Compressive Sampling Using Annihilating Filter-Based Low-Rank Interpolation

Abstract

Abstract-While the recent theory of compressed sensing provides an opportunity to overcome the Nyquist limit in recovering sparse signals, a solution approach usually takes the form of an inverse problem of an unknown signal, which is crucially dependent on specific signal representation. In this paper, we propose a drastically different two-step Fourier compressive sampling framework in a continuous domain that can be implemented via measurement domain interpolation, after which signal reconstruction can be done using classical analytic reconstruction methods. The main idea originates from the fundamental duality between the sparsity in the primary space and the low-rankness of a structured matrix in the spectral domain, showing that a low-rank interpolator in the spectral domain can enjoy all of the benefits of sparse recovery with performance guarantees. Most notably, the proposed low-rank interpolation approach can be regarded as a generalization of recent spectral compressed sensing to recover large classes of finite rate of innovations (FRI) signals at a near-optimal sampling rate. Moreover, for the case of cardinal representation, we can show that the proposed low-rank interpolation scheme will benefit from inherent regularization and an optimal incoherence parameter. Using a powerful dual certificate and the golfing scheme, we show that the new framework still achieves a near-optimal sampling rate for a general class of FRI signal recovery, while the sampling rate can be further reduced for a class of cardinal splines. Numerical results using various types of FRI signals confirm that the proposed low-rank interpolation approach offers significantly better phase transitions than conventional compressive sampling approaches.Index Terms-Compressed sensing, signals of finite rate of innovations, spectral compressed sensing, low rank matrix completion, dual certificates, golfing scheme.

A General Framework for Compressed Sensing and Parallel MRI Using Annihilating Filter Based Low-Rank Hankel Matrix

“…Indeed, the proof of Theorem 2.1 informs us that the explicit form of the Fourier samplesŷ[k] given bŷ is necessary and sufficient condition to have a rank-r Hankel matrix. Accordingly, we concluded that the signals with the finite rate of innovation (FRI) correspond to this class signals [25]. We further showed that for the case of the cardinal spline cases (where the knots {x j } are located on the uniform grid), the k-space data is also periodic; accordingly, a wrap-around Hankel matrix can be equivalently obtained from the periodic boundary condition [25].…”

“…Proof: See our companion paper [25]. Indeed, the proof of Theorem 2.1 informs us that the explicit form of the Fourier samplesŷ[k] given bŷ is necessary and sufficient condition to have a rank-r Hankel matrix.…”

“…Accordingly, we concluded that the signals with the finite rate of innovation (FRI) correspond to this class signals [25]. We further showed that for the case of the cardinal spline cases (where the knots {x j } are located on the uniform grid), the k-space data is also periodic; accordingly, a wrap-around Hankel matrix can be equivalently obtained from the periodic boundary condition [25]. Moreover, the spectral weighting comes from the DFT of the discrete whitening filter that has attenuating behaviour at high frequency regions, which makes the algorithm more robust to noise boosting.…”

“…Moreover, the spectral weighting comes from the DFT of the discrete whitening filter that has attenuating behaviour at high frequency regions, which makes the algorithm more robust to noise boosting. See [25] for more details. This cardinal spline model will be used throughout the paper for MR specific applications, where the unknown images should be reconstructed on a fixed grid.…”

“…Suppose that M ∈ C n 1 ×n 2 is a rank-r matrix whose SVD is U ΣV * . M is said to satisfy the standard incoherence condition with parameter μ if Proof: See our companion paper [25]. In compressed sensing MRI, the reconstruction image grid is always fixed; therefore, cardinal spline model that has knot locations on a fixed grid is more appropriate.…”