplitudes. The goal of multichannel blind deconvolution problem is retrieving the characteristics of the mentioned activities from the recorded signals. This problem is ill-posed without additional constraints, hence, different constraints are considered for this problem depending on the considered application. In this study, we propose a maximum likelihood framework for solving multichannel blind deconvolution problem when (1) the waveforms of the activities are time-limited signals, (2) the waveforms occur only a few times in the signal of each activity, or in other words, the occurrence times of the activities are sparse signals, and (3) there is no overlap between two consecutive occurrences of the waveform in the signal of each activity. The considered scenario can be adapted to several applications especially to neural recordings. We verify the efficiency of the proposed framework using simulations. Moreover, we apply the proposed framework on a real neural data set, and show how the obtained results can be employed to analyze the data from signal processing point of view.

Sparse Blind Deconvolution for Distributed Radar Autofocus Imaging

A common problem that arises in radar imaging systems, especially those mounted on mobile platforms, is antenna position ambiguity. Approaches to resolve this ambiguity and correct position errors are generally known as radar autofocus. Common techniques that attempt to resolve the antenna ambiguity generally assume an unknown gain and phase error afflicting the radar measurements. However, ensuring identifiability and tractability of the unknown error imposes strict restrictions on the allowable antenna perturbations. Furthermore, these techniques are often not applicable in near-field imaging, where mapping the position ambiguity to phase errors breaks down. In this paper, we propose an alternate formulation where the position error of each antenna is mapped to a spatial shift operator in the image-domain. Thus, the radar autofocus problem becomes a multichannel blind deconvolution problem, in which the radar measurements correspond to observations of a static radar image that is convolved with the spatial shift kernel associated with each antenna. To solve the reformulated problem, we also develop a block coordinate descent framework that leverages the sparsity and piecewise smoothness of the radar scene, as well as the one-sparse property of the two-dimensional shift kernels. We evaluate the performance of our approach using both simulated and experimental radar measurements and demonstrate its superior performance compared with state-of-the-art methods.

Identifiability of Multichannel Blind Deconvolution and Nonconvex Regularization Algorithm

In this paper, we consider the multichannel blind deconvolution problem, where we observe the output of channels $\mathbf {h}_{i}\in \mathbb {R}^n$ ( $i=1,...,N$ ) that all convolve with the same unknown input signal $\mathbf {x}\in \mathbb {R}^n$ . We wish to estimate the input signal and blur kernels simultaneously. Existing theoretical results showed that the original inputs are identifiable under subspace assumptions. However, the subspaces discussed before were randomly or generically chosen. Here, we propose deterministic subspace assumption, which is widely used in practice, and give some theoretical results. First of all, we derive tight sufficient condition for identifiability of signal and convolution kernels, which is only violated on a set of Lebesgue measure zero. Then, we present a nonconvex regularization algorithm by a lifting method and approximate the rank-one constraint via the difference of nuclear norm and Frobenius norm. The global minimizer of the proposed nonconvex algorithm is rank-one matrix under mild conditions on parameters and noise level. The stability result is also shown under the assumption that the inputs lie in a compact set. Besides, the computation of our regularization model is carried out and any limit point of iterations converges to a stationary point of our model. Finally, we provide numerical experiments to show that our nonconvex regularization model outperforms convex relaxation models, such as nuclear norm minimization and some nonconvex methods, such as alternating minimization method and spectral method.

Focused Blind Deconvolution

We introduce a novel multichannel blind deconvolution (BD) method that extracts sparse and front-loaded impulse responses from the channel outputs, i.e., their convolutions with a single arbitrary source. A crucial feature of this formulation is that it doesn't encode support restrictions on the unknowns, unlike most prior work on BD. The indeterminacy inherent to BD, which is difficult to resolve with a traditional L1 penalty on the impulse responses, is resolved in our method because it seeks a first approximation where the impulse responses are: "maximally white" -- encoded as the energy focusing near zero lag of the impulse-response auto-correlations; and "maximally front-loaded" -- encoded as the energy focusing near zero time of the impulse responses. Hence we call the method focused blind deconvolution (FBD). The focusing constraints are relaxed as the iterations progress. Note that FBD requires the duration of the channel outputs to be longer than that of the unknown impulse responses. A multichannel blind deconvolution problem that is appropriately formulated by sparse and front-loaded impulse responses arises in seismic inversion, where the impulse responses are the Green's function evaluations at different receiver locations, and the operation of a drill bit inputs the noisy and correlated source signature into the subsurface. We demonstrate the benefits of FBD using seismic-while-drilling numerical experiments, where the noisy data recorded at the receivers are hard to interpret, but FBD can provide the processing essential to separate the drill-bit (source) signature from the interpretable Green's function.

Fast and Guaranteed Blind Multichannel Deconvolution Under a Bilinear System Model

We consider the multichannel blind deconvolution problem where we observe the output of multiple channels that are all excited with the same unknown input. From these observations, we wish to estimate the impulse responses of each of the channels. We show that this problem is well-posed if the channels follow a bilinear model where the ensemble of channel responses is modeled as lying in a low-dimensional subspace but with each channel modulated by an independent gain. Under this model, we show how the channel estimates can be found by minimizing a quadratic function over a non-convex set. We analyze two methods for solving this non-convex program, and provide performance guarantees for each. The first is a method of alternating eigenvectors that breaks the program down into a series of eigenvalue problems. The second is a truncated power iteration, which can roughly be interpreted as a method for finding the largest eigenvector of a symmetric matrix with the additional constraint that it adheres to our bilinear model. As with most non-convex optimization algorithms, the performance of both of these algorithms is highly dependent on having a good starting point. We show how such a starting point can be constructed from the channel measurements. Our performance guarantees are non-asymptotic, and provide a sufficient condition on the number of samples observed per channel in order to guarantee channel estimates of certain accuracy. Our analysis uses a model with a “generic” subspace that is drawn at random, and we show the performance bounds hold with high probability. Mathematically, the key estimates are derived by quantifying how well the eigenvectors of certain random matrices approximate the eigenvectors of their mean. We also present a series of numerical results demonstrating that the empirical performance is consistent with the presented theory.

Improving Sparse Multichannel Blind Deconvolution with Correlated Seismic Data: Foundations and Further Results

For more than half a century, seismic deconvolution has been of a great interest in reflection seismics. Its goal is to remove the effect of the seismic wavelet, i.e., the waveform produced by the seismic source, from the data. Its importance to the industry is that it increases the resolution of the seismic image, yielding a more interpretable seismic section. From a theoretical standpoint, what is interesting is that neither the wavelet nor the characteristics of the subsurface are known, which makes the problem particularly challenging. Over the years, several seismic deconvolution approaches have been proposed, based on different assumptions about the seismic wavelet and the reflectivity series.

Blind Multichannel Deconvolution with Cascaded Kalman Filters

Blind multichannel deconvolution is a demanding technology in modern signal processing field. The proposed algorithm is based on two cascaded Kalman filter. The first Kalman filter performs the blind identification of the multichannel impulse response using the cross-correlation condition; the second Kalman filter carries out the deconvolution process. The two Kalman filters are unified into a common iteration period, so the blind multichannel deconvolution is adaptively executed. Simulation shows that the proposed algorithm works well.

Spectral Methods for Passive Imaging: Nonasymptotic Performance and Robustness

We study the problem of passive imaging through convolutive channels. A scene is illuminated with an unknown, unstructured source, and the measured response is the convolution of this source with multiple channel responses, each of which is time-limited. Spectral methods based on the commutativity of convolution, first proposed and analyzed in the 1990s, provide an elegant mathematical framework for attacking this problem. However, these now classical methods are very sensitive to noise, especially when working from relatively small sample sizes. In this paper, we show that a linear subspace model on the coefficients of the impulse responses of the channels can make this problem well-posed. We derive nonasymptotic error bounds for the generic subspace model by analyzing the spectral gap of the cross-correlation (CC) matrix of the channels relative to the perturbation introduced by noise. Numerical results show that this modified spectral method offers significant improvements over the classical method and outperforms other competing methods for multichannel blind deconvolution.