A convex approach to blind deconvolution with diverse inputs

Ahmed, Ali; Cosse, Augustin; Demanet, Laurent

doi:10.1109/camsap.2015.7383722

Cited by 19 publications

(30 citation statements)

References 11 publications

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“…We want to find out x l (or x) and D when y l (or y) and A l (or A) are given. Roughly, they correspond to the models in [5,1,8] respectively. Though all of the three models belong to the class of bilinear inverse problems, we will prove that simply solving linear least squares will give solutions to all those models exactly and robustly for invertible D and for several useful choices of A and A l .…”

Section: Our Contributionsmentioning

confidence: 99%

“…Therefore, our result is nearly optimal in terms of information theoretic limit. Compared with a similar setup in [1], we have a more efficient algorithm since [1] uses nuclear norm minimization to achieve exact recovery. However, the assumptions are slightly different, i.e., we assume that D is invertible and hence the result depends on D while [1] imposes "incoherence" on d by requiring…”

Section: Blind Deconvolution Via Diverse Inputsmentioning

confidence: 99%

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Self-Calibration and Bilinear Inverse Problems via Linear Least Squares

Ling¹,

Strohmer²

2018

SIAM J. Imaging Sci.

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Whenever we use devices to take measurements, calibration is indispensable. While the purpose of calibration is to reduce bias and uncertainty in the measurements, it can be quite difficult, expensive, and sometimes even impossible to implement. We study a challenging problem called self-calibration, i.e., the task of designing an algorithm for devices so that the algorithm is able to perform calibration automatically. More precisely, we consider the setup y = A(d)x+ε where only partial information about the sensing matrix A(d) is known and where A(d) linearly depends on d. The goal is to estimate the calibration parameter d (resolve the uncertainty in the sensing process) and the signal/object of interests x simultaneously. For three different models of practical relevance, we show how such a bilinear inverse problem, including blind deconvolution as an important example, can be solved via a simple linear least squares approach. As a consequence, the proposed algorithms are numerically extremely efficient, thus potentially allowing for real-time deployment. We also present a variation of the least squares approach, which leads to a spectral method, where the solution to the bilinear inverse problem can be found by computing the singular vector associated with the smallest singular value of a certain matrix derived from the bilinear system. Explicit theoretical guarantees and stability theory are derived for both techniques; and the number of sampling complexity is nearly optimal (up to a poly-log factor). Applications in imaging sciences and signal processing are discussed and numerical simulations are presented to demonstrate the effectiveness and efficiency of our approach.

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Section: Our Contributionsmentioning

confidence: 99%

Section: Blind Deconvolution Via Diverse Inputsmentioning

confidence: 99%

Self-Calibration and Bilinear Inverse Problems via Linear Least Squares

Ling¹,

Strohmer²

2018

SIAM J. Imaging Sci.

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“…This challenging problem appears in a variety of applications, such as audio processing [31], image processing [38], [36], neuroscience [41], spectroscopy [42], astronomy [13]. It also arises in wireless communications 1 [46] and is expected to play a central role in connection with the future Internet-of-Things [49]. Common to almost all approaches to tackle this problem is the assumption that we have multiple received signals at our disposal, often at least as many received signals as there are transmitted signals.…”

Section: Introductionmentioning

confidence: 99%

“…Ling and T. Strohmer are with the Department of Mathematics, University of California at Davis, Davis, CA 95616, USA (E-mail: syling@math.ucdavis.edu; strohmer@math.ucdavis.edu). 1 In wireless communications this is also known as "multiuser joint channel estimation and equalization." that under reasonable and practical conditions, it is indeed possible to recover the r transmitted signals and the associated channels in a robust, reliable, and efficient manner from just one single received signal.…”

Section: Introductionmentioning

confidence: 99%

Blind Deconvolution Meets Blind Demixing: Algorithms and Performance Bounds

Ling

Strohmer

2017

IEEE Trans. Inform. Theory

110

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Suppose that we have r sensors and each one intends to send a function g i (e.g. a signal or an image) to a receiver common to all r sensors. During transmission, each g i gets convolved with a function f i . The receiver records the function y, given by the sum of all these convolved signals. When and under which conditions is it possible to recover the individual signals g i and the blurring functions f i from just one received signal y? This challenging problem, which intertwines blind deconvolution with blind demixing, appears in a variety of applications, such as audio processing, image processing, neuroscience, spectroscopy, and astronomy. It is also expected to play a central role in connection with the future Internet-of-Things. We will prove that under reasonable and practical assumptions, it is possible to solve this otherwise highly ill-posed problem and recover the r transmitted functions g i and the impulse responses f i in a robust, reliable, and efficient manner from just one single received function y by solving a semidefinite program. We derive explicit bounds on the number of measurements needed for successful recovery and prove that our method is robust in the presence of noise. Our theory is actually sub-optimal, since numerical experiments demonstrate that, quite remarkably, recovery is still possible if the number of measurements is close to the number of degrees of freedom.

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“…However, model errors inevitably affect its physical implementation and can significantly degrade signal recovery, as first studied by Herman and Strohmer [3]. In particular, such model errors may arise from physical causes such as unknown convolution kernels [4,5,6] affecting the measurements; unknown attenuations or gains on the latter coefficients, e.g., pixel response non-uniformity [7] or fixedpattern noise in imaging systems; complex-valued (i.e., gain and phase) errors in sensor arrays [8,9,10].…”

Section: Introductionmentioning

confidence: 99%

Through the haze: a non-convex approach to blind gain calibration for linear random sensing models

Cambareri

Jacques

2018

Information and Inference: A Journal of the IMA

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Computational sensing strategies often suffer from calibration errors in the physical implementation of their ideal sensing models. Such uncertainties are typically addressed by using multiple, accurately chosen training signals to recover the missing information on the sensing model, an approach that can be resource-consuming and cumbersome. Conversely, blind calibration does not employ any training signal, but corresponds to a bilinear inverse problem whose algorithmic solution is an open issue. We here address blind calibration as a non-convex problem for linear random sensing models, in which we aim to recover an unknown signal from its projections on sub-Gaussian random vectors, each subject to an unknown positive multiplicative factor (or gain). To solve this optimisation problem we resort to projected gradient descent starting from a suitable, carefully chosen initialisation point. An analysis of this algorithm allows us to show that it converges to the exact solution provided a sample complexity requirement is met, i.e., relating convergence to the amount of information collected during the sensing process. Interestingly, we show that this requirement grows linearly (up to log factors) in the number of unknowns of the problem. This sample complexity is found both in absence of prior information, as well as when subspace priors are available for both the signal and gains, allowing a further reduction of the number of observations required for our recovery guarantees to hold. Moreover, in the presence of noise we show how our descent algorithm yields a solution whose accuracy degrades gracefully with the amount of noise affecting the measurements. Finally, we present some numerical experiments in an imaging context, where our algorithm allows for a simple solution to blind calibration of the gains in a sensor array.

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A convex approach to blind deconvolution with diverse inputs

Cited by 19 publications

References 11 publications

Self-Calibration and Bilinear Inverse Problems via Linear Least Squares

Self-Calibration and Bilinear Inverse Problems via Linear Least Squares

Blind Deconvolution Meets Blind Demixing: Algorithms and Performance Bounds

Through the haze: a non-convex approach to blind gain calibration for linear random sensing models

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