New criteria for blind signal separation

Blind deconvolution, which comprises simultaneous blur and image estimations, is a strongly ill-posed problem. It is by now well known that if multiple images of the same scene are acquired, this multichannel (MC) blind deconvolution problem is better posed and allows blur estimation directly from the degraded images. We improve the MC idea by adding robustness to noise and stability in the case of large blurs or if the blur size is vastly overestimated. We formulate blind deconvolution as an l(1) -regularized optimization problem and seek a solution by alternately optimizing with respect to the image and with respect to blurs. Each optimization step is converted to a constrained problem by variable splitting and then is addressed with an augmented Lagrangian method, which permits simple and fast implementation in the Fourier domain. The rapid convergence of the proposed method is illustrated on synthetically blurred data. Applicability is also demonstrated on the deconvolution of real photos taken by a digital camera.

We propose a method to estimate the number of partial discharge sources in multipath environment using the multichannel blind deconvolution method called the super-exponential deflation method. Blind deconvolution is a method of recovering transmitted signals from only observed signals. The deflation approach can extract a source signal which is non-Gaussian from the observed signals one by one. The deflation approach is repeated until all non-Gaussian signals are extracted, in which case the input signals become close to Gaussian noise. The eigenvalues of the spatial correlation matrix of the input signals indicate whether non-Gaussian signals are present or absent in the input signals. Thus, the proposed method estimates the number of sources by repeating the signal extraction and analyzing the eigenvalues using the criterion function to distinguish the presence or absence of the source signals in the observed signals. Appropriate antenna arrangement is an important factor in the effective separation of source signals. The aim of this study is to compare performance of antenna arrangements for estimating the number of PD sources.

We present a simple efficient local unsupervised learning algorithm for online adaptive multichannel blind deconvolution and separation of i.i.d. sources. Under mild conditions, there exits a stable inverse system so that the source signals can be exactly recovered from their convolutive mixtures. Based on the existence of the inverse filter, we construct a two-stage neural network which consists of blind equalization and source separation. In the blind equalization stage, we employ anti-Hebbian learning in the temporal domain for decorrelation. For blind separation, we can apply any existing algorithms. Extensive computer simulations confirm the validity and high performance of our proposed learning algorithm.

Identifiability is a key concern in ill-posed blind deconvolution problems arising in wireless communications and image processing. The single channel version of the problem is the most challenging and there have been efforts to use sparse models for regularizing the problem. Identifiability of the sparse blind deconvolution problem is analyzed and it is established that a simple sparsity assumption in the canonical basis is insufficient for unique recovery; a surprising negative result. The proof technique involves lifting the deconvolution problem into a rank one matrix recovery problem and analyzing the rank two nullspace of the resultant linear operator. A DoF (degrees of freedom) wise tight parametrized subset of this rank two null-space is constructed to establish the results.

Convolutive blind source separation by efficient blind deconvolution and minimal filter distortion

In many real-life situations, we need to reconstruct a blurred image in situations when no information about the blurring is available. This problem is known as the problem of blind deconvolution. There exist techniques for solving this problem, but these techniques are not rotation-invariant. Thus, the result of using this technique may change with rotation. So, if we rotate the image a little bit, the method, in general, leads to a different deconvolution result. Therefore, even when the original reconstruction is optimal, the reconstruction of a rotated image will be different and, thus, not optimal. To improve the quality of image decomposition, it is desirable to modify the current state-of-the art techniques by making them rotation-invariant. In this paper, we show how this can be done, and we show that this indeed improves the quality of 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.

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.

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 Signal Separation II. Linear, Convolutive Combinations: II. Linear, Convolutive Combinations

Publication in the conference proceedings of EUSIPCO, Tampere, Finland, 2000

Publication in the conference proceedings of EUSIPCO, Toulouse, France, 2002

In view of band-limited seismic data with a low dominant frequency and short data records, this paper proposes a new approach to the seismic blind deconvolution problem, a new criterion for blind deconvolution is constructed, by which we can measure the independency of output signals. The optimization program for new criterion of blind deconvolution is performed by applying Neidell's wavelet model to the inverse filter, which then makes the optimization program for multivariate reduce to univariate case. The proposed method can broaden the spectrum of seismic data effectively resulting in the improved resolution even in the case of band-limited seismic data with a low dominant frequency and short data records. The simulation test and actual seismic data processing results show that the correctness of the method.

The cross-correlation is an effective tool for estimating a time delay between two signals. Unfortunately, in practical world, there always exist problems of multi-path time delay estimation (MTDE). The cross-correlation fails to deal with the MTDE case. In this paper, a novel technique for high-resolution multi-path time delay estimation (HMTDE) is proposed. The new method is based on 2D blind deconvolution and the slice selection on the third order cross-moment (TOCM) of the two signals. To get a high resolution estimation of the multi-path time delay, it is necessary to remove the component of transmitted signal. Instead of solving a one-dimensional blind deconvolution problem in original signals domain, the proposed method solves a two-dimensional blind deconvolution problem in their TOCM domain. It recovers a scaled and time-shifted version of the reflectivity sequence with the maximum time delay slices (MTDS) of TOCM of the reflectivities corresponding to the two signals. The only information of the transmitted signal needed in this method is its third order moment(TOM), which can be estimated only using the two received signals. It is well known that higher order statistics (HOS) carry more information and be insensitive to additive Gaussian noise than does the second order statistics (SOS). These properties lead to a robust and unique solution of HMTDE by using our TOCM based approach.

Single-channel signal separation and deconvolution aims to separate and deconvolve individual sources from a single-channel mixture. Single-channel signal separation and deconvolution is a challenging problem in which no prior knowledge of the mixing filters is available. Both individual sources and mixing filters need to be estimated. In addition, a mixture may contain non-stationary noise which is unseen in the training set. We propose a synthesizing-decomposition (S-D) approach to solve the single-channel separation and deconvolution problem. In synthesizing, a generative model for sources is built using a generative adversarial network (GAN). In decomposition, both mixing filters and sources are optimized to minimize the reconstruction error of the mixture. The proposed S-D approach achieves a peak-to-noise-ratio (PSNR) of 18.9 dB and 15.4 dB in image inpainting and completion, outperforming a baseline convolutional neural network PSNR of 15.3 dB and 12.2 dB, respectively and achieves a PSNR of 13.2 dB in source separation together with deconvolution, outperforming a convolutive non-negative matrix factorization (NMF) baseline of 10.1 dB.