arXiv subjects

A. Fannjiang

Publications and source records attributed to A. Fannjiang.

9 recordsLinked to original sources

arXiv2020

Blind Ptychography: Uniqueness and Ambiguities

Ptychography with an unknown mask and object is analyzed for general ptychographic measurement schemes that are strongly connected and possess an anchor. Under a mild constraint on the mask phase, it is proved that the masked object estimate must be the product of a block phase factor and the true masked object. This local uniqueness manifests itself in the phase drift equation that determines the ambiguity at different locations connected by ptychographic shifts. The proposed mixing schemes effectively connects the ambiguity throughout the whole domain such that a distinct ambiguity profile arises and consequently possess the global uniqueness that the block phases have an affine profile and that the object and mask can be simultaneously recovered up to a constant scaling factor and an affine phase factor.

eess.IV↗

arXiv2020

Fixed Point Analysis of Douglas-Rachford Splitting for Ptychography and Phase Retrieval

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.

math.NA↗

arXiv2018

Blind Ptychography by Douglas-Rachford Splitting

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.

cs.CV↗

arXiv2016

Phase Retrieval by Linear Algebra

The null vector method, based on a simple linear algebraic concept, is proposed as a solution to the phase retrieval problem. In the case with complex Gaussian random measurement matrices, a non-asymptotic error bound is derived, yielding an asymptotic regime of accurate approximation comparable to that for the spectral vector method.

cs.IT↗

arXiv2012

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 super-resolution 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 super-resolution.

cs.IT↗

arXiv2011

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 {\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.

cs.IT↗

arXiv2005

Quenching of Reaction by Cellular Flows

We consider a reaction-diffusion equation in a cellular flow. We prove that in the strong flow regime there are two possible scenario for the initial data that is compactly supported and the size of the support is large enough. If the flow cells are large compared to the reaction length scale, propagating fronts will always form. For the small cell size, any finitely supported initial data will be quenched by a sufficiently strong flow. We estimate that the flow amplitude required to quench the initial data of support $L_0$ is $A>CL_0^4\ln(L_0)$. The essence of the problem is the question about the decay of the $L^\infty$ norm of a solution to the advection-diffusion equation, and the relation between this rate of decay and the properties of the Hamiltonian system generated by the two-dimensional incompressible fluid flow.

math.AP↗

arXiv2005

Relaxation Time of Quantized Toral Maps

We introduce the notion of the relaxation time for noisy quantum maps on the 2d-dimensional torus - a generalization of previously studied dissipation time. We show that relaxation time is sensitive to the chaotic behavior of the corresponding classical system if one simultaneously considers the semiclassical limit ($\hbar$ -> 0) together with the limit of small noise strength ($\ep$ -> 0). Focusing on quantized smooth Anosov maps, we exhibit a semiclassical regime $\hbar<\ep^{E}$ << 1 (where E>1) in which classical and quantum relaxation times share the same asymptotics: in this regime, a quantized Anosov map relaxes to equilibrium fast, as the classical map does. As an intermediate result, we obtain rigorous estimates of the quantum-classical correspondence for noisy maps on the torus, up to times logarithmic in $\hbar^{-1}$. On the other hand, we show that in the ``quantum regime'' $\ep$ << $\hbar$ << 1, quantum and classical relaxation times behave very differently. In the special case of ergodic toral symplectomorphisms (generalized ``Arnold's cat'' maps), we obtain the exact asymptotics of the quantum relaxation time and precise the regime of correspondence between quantum and classical relaxations.

math-ph↗

arXiv2004

Dissipation time and decay of correlations

We consider the effect of noise on the dynamics generated by volume-preserving maps on a d-dimensional torus. The quantity we use to measure the irreversibility of the dynamics is the dissipation time. We focus on the asymptotic behaviour of this time in the limit of small noise. We derive universal lower and upper bounds for the dissipation time in terms of various properties of the map and its associated propagators: spectral properties, local expansivity, and global mixing properties. We show that the dissipation is slow for a general class of non-weakly-mixing maps; on the opposite, it is fast for a large class of exponentially mixing systems which include uniformly expanding maps and Anosov diffeomorphisms.

math.DS↗