Developing methods to process irregularly structured data is crucial in applications like gene-regulatory, brain, power, and socioeconomic networks. Graphs have been the go-to algebraic tool for modeling the structure via nodes and edges capturing their interactions, leading to the establishment of the fields of graph signal processing (GSP) and graph machine learning (GML). Key graph-aware methods include Fourier transform, filtering, sampling, as well as topology identification and spatiotemporal processing. Although versatile, graphs can model only pairwise dependencies in the data. To this end, topological structures such as simplicial and cell complexes have emerged as algebraic representations for more intricate structure modeling in data-driven systems, fueling the rapid development of novel topological-based processing and learning methods. This paper first presents the core principles of topological signal processing through the Hodge theory, a framework instrumental in propelling the field forward thanks to principled connections with GSP-GML. It then outlines advances in topological signal representation, filtering, and sampling, as well as inferring topological structures from data, processing spatiotemporal topological signals, and connections with topological machine learning. The impact of topological signal processing and learning is finally highlighted in applications dealing with flow data over networks, geometric processing, statistical ranking, biology, and semantic communication.

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In this work, we deal with the problem of reconstructing a complete bandlimited graph signal from partially sampled noisy measurements. For a known graph structure, an efficient greedy algorithm is presented to partition the graph nodes into disjoint subsets such that sampling the graph signal from any subset leads to a sufficiently accurate reconstruction. Furthermore, we consider a scenario where the graph is massive and data processing centrally is no longer practical. To overcome this issue, a distributed framework is proposed that allows us to implement partitioning algorithms in a parallelized fashion. Finally, we provide numerical simulation results on synthetic and real-world data to show that our proposals outperform the state-of-the-art.

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In this paper, we propose a new method for joint ranging and Phase Offset (PO) estimation of multiple transponder-equipped aviation vehicles (TEAVs), including Manned Aerial Vehicles (MAVs) and Unmanned Aerial Vehicles (UAVs). The proposed method employs the overlapping uncoordinated Automatic Dependent Surveillance-Broadcast (ADS-B) packets broadcasted by the TEAVs for joint range and PO estimation prior to ADS-B packet decoding; thus, it can improve air safety when packet decoding is infeasible due to packet collision. Moreover, it enables coherent detection of ADS-B packets, which can result in more reliable multiple target tracking in aviation systems using cooperative sensors for sense and avoid. By minimizing the Kullback-Leibler Divergence (KLD), we show that the received complex baseband signal, coming from K uncoordinated TEAVs, which is corrupted by Additive White Gaussian Noise (AWGN) at a single antenna receiver can be approximated by an independent and identically distributed (i.i.d.) Gaussian Mixture (GM) with 2^K mixture components in the two-dimensional plane. The proposed estimator employs the Expectation-Maximization (EM) algorithm to estimate the modes of the 2D Gaussian mixture followed by a reordering estimation technique to jointly estimate range and PO. Simulation results show that the proposed joint estimator outperforms excising methods, such as the time segmentation method and the blind adaptive beamforming.

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Identifying overlapping communities from data is crucial for grasping the complex structure and dynamics of networks, amongst others in fields such as computational neuroscience. Research using fMRI has demonstrated that brain regions can change their functional network membership over time using temporal independent component analysis (tICA). However, reproducibility of such overlapping communities remains a challenge. Recently, several alternative approaches have been proposed to identify such overlapping communities. While results are promising, less is known about the model and assumptions that underlie these approaches. This paper shows that the bilinear model, combined with the assumption of quasi-stationary and uncorrelated sources, underlies novel methods for identifying overlapping brain networks. Furthermore, we propose a new algorithm, and through simulations, we investigate the robustness of our algorithm and several existing methods to solve the problem in noisy conditions with few available data samples. We conclude that quasi-stationary blind source separation-based techniques can have a promising advantage over tICA in terms of identifiability of overlapping communities and thus have the potential to contribute towards greater reproducibility of results.

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Four-dimensional ultrasound imaging of complex biological systems such as the brain is technically challenging because of the spatiotemporal sampling requirements. We present computational ultrasound imaging (cUSi), an imaging method that uses complex ultrasound fields that can be generated with simple hardware and a physical wave prediction model to alleviate the sampling constraints. cUSi allows for high-resolution four-dimensional imaging of brain hemodynamics in awake and anesthetized mice.

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We consider the problem of recovering complex-valued block sparse signals with unknown borders. Such signals arise naturally in numerous applications. Several algorithms have been developed to solve the problem of unknown block partitions. In pattern-coupled sparse Bayesian learning (PCSBL), each coefficient involves its own hyperparameter and those of its immediate neighbors to exploit the block sparsity. Extended block sparse Bayesian learning (EBSBL) assumes the block sparse signal consists of correlated and overlapping blocks to enforce block correlations. We propose a simpler alternative to EBSBL and reveal the underlying relationship between the proposed method and a particular case of EBSBL. The proposed algorithm uses the fact that immediate neighboring sparse coefficients are correlated. The proposed model is similar to classical sparse Bayesian learning (SBL). However, unlike the diagonal correlation matrix in conventional SBL, the unknown correlation matrix has a tridiagonal structure to capture the correlation with neighbors. Due to the entanglement of the elements in the inverse tridiagonal matrix, instead of a direct closed-form solution, an approximate solution is proposed. The alternative algorithm avoids the high dictionary coherence in EBSBL, reduces the unknowns of EBSBL, and is computationally more efficient. The sparse reconstruction performance of the algorithm is evaluated with both correlated and uncorrelated block sparse coefficients. Simulation results demonstrate that the proposed algorithm outperforms PCSBL and correlation-based methods such as EBSBL in terms of reconstruction quality. The numerical results also show that the proposed correlated SBL algorithm can deal with isolated zeros and nonzeros as well as block sparse patterns.

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Computational ultrasound imaging (cUSi) offers high-resolution 3D imaging with simpler hardware by relying on computational power. Central to cUSi is a large model matrix that stores all pulse-echo signals. For 3D imaging this matrix easily surpasses 1 terabyte, hindering in-memory storage and real-time processing. This paper presents a solution for cUSi through an aberrating layer by introducing a virtual array concept, which uses transfer functions to map data from the real to a virtual array, enabling the use of conventional reconstruction techniques like delay-and-sum (DAS). We demonstrate the mathematical similarity of this approach to using a full model matrix and validate it with promising imaging results.@en

The main focus of this paper is an active sensing application that involves selecting transmit and receive sensors to optimize the Cramér-Rao bound (CRB) on target parameters. Although the CRB is non-convex in the transmit and receive selection, we demonstrate that it is convex in the virtual array weight vector, which describes the multiplicity of the virtual array elements. Based on this finding, we propose a novel algorithm that optimizes the virtual array weight vector first and then finds a matching transceiver array. This greatly enhances the efficiency of the transmit and receive sensor selection problem.

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In this paper, we present a novel convolution theorem which encompasses the well known convolution theorem in (graph) signal processing as well as the one related to time-varying filters. Specifically, we show how a node-wise convolution for signals supported on a graph can be expressed as another node-wise convolution in a frequency domain graph, different from the original graph. This is achieved through a parameterization of the filter coefficients following a basis expansion model. After showing how the presented theorem is consistent with the already existing body of literature, we discuss its implications in terms of non-stationarity. Finally, we propose a data-driven algorithm based on subspace fitting to learn the frequency domain graph, which is then corroborated by experimental results on synthetic and real data.

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A new method for joint ranging and Phase Offset (PO) estimation of multiple drones/aircrafts is proposed in this paper. The proposed method employs the superimposed uncoordinated Automatic Dependent Surveillance-Broadcast (ADS-B) packets broadcasted by drones/aircrafts for joint range and PO estimation. It jointly estimates range and PO prior to ADS-B packet decoding; thus, it can improve air safety when packet decoding is infeasible due to packet collision. Moreover, it enables coherent detection of ADS-B packets, which can result in more reliable multiple target tracking in aviation systems using cooperative sensors for detect and avoid (DAA). By minimizing the Kullback-Leibler Divergence (KLD) statistical distance measure, we show that the received complex baseband signal coming from K uncoordinated drones/aircrafts corrupted by Additive White Gaussian Noise (AWGN) at a single antenna receiver can be approximated by an independent and identically distributed (i.i.d.) Gaussian Mixture (GM) with 2^{K} mixture components in the two-dimensional (2D) plane. While direct joint Maximum Likelihood Estimation (MLE) of range and PO from the derived GM Probability Density Function (PDF) leads to an intractable maximization, our proposed method employs the Expectation-Maximization (EM) algorithm to estimate the modes of the 2D Gaussian mixture followed by a reordering estimation technique through combinatorial optimization to estimate range and PO. An extension to a multiple antenna receiver is also investigated in this article. While the proposed estimator can estimate the range of multiple drones/aircrafts with a single receive antenna, a larger number of drones/aircrafts can be supported with higher accuracy by the use of multiple antennas at the receiver. The effectiveness of the proposed estimator is supported by simulation results. We show that the proposed estimator can jointly estimate the range of multiple drones/aircrafts accurately.

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We consider the scenario of finding the transfer function of an aberrating layer in front of a receiving ultrasound (US) array, assuming a separate non-aberrated transmit source. We propose a method for blindly estimating this transfer function without exact knowledge of the ultrasound sources or acoustic contrast image, and without directly measuring the transfer function using a separate controlled calibration experiment. Instead, the measurement data of many unknown random images is collected, such as from blood flow, and its second-order statistics are exploited. A measurement model is formulated that explicitly defines the layer's transfer function. A covariance domain problem is then defined to eliminate the image variable, and it is solved for the layer's transfer function using manifold-based optimization. The proposed approach and calibration algorithm are evaluated on a range of challenging and realistic simulations using the k-Wave toolbox. Our results show that, given a sufficiently efficient parameterization of the layer's transfer function, and by jointly estimating the transfer function at multiple frequencies, the proposed algorithm is able to obtain an accurate estimate. Subsequent simulated imaging experiments using the obtained transfer function also show increased imaging performance in various aberrating layers, including a skull layer.

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In this work, we consider the self-calibration problem of joint calibration and direction-of-Arrival (DOA) estimation using acoustic sensor arrays. Unlike many previous iterative approaches, we propose solvers that can be readily used for both linear and non-linear arrays for jointly estimating the sensor gain, phase errors, and the source DOAs. We derive these algorithms for both the conventional element-space and covariance data models. We focus on sparse and regular arrays formed using scalar sensors as well as vector sensors. The developed algorithms are obtained by transforming the underlying non-linear calibration model into a linear model, and subsequently by using convex relaxation techniques to estimate the unknown parameters. We also derive identifiability conditions for the existence of a unique solution to the self-calibration problem. To demonstrate the effectiveness of the developed techniques, numerical experiments, and comparisons to the state-of-The-Art methods are provided. Finally, the results from an experiment that was performed in an anechoic chamber using an acoustic vector sensor array are presented to demonstrate the usefulness of the proposed self-calibration techniques.

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Differential orthogonal signal-division multiplexing (OSDM) is attractive for underwater acoustic (UWA) communications because it can eliminate channel estimation, resulting in a substantial reduction of complexity at the receiver. However, when the channel is time-varying, it may suffer from serious inter-vector interference (IVI), which is similar to inter-carrier interference (ICI) in differential orthogonal frequency-division multiplexing (OFDM). To mitigate this degradation of system performance, this paper provides a novel two-hop differential OSDM system based on IVI self-cancellation. Although this method improves system reliability at the cost of losing data rate, it is easy to implement in UWA modems. Finally, numerical simulations demonstrate the effectiveness of the proposed two-hop differential OSDM system over time-varying UWA channels.@en

This article focuses on greedy approaches to select the most informative k sensors from N candidates to maximize the Fisher information, i.e., the determinant of the Fisher information matrix (FIM), which indicates the volume of the information ellipsoid (VIE) constructed by the FIM. However, it is a critical issue for conventional greedy approaches to quantify the Fisher information properly when the FIM of the selected subset is rank-deficient in the first (n-1) steps, where n is the problem dimension. In this work, we propose a new metric, i.e., the Fisher information intensity (FII), to quantify the Fisher information contained in the subset S with respect to that in the ground set N specifically in the subspace spanned by the vectors associated with S. Based on the FII, we propose to optimize the ratio between VIEs corresponding to S and N. This volume ratio is composed of a nonzero (i.e., the FII) and a zero part. Moreover, the volume ratio can be easily calculated based on a change of basis. A cost function is developed based on the volume ratio and proven monotone submodular. A greedy algorithm and its fast version are proposed accordingly to guarantee a near-optimal solution with a complexity of O Nkn-3 and O Nkn2, respectively. Numerical results demonstrate the superiority of the proposed algorithms under various measurement settings.

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Sensor networks are omnipresent in our daily lives. In this chapter, we deal with the problem of organizing such a sensor network, or in other words, how to best place the different sensors. The sensing applications we consider in this chapter yield the estimation and detection of an unknown variable that can be indirectly measured by different sensors. We assume that the number of sensors is limited, e.g., because of cost considerations, and hence the problem boils down to optimally placing the smallest number of sensors such that a certain inference performance can be guaranteed. Different types of inference metrics for both estimation and detection will be studied. A major contribution of this work involves the treatment of not only conditionally independent measurements yet also conditionally dependent ones, which makes the problem far from trivial. Our results will be corroborated by a running example on field estimation and detection.@en

It has been previously demonstrated that applying an aberrating mask for 2D compressive imaging using a low number of sensors (elements) can significantly improve image resolution, as evaluated via the point spread function. Here we investigate the potential to apply a similar approach for 3D flow monitoring. We conducted a 3D k-Wave simulation using a 5x5 sensor array coupled to a physical coding mask, performing B-mode and power Doppler imaging on a 3D carotid artery flow model. An approximately three times smaller lateral PSF was achieved at the cost of increased background clutter level and slightly increased axial PSF. A better definition of the vessel border and finer flow speckle were observed in power Doppler imaging. Our results suggest that 3D compressive imaging using a very low sensor count of 25 with spatial coding mask has the potential to monitor 3D carotid artery flow.@en

Integrated sidelobe level is a useful measure to quantify robustness of a waveform-filter pair to unknown range clutter and multiple closely located targets. Sidelobe suppression on receive will incur a loss in the signal to noise ratio after pulse compression. We derive a pulse compression filter that has the greatest integrated sidelobe suppression possible for a given acceptable signal to noise ratio loss. The solution is given in a closed form, which can be adjusted using a single parameter to chose between greater sidelobe or interference and noise suppression. We verify the derived filter using simulations, comparing it to other proposed mismatched filter designs. To expand the robustness of the filter, we additionally investigate noise uncertainty robustness. We derive two robustness measures for noise uncertainty and analyze the performance through simulation.@en

Graph-based learning and estimation are fundamental problems in various applications involving power, social, and brain networks, to name a few. While learning pair-wise interactions in network data is a well-studied problem, discovering higher-order interactions among subsets of nodes is still not yet fully explored. To this end, encompassing and leveraging (non)linear structural equation models as well as vector autoregressions, this paper proposes autoregressive graph Volterra models (AGVMs) that can capture not only the connectivity between nodes but also higher-order interactions presented in the networked data. The proposed overarching model inherits the identifiability and expressibility of the Volterra series. Furthermore, two tailored algorithms based on the proposed AGVM are put forth for topology identification and link prediction in distribution grids and social networks, respectively. Real-data experiments on different real-world collaboration networks highlight the impact of higher-order interactions in our approach, yielding discernible differences relative to existing methods.@en

We consider the problem of recovering block-sparse signals with unknown boundaries. Such signals arise naturally in various applications. Recent literature introduced a pattern-coupled or clustered Gaussian prior, in which each coefficient involves its own hyperparameter as well as its immediate neighbors' hyperparameters. Some methods use a hierarchical distribution making the solution vulnerable to the parameter choice. Besides, these methods mainly rely on the expectation-maximization (EM) algorithm and either require a suboptimal solution or an approximation of the hyperparameters. To address these difficulties, we propose to solve the pattern coupling problem via fixed point iterations instead of the EM algorithm. The proposed algorithm does not require any further assumptions on the hyperparameters and provides a simple update rule for the hyperparameters. Although the fixed point iterations method is an empirical strategy, it provides a fast convergence rate. The proposed algorithm is tested on a simple direction of arrival (DOA) and amplitude estimation problem. From our simulations, we see that the proposed method achieves similar reconstruction results with the state-of-the-art; however, the proposed method is faster than the existing counterparts.@en

Sensor selection is a useful method to help reduce computational, hardware, and power requirements while maintaining acceptable performance. Although minimizing the Cramér-Rao bound has been adopted previously for sparse sensing, it did not consider multiple targets and unknown target directions. We propose to tackle the sensor selection problem for direction of arrival estimation using the worst-case Cramér-Rao bound of two uncorrelated equal power sources on planar arrays. We cast the problem as a convex semi-definite program and retrieve the binary selection by randomized rounding. We illustrate the proposed method through numerical examples related to planar arrays. We show that our method selects a combination of edge and center elements, which contrasts with solutions obtained by minimizing the single-target Cramér-Rao bound.@en