A. Fannjiang

A. Fannjiang, W. Liao

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 {\em 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 shares, to a lesser degree, BLOOMP's amazing attribute with respect to dynamic range. Detailed comparisons with existing algorithms such as Spectral Iterative Hard Thresholding (SIHT) and the frame-adapted BP are given.

A. Fannjiang, Z. Zhang

Blind ptychography is the scanning version of coherent diffractive imaging which seeks to recover both the object and the probe simultaneously. Based on alternating minimization by Douglas-Rachford splitting, AMDRS is a blind ptychographic algorithm informed by the uniqueness theory, the Poisson noise model and the stability analysis. Enhanced by the initialization method and the use of a randomly phased mask, AMDRS converges globally and geometrically. Three boundary conditions are considered in the simulations: periodic, dark-field and bright-field boundary conditions. The dark-field boundary condition is suited for isolated objects while the bright-field boundary condition is for non-isolated objects. The periodic boundary condition is a mathematically convenient reference point. Depending on the avail- ability of the boundary prior the dark-field and the bright-field boundary conditions may or may not be enforced in the reconstruction. Not surprisingly, enforcing the boundary condition improves the rate of convergence, sometimes in a significant way. Enforcing the bright-field condition in the reconstruction can also remove the linear phase ambiguity.