Exact Recovery of Multichannel Sparse Blind Deconvolution via Gradient Descent

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 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.

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.

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.

Stepped chirp signal is widely employed in high resolution radar systems. Conventional spectrum reconstruction methods are sensitive to system errors. In this paper, a novel range profile synthesis algorithm is proposed based on multichannel blind deconvolution (MBD) theory. The received signals are modelled as the outputs of a series of linear time-invariant (LTI) systems with a common unknown input. In the presence of system errors, the impulse responses of the LTI systems are also unknown. Thus, range profile synthesis is formulated as a MBD problem. Regularizations are introduced to solve the ill-posedness. Alternating optimization method is used to solve the problem. Simulation results show that the proposed method is able to provide accurate range profile estimations. (4 pages)

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.

The problem of multichannel blind signal deconvolution is considered. The mixing system is supposed to be stable and invertible and the input signals, also called sources, are assumed zero-mean independent and identically distributed (IID) random signals. Using the hypothesis that sources are statistically independent, we propose a generalization to the convolutive case of some separation criteria available in the instantaneous one. Hence, we obtain a new generalized class of criteria for signal deconvolution.

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 functional 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 a 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.

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.

We developed a sparse multichannel blind deconvolution (SMBD) method. The method is a modification of the multichannel blind deconvolution technique often called Euclid deconvolution, in which the multichannel impulse response of the earth is estimated by solving an homogeneous system of equations. Classical Euclid deconvolution is unstable in the presence of noise and requires the correct estimation of the length of the seismic wavelet. The proposed method, on the other hand, can tolerate moderate levels of noise and does not require a priori knowledge of the length of the wavelet. SMBD solves the homogeneous system of equations arising in Euclid deconvolution by imposing sparsity on the unknown multichannel impulse response. Trivial solutions to the aforementioned homogeneous system of equations are avoided by seeking sparse solutions on the unit sphere. We tested SMBD with synthetic and real data examples. Synthetic examples were used to judge the viability of the method in terms of noise. We found that SMBD gives reasonable estimates of the wavelet and reflectivity series for [Formula: see text]. The results clearly deteriorated when we tried to work on data that were severely contaminated by noise. A real marine data set was also used to test SMBD. In this case, the estimated wavelet was compared with a wavelet estimated by averaging first breaks. The estimated wavelet showed a noticeable resemblance to the average first break with normalized correlation coefficient of 0.92.

Previous research presented by the author and others into maximum-likelihood image restoration for incoherent imagery is extended to consider problems of blind deconvolution in which the impulse response of the system is assumed to be unknown. Potential applications that motivate this study are wide-field and confocal fluorescence microscopy, although applications in astronomy and infrared imaging are foreseen as well. The methodology incorporates the iterative expectation-maximization algorithm. Although the precise impulse response is assumed to be unknown, some prior knowledge about characteristics of the impulse response is used. In preliminary simulation studies that are presented, the circular symmetry and the band-limited nature of the impulse response are used as such. These simulations demonstrate the potential utility and present limitations of these methods.

Multichannel blind deconvolution (MBD) techniques allow recovery of individual source signals from observed mixtures, exploiting only the assumption of mutual independence of sources. Generally, complex signals are recovered with an arbitrary phase rotation. In this paper, we propose two constrained MBD algorithms to separate complex sources and remove the phase ambiguities simultaneously when the sources have independent in-phase and quadrature (I/Q) components. The algorithms apply the MBD techniques to real-valued signals, which are composed by the real and imaginary parts of the complex-valued sources, and impose constraint on the structure of separating filter to avoid the so-called I/Q associated problem. Simulation results demonstrate that the proposed algorithms are effective in recovering source phases without affecting source separation.

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

Single-image blind deconvolution is one of the most challenging fields in image processing which restores a sharp image from its blurred version. Nowadays blind deconvolution algorithms have made significant progress. However, the restoration of blurred images with little scale edges and periodic textures is still a hard work. To solve this problem, this paper proposes a new normalized sparse regularization blind deconvolution algorithm, which uses a gradient saliency map to prohibit the image small structures on image blurry kernel estimation. Firstly, salient detection is performed to select the important area which conforms with the human vision system and generates a binary mask to screen out useful gradients. Secondly, the normalized sparse regularization blind deconvolution method is applied to obtain accurate blur kernel and recover the sharp image. Finally, the experiment results show that the algorithm can effectively deblur the degraded image on different scenarios.

In this paper we consider image reconstruction from multichannel phased array MRI data without prior knowledge of the coil sensitivity functions. A new framework based on multichannel blind deconvolution (MBD) is developed for joint estimation of the image function and the sensitivity functions in k-space. By exploiting the smoothness of the estimated functions in the spatial domain, we develop a regularization approach in conjunction with MBD to obtain good reconstruction of the image function. Experimental results using simulated and real data demonstrate that the proposed reconstruction algorithm can better removes the sensitivity weighting in the reconstructed images compared to the sum-of-squares (SoS) approach.