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1. Phase retrieval of real-valued signals in a shift-invariant space

   https://doi.org/10.1016/j.acha.2018.11.002  

   Chen, Yang ; Cheng, Cheng ; Sun, Qiyu ; Wang, Haichao ( July 2020 , Applied and computational harmonic analysis)

   In this paper, we consider an infinite-dimensional phase retrieval problem to reconstruct real-valued signals living in a shift-invariant space from their phaseless samples taken either on the whole line or on a discrete set with finite sampling density. We characterize all phase retrievable signals in a real-valued shift-invariant space using their nonseparability. For nonseparable signals generated by some function with support length L, we show that they can be well approximated, up to a sign, from their noisy phaseless samples taken on a discrete set with sampling density 2L-1 . In this paper, we also propose an algorithm with linear computational complexity to reconstruct nonseparable signals in a shift-invariant space from their phaseless samples corrupted by bounded noises. 

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2. A novel scheme for support identification and iterative sampling of bandlimited graph signals

   https://doi.org/10.1109/GlobalSIP.2018.8646488  

   Hashemi, Abolfazl ; Shafipour, Rasoul ; Vikalo, Haris ; Mateos, Gonzalo ( February 2019 , 2018 IEEE Global Conference on Signal and Information Processing (GlobalSIP))

   We study the problem of sampling and reconstruction of bandlimited graph signals where the objective is to select a node subset of prescribed cardinality that ensures interpolation of the original signal with the lowest reconstruction error. We propose an efficient iterative selection sampling approach and show that in the noiseless case the original signal is exactly recovered from the set of selected nodes. In the case of noisy measurements, a bound on the reconstruction error of the proposed algorithm is established. We further address the support identification of the bandlimited signal with unknown support and show that under a pragmatic sufficient condition, the proposed framework requires minimal number of samples to perfectly identify the support. The efficacy of the proposed methods is illustrated through numerical simulations on synthetic and real-world graphs. 

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3. Stable recovery of planar regions with algebraic boundaries in Bernstein form

   https://doi.org/10.1007/s10444-021-09843-0  

   Conti, Costanza ; Cotronei, Mariantonia ; Labate, Demetrio ; Molina, Wilfredo ( April 2021 , Advances in Computational Mathematics)

   null (Ed.)

   Abstract We present a new method for the stable reconstruction of a class of binary images from a small number of measurements. The images we consider are characteristic functions of algebraic domains, that is, domains defined as zero loci of bivariate polynomials, and we assume to know only a finite set of uniform samples for each image. The solution to such a problem can be set up in terms of linear equations associated to a set of image moments. However, the sensitivity of the moments to noise makes the numerical solution highly unstable. To derive a robust image recovery algorithm, we represent algebraic polynomials and the corresponding image moments in terms of bivariate Bernstein polynomials and apply polynomial-generating, refinable sampling kernels. This approach is robust to noise, computationally fast and simple to implement. We illustrate the performance of our reconstruction algorithm from noisy samples through extensive numerical experiments. Our code is released open source and freely available. 

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4. Heterogeneous multireference alignment: A single pass approach

   https://doi.org/10.1109/CISS.2018.8362313  

   Boumal, Nicolas ; Bendory, Tamir ; Lederman, Roy R. ; Singer, Amit ( March 2018 , Information Sciences and Systems (CISS), 2018 52nd Annual Conference on)

   Multireference alignment (MRA) is the problem of estimating a signal from many noisy and cyclically shifted copies of itself. In this paper, we consider an extension called heterogeneous MRA, where K signals must be estimated, and each observation comes from one of those signals, unknown to us. This is a simplified model for the heterogeneity problem notably arising in cryo-electron microscopy. We propose an algorithm which estimates the K signals without estimating either the shifts or the classes of the observations. It requires only one pass over the data and is based on low-order moments that are invariant under cyclic shifts. Given sufficiently many measurements, one can estimate these invariant features averaged over the K signals. We then design a smooth, non-convex optimization problem to compute a set of signals which are consistent with the estimated averaged features. We find that, in many cases, the proposed approach estimates the set of signals accurately despite non-convexity, and conjecture the number of signals K that can be resolved as a function of the signal length L is on the order of √L. 

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5. Graph Topology Learning and Signal Recovery Via Bayesian Inference

   https://doi.org/10.1109/DSW.2019.8755601  

   Ramezani-Mayiami, Mahmoud ; Hajimirsadeghi, Mohammad ; Skretting, Karl ; Blum, Rick S. ; Vincent Poor, H. ( June 2019 , 2019 IEEE Data Science Workshop (DSW))

   The estimation of a meaningful affinity graph has become a crucial task for representation of data, since the underlying structure is not readily available in many applications. In this paper, a topology inference framework, called Bayesian Topology Learning, is proposed to estimate the underlying graphtopologyfromagivensetofnoisymeasurementsofsignals. It is assumed that the graph signals are generated from GaussianMarkovRandomFieldprocesses. First,usingafactor analysis model, the noisy measured data is represented in a latent space and its posterior probability density function is found. Thereafter, by utilizing the minimum mean square error estimator and the Expectation Maximization (EM) procedure, a filter is proposed to recover the signal from noisy measurements and an optimization problem is formulated to estimatetheunderlyinggraphtopology. Theexperimentalresults show that the proposed method has better performance whencomparedtothecurrentstate-of-the-artalgorithmswith different performance measures. 

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