Blind Gain and Phase Calibration via Sparse Spectral Methods

Abstract

Blind gain and phase calibration (BGPC) is a bilinear inverse problem involving the determination of unknown gains and phases of the sensing system, and the unknown signal, jointly. BGPC arises in numerous applications, e.g., blind albedo estimation in inverse rendering, synthetic aperture radar autofocus, and sensor array auto-calibration. In some cases, sparse structure in the unknown signal alleviates the ill-posedness of BGPC. Recently there has been renewed interest in solutions to BGPC with careful analysis of error bounds. In this paper, we formulate BGPC as an eigenvalue/eigenvector problem, and propose to solve it via power iteration, or in the sparsity or joint sparsity case, via truncated power iteration. Under certain assumptions, the unknown gains, phases, and the unknown signal can be recovered simultaneously. Numerical experiments show that power iteration algorithms work not only in the regime predicted by our main results, but also in regimes where theoretical analysis is limited. We also show that our power iteration algorithms for BGPC compare favorably with competing algorithms in adversarial conditions, e.g., with noisy measurement or with a bad initial estimate.

Multichannel Sparse Blind Deconvolution on the Sphere

“…We compare MGD (with random initialization) with three blind calibration algorithms that solve MSBD in the frequency domain: truncated power iteration [33] (initialized with f (0) = e 1 and x (0) i = 0), an off-the-grid algebraic method [62] (simplified from [63]), and an off-the-grid optimization approach [64].…”

“…The inequality (33) is due to the fact that (I + A) −1 − I ≤ (I + A) −1 A ≤ A 1− A for A < 1. The last line (34) follows from (31)…”

“…We would like to emphasize that, while earlier papers on MBD [16][17][18][19] consider a linear convolution model, more recent guaranteed methods for MSBD [32,33] consider a circular convolution model. By zero padding the signal and the filter, one can rewrite a linear convolution as a circular convolution.…”

“…Unlike the second order methods in previous works [40,42], we take advantage of recent advances in the understanding of first order methods [43,44], and prove that: (1) Manifold gradient descent with random initialization can escape saddle point almost surely; (2) Perturbed manifold gradient descent (PMGD) recovers a signed shifted version of the ground truth in polynomial time with high probability. This is the first guaranteed algorithm for MSBD that does not rely on restrictive assumptions on f (e.g., dominant entry [32], spectral flatness [33]), or on {x i } N i=1 (e.g., jointly sparse, Gaussian random dictionary [33]).…”

Unfolding Neural Networks for Compressive Multichannel Blind Deconvolution

“…Prior works have proposed computationally efficient and robust algorithms to recover the filters from the measurements. Examples are 1-norm methods [8][9][10], the sparse dictionary calibration [11,12] and truncated power iteration methods [13], and convolutional dictionary learning [14]. The works in [8,15,16] establish theoretical guarantees on identifying the S-MBD problem.…”