Blind Deconvolution using Convex Programming

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December 23, 2021
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Blind deconvolution refers to recovering signals by observing only their convolution. Blind deconvolution is a fundamental problem in signal processing, wireless communication, and systems theory. This problem arises in the context of blind channel estimation in wireless communications, passive imaging; for example, to estimate earth layer prole for oil exploration, seismic interferometry, synthetic aperture radar imaging, and many other applications. Unfortunately, blind deconvolution is ill-posed inverse problem meaning that many dierent pairs of signals can generate same convolution making it impossible to choose the truth among them. We re frame the non-linear inverse problem as a linear rank-1 matrix recovery problem and use the geometry along with statistics to prove that the solution of the resulting convex program recovers the truth exactly! We also give a recovery algorithm and numerical experiments to validate our claims. Wireless communication in an unknown medium, passive imaging, channel equalization, and image restoration in computer vision are some of the motivating applications of blind deconvolution. In the figure, original 256?256 shapes image is shown in (a) and blurring kernel with a support size of 65 pixels is shown in (b), the locations of which are assumed to be known. The result of convolving (a) and (b) is depicted in (c). Check out our related papers on this topic and links to our numerical solvers below.

Applications

Wireless communication in an unknown medium, passive imaging, channel equalization, and image restoration in computer vision are some of the motivating applications of blind deconvolution.

[2] A. Ahmed, and L. Demanet Leveraging Diversity and Sparsity in Blind Deconvolution, arXiv preprint arXiv:1610.06098, 2016.?