The Alternating Direction Method of Multipliers (ADMM) is uniquely suited for large-scale signal restoration problems, owing to its parallelizability. ADMM has been successfully deployed into several imaging modalities, including blind ptychography, phase retrieval, computer tomography, network unrolling, and magnetic resonance imaging. It is not an overstatement to say that while global convergence guarantees of ADMM under convex objective losses cement its utility, non-convex objectives offer superior reconstruction quality at the expense of these guarantees. In this paper, we break new ground with the development of invex ADMM which retains global optima guarantees for a novel class of non-convex, invex losses - an invex function is a mapping where any critical point is a global minimizer. More specifically, we propose a family of invex losses for which ADMM converges to the global minima, along with their guarantees. Our experimental evaluation and theoretical results shows that the proposed approach is very promising, and can aid in extending existing convergences guarantees of ADMM beyond traditional convex settings.

Signal restoration is an important constrained optimization problem with significant applications in various domains. Although non-convex constrained optimization problems have been shown to perform better than convex counterparts in terms of reconstruction quality, convex constrained optimization problems have been preferably for its global optima guarantees. Despite the success of non-convex methods in a large number of applications, it is not an overstatement to say that there is little or no hope for non-convex problems to ensure global optima. In this paper, for the first time, we develop invex constrained optimization theory to mitigate the loss of guarantees for global optima in non-convex constrained inverse problems, where the invex function is a mapping where any critical point is a global minimizer. We also develop relevant theories to extend the global optima guarantee to a set of quasi-invex functions - the largest optimizable mappings. More specifically, we propose a family of invex/quasi-invex of functions for handling constrained inverse problems using the non-convex setting along with guarantees for their global optima. Our experimental evaluation shows that the proposed approach is very promising and can aid in extending existing convex optimization algorithms, such as the alternating direction method of multipliers, and accelerated proximal gradient methods.

Retrieving a signal from its triple correlation spectrum, also called bispectrum, arises in a wide range of signal processing problems. Conventional methods do not provide an accurate inversion of bispectrum to the underlying signal. In this paper, we present an approach that uniquely recovers signals with finite spectral support (band-limited signals) from at least 3B measurements of its bispectrum function (BF), where B is the signal’s bandwidth. Our approach also extends to time-limited signals. We propose a two-step trust region algorithm that minimizes a non-convex objective function. First, we approximate the signal by a spectral algorithm and then refine the attained initialization based on a sequence of gradient iterations. Numerical experiments suggest that our proposed algorithm is able to estimate band-/time-limited signals from its BF for both complete and undersampled observations.

We investigate the theory of affine groups in the context of designing radar waveforms that obey the desired wideband ambiguity function (WAF). The WAF is obtained by correlating the signal with its time-dilated, Doppler-shifted, and delayed replicas. We consider the WAF definition as a coefficient function of the unitary representation of the group a • x + b. This is essentially an algebraic problem applied to the radar waveform design. Prior works on this subject largely analyzed narrow-band ambiguity functions. Here, we show that when the underlying wideband signal of interest is a pulse or pulse train, a tight frame can be built to design that waveform. Specifically, we design the radar signals by minimizing the ratio of bounding constants of the frame in order to obtain lower sidelobes in the WAF. This minimization is performed by building a codebook based on difference sets in order to achieve the Welch bound. We show that the tight frame so obtained is connected with the wavelet transform that defines the WAF.

Image reconstruction enhanced by regularizers, e.g., to enforce sparsity, low rank or smoothness priors on images, has many successful applications in vision tasks such as computer photography, biomedical and spectral imaging. It has been well accepted that non-convex regularizers normally perform better than convex ones in terms of the reconstruction quality. But their convergence analysis is only established to a critical point, rather than the global optima. To mitigate the loss of guarantees for global optima, we propose to apply the concept of invexity and provide the first list of proved invex regularizers for improving image reconstruction. Moreover, we establish convergence guarantees to global optima for various advanced image reconstruction techniques after being improved by such invex regularization. To the best of our knowledge, this is the first practical work applying invex regularization to improve imaging with global optima guarantees. To demonstrate the effectiveness of invex regularization, numerical experiments are conducted for various imaging tasks using benchmark datasets.

This paper presents a uniqueness result which states that a band- limited signal can be recovered from at least 3B measurements where B is the bandwidth from the radar ambiguity function (AF). This function is a two-dimensional mapping of the propagation delay and Doppler frequency. This formal model represents the distortion of a returned pulse due to the receiver matched filter. To estimate a time/band-limited signal from its radar AF, a trust region algorithm that minimizes a smoothed non-convex least-squares objective function is proposed. The method consists of two steps. First, we approximate the signal by an iterative spectral algorithm. Then, the attained initialization is refined based upon a sequence of gradient iterations. To the best of our knowledge this work is seminal in the sense of solving the radar phase retrieval problem for band-limited signals. Simulations results suggest that the proposed algorithm is able to estimate band-limited signals from its radar AF for both complete and incomplete radar cases. The AF is incomplete when only few shifts are considered. Numerical results show that the proposed algorithm estimates the signal with mean-square-error of 5 × 10-2for both complete and incomplete noisy cases.

We consider the recovery of a complex band-limited radar waveform from the magnitude of the fractional Fourier transform (FrFT) formulation of its ambiguity function (AF). This is essentially a phase retrieval (PR) problem applied to radar waveform design. The FrFT-based AF is mathematically obtained by correlating the signal with its frequency-rotated, Doppler-shifted, and delayed replicas. It completely characterizes the radar's capability to discriminate closely-spaced targets in the delay-Doppler plane. Unlike prior works which largely involved analytical approaches, our method WaveMax formulates the recovery of the waveform via the FrFT-based AF PR as a convex optimization problem. Specifically, we retrieve the signal by solving a basis pursuit that requires a designed approximation of the radar signal obtained by extracting the leading eigenvector of a matrix depending on the AF. Our theoretical analysis shows that unique waveform reconstruction is possible using signal samples no more than thrice the number of signal frequencies or time samples. Numerical experiments demonstrate that our method recovers band-limited signals from both even-sparse and random samples of the AFs with a mean squared error of$1\times 10^{-6}$and$5\times 10^{-2}$for full noiseless samples and sparse noisy samples, respectively.

Super-resolution phase retrieval is an inverse problem that appears in diffractive optical imaging (DOI) and consists in estimating a high-resolution image from low-resolution phaseless measurements. DOI has three diffraction zones where the data can be acquired, known as near, middle, and far fields. Recent works have studied super-resolution phase retrieval under a setup that records coded diffraction patterns at the near and far fields. However, the attainable resolution of the image is mainly governed by the sensor characteristics, whose cost increases in proportion to the resolution. Also, these methodologies lack theoretical analysis. Hence, this work derives super-resolution models from low-resolution coded phaseless measurements at any diffraction zone that in contrast to prior contributions, the attainable resolution of the image is determined by the resolution of the coded aperture. For the proposed models, the existence of a unique solution (up to a global unimodular constant) is guaranteed with high probability, which can be increased by designing the coded aperture. Therefore, a strategy that designs the spatial distribution of the coded aperture is developed. Additionally, a super-resolution phase retrieval algorithm that minimizes a smoothed nonconvex least-squares objective function is proposed. The method first approximates the image by a spectral algorithm, which is then refined based upon a sequence of alternate steps. Simulation results show that the proposed algorithm overcomes state-of-the-art methods in reconstructing the high-resolution image. In addition, the reconstruction quality using designed coded apertures is higher than that of the non-designed ensembles.

Phase retrieval is an inverse problem that consists in estimating a scene from diffraction intensities. This problem appears in optical imaging, which has three main diffraction zones where the measurements can be acquired, i.e., near, middle and far. Recent works have theoretically solved this inverse problem for the far zone, creating redundancy in the measurement process by including a coded aperture, which allows to modulate the scene and acquire coded diffraction patterns (CDP). However, in the state-of-the-art, the PR problem has not been theoretically studied for CDP at the near and middle zones. Moreover, the structure of the coded aperture is selected at random, leading to suboptimal estimations. Indeed, some of the coding elements employed in the literature are unfeasible because they increase the power of the scene in the modulation process. This paper provides theoretical guarantees for the recovery of a scene from CDP acquired at the three diffraction zones using admissible modulations. Based on the theoretical results, it is established that the image reconstruction quality directly depends on the coded aperture structure; therefore, a design strategy is proposed. In fact, when the scene can be sparsely represented in some basis, its support can be better estimated for a suitable choice of the coding elements in the modulation process. Experimental results show that the scene is successfully recovered by using designed coded apertures with up to 40% less measurements compared to non-designed ensembles.

Phase retrieval is a kind of ill-posed inverse problem, which is present in various applications, such as optics, astronomical imaging, and X-ray crystallography. Mathematically this inverse problem consists on recovering an unknown signal x ∈ Rn/Cnfrom a set of absolute square projections yk= |(ak, x)|2, k = 1,··· , m, where ak are the sampling vectors. However, the square absolute function is in general nonconvex and non-differentiable, which are desired properties in order to solve the problem, when traditional convex optimization algorithms are used. Therefore, this paper introduces a special differentiable function, known as smoothing function, in order to solve the phase retrieval problem by using the smoothing projected gradient (SPG) method. Moreover, to accelerate the convergence of this algorithm, this paper uses a nonlinear conjugate gradient method applied to the smoothing function as the search direction. Simulation results are provided to validate its efficiency on existing algorithms for phase retrieval. It is shown that compared with recently developed algorithms, the proposed method is able to accelerate the convergence.

X-ray crystallography is an experimental technique to estimate the 3D atomic positions of the elements present in a crystal. This technique constructs the 3D structure from the phase of diffracted and patterned X-rays (DPX). Multiple intensity DPX measurements are acquired to solve the phase retrieval problem. The feasibility of implementing this technique depends on solving the phase retrieval problem using expensive multiple valued patterns and the Truncated Wirtinger Flow Algorithm. This paper presents a Stochastic Truncated Wirtinger Flow Algorithm (STWF) which solves the phase retrieval problem based on DPX measurements low-cost boolean block-unblock coded apertures. Several simulations are realized to demonstrate the convergence of the STWF algorithm and the optimal parameters of the boolean coded apertures. The results indicate that given the DPX measurements, the quality of reconstructed phase images using STWF attained up 24:63dB of PSNR.