Structured low-rank recovery of piecewise constant signals with performance guarantees

Abstract

We derive theoretical guarantees for the exact recovery of piecewise constant two-dimensional images from a minimal number of non-uniform Fourier samples using a convex matrix completion algorithm. We assume the discontinuities of the image are localized to the zero level-set of a bandlimited function, which induces certain linear dependencies in Fourier domain, such that a multifold Toeplitz matrix built from the Fourier data is known to be low-rank. The recovery algorithm arranges the known Fourier samples into the structured matrix then attempts recovery of the missing Fourier data by minimizing the nuclear norm subject to structure and data constraints. This work adapts results by Chen and Chi on the recovery of isolated Diracs via nuclear norm minimization of a similar multifold Hankel structure. We show that exact recovery is possible with high probability when the bandlimited function describing the edge set satisfies an incoherency property. Finally, we demonstrate the algorithm on the recovery of undersampled MRI data.

Multi-shot sensitivity-encoded diffusion data recovery using structured low-rank matrix completion (MUSSELS)

“…Recently this idea was also extended to calibration-less parallel imaging reconstruction using structured low-rank matrix completion (31,33,35,55). The fundamental performance guarantees and the sampling requirement of the structured low-rank approach for various FRI models have also been theoretically addressed (36,56).…”

Convex Recovery of Continuous Domain Piecewise Constant Images From Nonuniform Fourier Samples

“…We can give incoherency measure of an edge set a more precise geometric interpretation based on the minimum separation distance of a set of admissible nodes. We generalize a bound on the condition number of Vandermonde matrices derived in [27] to the case of the Vandermonde-like matrix (21), and use this to derive a bound for the incoherence parameter ρ.…”

MRI artifact correction using sparse + low‐rank decomposition of annihilating filter‐based hankel matrix

“…In particular, when can be modelled as finite rate of innovation signals (e.g., stream of Diracs, or differentiated Diracs, etc. ), then can be a finite length impulse response filter .…”

“…Similar approaches have been independently proposed for two‐dimensional (2D) super‐resolution MRI , accelerated MRI and the 2D image zooming interpolation . The fundamental performance guarantees for this approach have been also extensively studied, which demonstrates near optimal performance for recovery of various signals with the finite rate of innovations from sparse measurements .…”