Wenjing Liao

Coherence Pattern-Guided Compressive Sensing with Unresolved Grids

Highly coherent sensing matrices arise in discretization of continuum imaging problems such as radar and medical imaging when the grid spacing is below the Rayleigh threshold. Algorithms based on techniques of band exclusion (BE) and local optimization (LO) are proposed to deal with such coherent sensing matrices. These techniques are embedded in the existing compressed sensing algorithms, such as Orthogonal Matching Pursuit (OMP), Subspace Pursuit (SP), Iterative Hard Thresholding (IHT), Basis Pursuit (BP), and Lasso, and result in the modified algorithms BLOOMP, BLOSP, BLOIHT, BP-BLOT, and Lasso-BLOT, respectively. Under appropriate conditions, it is proved that BLOOMP can reconstruct sparse, widely separated objects up to one Rayleigh length in the Bottleneck distance independent of the grid spacing. One of the most distinguishing attributes of BLOOMP is its capability of dealing with large dynamic ranges. The BLO-based algorithms are systematically tested with respect to four performance metrics: dynamic range, noise stability, sparsity, and resolution. With respect to dynamic range and noise stability, BLOOMP is the best performer. With respect to sparsity, BLOOMP is the best performer for high dynamic range, while for dynamic range near unity BP-BLOT and Lasso-BLOT with the optimized regularization parameter have the best performance. In the noiseless case, BP-BLOT has the highest resolving power up to certain dynamic range. The algorithms BLOSP and BLOIHT are good alternatives to BLOOMP and BP/Lasso-BLOT: they are faster than both BLOOMP and BP/Lasso-BLOT and share, to a lesser degree, BLOOMPs amazing attribute with respect to dynamic range. Detailed comparisons with the algorithms Spectral Iterative Hard Thresholding (SIHT) and the frame-adapted BP demonstrate the superiority of the BLO-based algorithms for the problem of sparse approximation in terms of highly coherent, redundant dictionaries.

Super-resolution by compressive sensing algorithms

In this work, super-resolution by 4 compressive sensing methods (OMP, BP, BLOOMP, BP-BLOT) with highly coherent partial Fourier measurements is comparatively studied. An alternative metric more suitable for gauging the quality of spike recovery is introduced and based on the concept of filtration with a parameter representing the level of tolerance for support offset. In terms of the filtered error norm only BLOOMP and BP-BLOT can perform grid-independent recovery of well separated spikes of Rayleigh index 1 for arbitrarily large superresolution factor. Moreover both BLOOMP and BP-BLOT can localize spike support within a few percent of the Rayleigh length. This is a weak form of super-resolution. Only BP-BLOT can achieve this feat for closely spaced spikes separated by a fraction of the Rayleigh length, a strong form of superresolution.

Mismatch and resolution in compressive imaging

Highly coherent sensing matrices arise in discretization of continuum problems such as radar and medical imaging when the grid spacing is below the Rayleigh threshold as well as in using highly coherent, redundant dictionaries as sparsifying operators. Algorithms (BOMP, BLOOMP) based on techniques of band exclusion and local optimization are proposed to enhance Orthogonal Matching Pursuit (OMP) and deal with such coherent sensing matrices. BOMP and BLOOMP have provably performance guarantee of reconstructing sparse, widely separated objects independent of the redundancy and have a sparsity constraint and computational cost similar to OMPs. Numerical study demonstrates the effectiveness of BLOOMP for compressed sensing with highly coherent, redundant sensing matrices.

Fourier phasing with phase-uncertain mask

Fourier phasing is the problem of retrieving Fourier phase information from Fourier intensity data. The standard Fourier phase retrieval (without a mask) is known to have many solutions which cause the standard phasing algorithms to stagnate and produce wrong or inaccurate solutions. In this paper Fourier phase retrieval is carried out with the introduction of a randomly fabricated mask in measurement and reconstruction. Highly probable uniqueness of solution, up to a global phase, was previously proved with exact knowledge of the mask. Here the uniqueness result is extended to the case where only rough information about the masks phases is assumed. The exponential probability bound for uniqueness is given in terms of the uncertainty-to-diversity ratio (UDR) of the unknown mask. New phasing algorithms alternating between the object update and the mask update are systematically tested and demonstrated to have the capability of recovering both the object and the mask (within the object support) simultaneously, consistent with the uniqueness result. Phasing with a phase-uncertain mask is shown to be robust with respect to the correlation in the mask as well as the Gaussian and Poisson noises.

MUSIC for Multidimensional Spectral Estimation: Stability and Super-Resolution

This paper presents a performance analysis of the MUltiple SIgnal Classification (MUSIC) algorithm applied on D dimensional single-snapshot spectral estimation while s true frequencies are located on the continuum of a bounded domain. Inspired by the matrix pencil form, we construct a D-fold Hankel matrix from the measurements and exploit its Vandermonde decomposition in the noiseless case. MUSIC amounts to identifying a noise subspace, evaluating a noise-space correlation function, and localizing frequencies by searching the s smallest local minima of the noise-space correlation function. In the noiseless case, (2s)D measurements guarantee an exact reconstruction by MUSIC as the noise-space correlation function vanishes exactly at true frequencies. When noise exists, we provide an explicit estimate on the perturbation of the noise-space correlation function in terms of noise level, dimension D, the minimum separation among frequencies, the maximum and minimum amplitudes while frequencies are separated by 2 Rayleigh Length (RL) at each direction. As a by-product the maximum and minimum non-zero singular values of the multidimensional Vandermonde matrix whose nodes are on the unit sphere are estimated under a gap condition of the nodes. Under the 2-RL separation condition, if noise is i.i.d. Gaussian, we show that perturbation of the noise-space correlation function decays like √(log(#(N))/#(N)) as the sample size #(N) increases. When the separation among frequencies drops below 2 RL, our numerical experiments show that the noise tolerance of MUSIC obeys a power law with the minimum separation of frequencies.

Compressed sensing phase retrieval

Theory of absolute uniqueness for phase retrieval with random illumination is presented. Suitable random illumination eliminates all sources of ambiguity, trivial and nontrivial. As a result, random-illumination-aided phase retrieval algorithms can accurately recover objects with a below-Nyquist sampling rate close to the minimum and reduce the number of iterations by order of magnitude.

Learning adaptive multiscale approximations to data and functions near low-dimensional sets

In the setting where a data set in ℝD consists of samples from a probability measure ρ concentrated on or near an unknown d-dimensional set M, with D large but d ≪ D, we consider two sets of problems: geometric approximation of M and regression of a function f on M. In the first case we construct multiscale low-dimensional empirical approximations of M, which are adaptive when M has geometric regularity that may vary at different locations and scales, and give performance guarantees. In the second case we exploit these empirical geometric approximations to construct multiscale approximations to f on M, which adapt to the unknown regularity of f even when this varies at different scales and locations. We prove guarantees showing that we attain the same learning rates as if f was defined on a Euclidean domain of dimension d, instead of an unknown manifold M. All algorithms have complexity O(n log n), with constants scaling linearly in D and exponentially in d.

MUSIC for joint frequency estimation: Stability with compressive measurements

This paper studies the application of MUtiple Signal Classification (MUSIC) algorithm on Multiple Measurement Vector (MMV) problem for the purpose of frequency parameter estimation while s true frequencies are located in the continuum of a bounded domain and sensors are randomly selected from a Uniform Linear Array (ULA). The MUSIC algorithm amounts to identifying a noise subspace from measurements, forming a noise-space correlation function and searching the s smallest local minima of the noise-space correlation function. Under the assumption that the true frequencies are separated by at least one Rayleigh Length (RL), we show that with high probability the noise-space correlation function is stably perturbed by noise if the number of sensors n ~ O(s) up to a logarithmic factor by means of a compressive version of discrete Ingham inequalities. As the theory implies, our numerical experiments demonstrate that the reconstruction error of MUSIC with n random sensors makes little difference once n is above a point of transition.