Douglas--Rachford splitting (DRS) methods based on the proximal point algorithms for the Poisson and Gaussian log-likelihood functions are proposed for ptychography and phase retrieval. Fixed point analysis shows that the DRS iterated sequences are always bounded explicitly in terms of the step size and that the fixed points are attracting if and only if the fixed points are regular solutions. This alleviates two major drawbacks of the classical Douglas--Rachford algorithm: slow convergence when the feasibility problem is consistent and divergent behavior when the feasibility problem is inconsistent. Fixed point analysis also leads to a simple, explicit expression for the optimal step size in terms of the spectral gap of an underlying matrix. When applied to the challenging problem of blind ptychography, which seeks to recover both the object and the probe simultaneously, alternating minimization with the DRS inner loops, even with a far from optimal step size, converges geometrically under the nearly minimum conditions established in the uniqueness theory.

The problem of imaging point objects can be formulated as estimation of an unknown atomic measure from its M+1 consecutive noisy Fourier coefficients. The standard resolution of this inverse problem is 1/M and super-resolution refers to the capability of resolving atoms at a higher resolution. When any two atoms are less than 1/M apart, this recovery problem is highly challenging and many existing algorithms either cannot deal with this situation or require restrictive assumptions on the sign of the measure. ESPRIT is an efficient method which does not depend on the sign of the measure. This paper provides an explicit error bound on the support matching distance of ESPRIT in terms of the minimum singular value of Vandermonde matrices. When the support consists of multiple well-separated clumps and noise is sufficiently small, the support error by ESPRIT scales like SRF2λ-2×Noise, where the Super-Resolution Factor (SRF) governs the difficulty of the problem and λ is the cardinality of the largest clump. Our error bound matches the min-max rate of a special model with one clump of closely spaced atoms up to a factor of M in the small noise regime, and therefore establishes the near-optimality of ESPRIT. Our theory is validated by numerical experiments.

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, BLOOMP's 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.

The linear inverse source and scattering problems are studied from the perspective of compressed sensing. By introducing the sensor as well as target ensembles, the maximum number of recoverable targets is proved to be at least proportional to the number of measurement data modulo a log-square factor with overwhelming probability. Important contributions include the discoveries of the threshold aperture, consistent with the classical Rayleigh criterion, and the incoherence effect induced by random antenna locations. The predictions of theorems are confirmed by numerical simulations.

The problem of imaging extended targets (sources or scatterers) is formulated in the framework of compressed sensing with emphasis on subwavelength resolution. The proposed formulation of the problems of inverse source/scattering is essentially exact and leads to the random partial Fourier measurement matrix in the case of periodic targets. In the case of square-integrable targets, the proposed sampling scheme in the Littlewood–Paley wavelet basis block-diagonalizes the scattering matrix with each block in the form of a random partial Fourier matrix corresponding to each dyadic scale of the target. The resolution issue is analyzed from two perspectives: stability and the signal-to-noise ratio (SNR). The subwavelength modes are shown to be typically unstable unless the measurement is carried out in near field. The number of the stable modes typically increases as the negative dth (the dimension of the target) power of the distance between the target and the sensors/source (in the unit of wavelength). he resolution limit is shown to be inversely proportional to the SNR in the high SNR limit. Numerical simulations are provided to validate the theoretical predictions.

This paper presents a study of stability, resolution, and detection for broad-band synthetic aperture (SA) imaging in Rician fading media. The stability condition $BNK^2/(K+1)\gg M$ is derived, where K is the Rician factor, B is the effective number of coherence bands, N is the effective number of array elements, and M is the number of (widely separated) targets. The imaging method is tested numerically with randomly distributed discrete scatterers, and comparisons with the imaging with the full response matrix (RM) are made. The resolution study reveals several interesting effects: First, given the same measurement resources, SA imaging has better resolution performance, although less stable, than RM imaging; second, for both imaging methods, the cross-range resolution measure (i.e., “full width at half maximum”) decreases with the aperture (N fixed) and the probe spacing (the total aperture fixed) while the range resolution increases with both parameters. A statistical scheme is introduced to reduce the uncertainty when the stability regime is not realized.