Disjoint Sources Research Articles

Deep learning-enhanced blind separation of incoherent and spatially disjoint sound sources

Separating incoherent sound sources within complex acoustical fields presents a significant challenge in acoustic imaging. Existing methods, such as Principal Component Analysis (PCA) applied to the Cross-Spectral Matrix (CSM), yield 'virtual' sources based on statistical orthogonality. However, this approach often fails to accurately identify distinct physical sources, primarily due to its reliance on statistical orthogonality solely. A state-of-the-art method involves computing a rotation matrix to enforce criteria such as least spatial entropy, or spatial orthogonality among sources, a process that, while effective, significantly increases computational complexity and time. This work introduces a hybrid approach combining PCA and deep learning for predicting spatially disjoint source maps from virtual sources. By simulating sound sources in random quantities and locations, we train a neural network tailored to this task. We address the order mismatch between PCA-derived virtual sources and pre-simulated labels by framing source separation as a set prediction problem, utilizing the Hungarian loss for efficient mismatch resolution. This method simplifies the separation process, offering faster post-training computations and eliminating the need for complex optimizations. Validation in simulated environments and real-world datasets has shown the model's effectiveness in source separation for acoustic imaging, indicating the potential of integrating deep learning with existing methods.

Restoration towards decomposition: A simple approach for domain generalization

Domain generalization (DG) aims to train a model capable of generalizing to unseen target domains by utilizing data from multiple disjoint source domains. Previous domain generalization methods primarily utilize data augmentation and domain-invariant feature learning. However, since the target domain is unknown, data augmentation methods struggle to generate images or features that closely resemble the target domain. In addition, existing domain-invariant feature learning methods are incapable of achieving complete decoupling, resulting in misalignment of feature distribution between the source and target domains. Hence, in this work, we propose a new perspective to address domain generalization by focusing on learning the ability to transform the target domain visual representations into those of the source domain. If the style of the unknown target domain can be transformed into the style of the known source domain, the adverse effects of domain shift can be effectively mitigated. Following this line of thought, we have developed a meta-learning task named “feature restoration” aimed at training a model to acquire this capability. Specifically, feature restoration involves replacing the domain-specific components within the visual representations of the target domain with those of the source domain. Experimental results on five DG benchmarks reveal that our method can achieve state-of-the-art performance. Ablation studies and visualization results further demonstrate the rationality and effectiveness of our design.

A minimum semi-degree sufficient condition for one-to-many disjoint path covers in semicomplete digraphs

Let D be a digraph. We define the minimum semi-degree of D as δ0(D):=min⁡{δ+(D),δ−(D)}. Let k be a positive integer, and let S={s} and T={t1,…,tk} be any two disjoint subsets of V(D). A set of k internally disjoint paths joining source set S and sink set T that cover all vertices of D are called a one-to-many k-disjoint directed path cover (k-DDPC for short) of D. A digraph D is semicomplete if for every pair x,y of vertices of it, there is at least one arc between x and y.In this paper, we prove that every semicomplete digraph D of sufficiently large order n with δ0(D)≥⌈(n+k−1)/2⌉ has a one-to-many k-DDPC joining any disjoint source set S and sink set T, where S={s},T={t1,…,tk}.

Generalized Meta-FDMixup: Cross-Domain Few-Shot Learning Guided by Labeled Target Data.

The vanilla Few-shot Learning (FSL) learns to build a classifier for a new concept from one or very few target examples, with the general assumption that source and target classes are sampled from the same domain. Recently, the task of Cross-Domain Few-Shot Learning (CD-FSL) aims at tackling the FSL where there is a huge domain shift between the source and target datasets. Extensive efforts on CD-FSL have been made via either directly extending the meta-learning paradigm of vanilla FSL methods, or employing massive unlabeled target data to help learn models. In this paper, we notice that in the CD-FSL task, the few labeled target images have never been explicitly leveraged to inform the model in the training stage. However, such a labeled target example set is very important to bridge the huge domain gap. Critically, this paper advocates a more practical training scenario for CD-FSL. And our key insight is to utilize a few labeled target data to guide the learning of the CD-FSL model. Technically, we propose a novel Generalized Meta-learning based Feature-Disentangled Mixup network, namely GMeta-FDMixup. We make three key contributions of utilizing GMeta-FDMixup to address CD-FSL. Firstly, we present two mixup modules - mixup-P and mixup-M that help facilitate utilizing the unbalanced and disjoint source and target datasets. These two novel modules enable diverse image generation for training the model on the source domain. Secondly, to narrow the domain gap explicitly, we contribute a novel feature disentanglement module that learns to decouple the domain-irrelevant and domain-specific features. By stripping the domain-specific features, we alleviate the negative effects caused by the domain inductive bias. Finally, we repurpose a new contrastive learning module, dubbed ConL. ConL prevents the model from only capturing category-related features via introducing contrastive loss. Thus, the generalization ability on novel categories is improved. Extensive experimental results on two benchmarks show the superiority of our setting and the effectiveness of our method. Code and models will be released.

A sufficient condition for the unpaired k-disjoint path coverability of interval graphs

Given disjoint source and sink sets, $$S = \{ s_1,\ldots ,s_k\}$$ and $$T = \{t_1,\ldots , t_k\}$$ , in a graph G, an unpaired k-disjoint path cover joining S and T is a set of pairwise vertex-disjoint paths $$\{ P_1, \ldots , P_k\}$$ that altogether cover every vertex of the graph, in which $$P_i$$ is a path from source $$s_i$$ to some sink $$t_j$$ . In this paper, we prove that if the scattering number, $${{\,\mathrm{sc}\,}}(G)$$ , of an interval graph G of order $$n \ge 2k$$ is less than or equal to $$-k$$ , there exists an unpaired k-disjoint path cover joining S and T in G for any possible configurations of source and sink sets S and T of size k each. The bound $${{\,\mathrm{sc}\,}}(G) \le -k$$ is tight; moreover, the proof directly leads to a quadratic algorithm for building an unpaired k-disjoint path cover.

Inverse problems for nonlinear hyperbolic equations with disjoint sources and receivers

AbstractThe article studies inverse problems of determining unknown coefficients in various semi-linear and quasi-linear wave equations given the knowledge of an associated source-to-solution map. We introduce a method to solve inverse problems for nonlinear equations using interaction of three waves that makes it possible to study the inverse problem in all globally hyperbolic spacetimes of the dimension$n+1\geqslant 3$and with partial data. We consider the case when the set$\Omega _{\mathrm{in}}$, where the sources are supported, and the set$\Omega _{\mathrm{out}}$, where the observations are made, are separated. As model problems we study both a quasi-linear equation and a semi-linear wave equation and show in each case that it is possible to uniquely recover the background metric up to the natural obstructions for uniqueness that is governed by finite speed of propagation for the wave equation and a gauge corresponding to change of coordinates. The proof consists of two independent components. In the geometric part of the article we introduce a novel geometrical object, the three-to-one scattering relation. We show that this relation determines uniquely the topological, differential and conformal structures of the Lorentzian manifold in a causal diamond set that is the intersection of the future of the point$p_{in}\in \Omega _{\mathrm{in}}$and the past of the point$p_{out}\in \Omega _{\mathrm{out}}$. In the analytic part of the article we study multiple-fold linearisation of the nonlinear wave equation using Gaussian beams. We show that the source-to-solution map, corresponding to sources in$\Omega _{\mathrm{in}}$and observations in$\Omega _{\mathrm{out}}$, determines the three-to-one scattering relation. The methods developed in the article do not require any assumptions on the conjugate or cut points.

A Lagrangian Multiplier Method for TDOA and FDOA Positioning of Multiple Disjoint Sources with Distance and Velocity Correlation Constraints

This paper considers the source localization problem using time differences of arrival (TDOA) and frequency differences of arrival (FDOA) for multiple disjoint sources moving together with constraints on their distances and velocity correlation. To make full use of the synergistic improvement of multiple source localization, the constraints on all sources are combined together to obtain the optimal result. Unlike the existing methods that can achieve the normal Cramér-Rao lower bound (CRLB), our object is to further improve the accuracy of the estimation with constraints. On the basis of maximum likelihood criteria, a Lagrangian estimator is developed to solve the constrained optimization problem by iterative algorithm. Specifically, by transforming the inequality constraints into exponential functions, Lagrangian multipliers can be used to determine the source locations via Newton’s method. In addition, the constrained CRLB for source localization with distance and velocity correlation constraints is also derived. The estimated accuracy of the source positions and velocities is shown to achieve the constrained CRLB. Simulations are included to confirm the advantages of the proposed method over the existing methods.

Characterization of interval graphs that are unpaired 2-disjoint path coverable

Given disjoint source and sink sets, S={s1,…,sk} and T={t1,…,tk}, in a graph G, an unpaired k-disjoint path cover joining S and T is a set of pairwise vertex-disjoint paths {P1,…,Pk} that altogether cover every vertex of the graph, in which Pi is a path from source si to some sink tj. In terms of a generalized scattering number, named an r-scattering number, we characterize interval graphs that have an unpaired 2-disjoint path cover joining S and T for any possible configurations of source and sink sets S and T of size 2 each. Also, it is shown that the r-scattering number of an interval graph can be computed in polynomial time.

A Novel Estimator for TDOA and FDOA Positioning of Multiple Disjoint Sources in the Presence of Calibration Emitters

Multiple-target localization is extensively applied in wireless connected networks. However, sensor location uncertainty is known to degrade significantly the target localization accuracy. Fortunately, calibration emitters such as unmanned aerial vehicles (UAV) with known location can be used to reduce the loss in localization accuracy due to sensor location errors. This paper is devoted to the use of UAV calibration emitters for time differences of arrival (TDOA) and frequency differences of arrival (FDOA) positioning of multiple targets. The study starts with deriving the Cramer-Rao bound (CRB) for TDOA/FDOA-based target location estimate when several UAV calibration signals are available. Subsequently, the paper presents an iterative constrained weighted least squares (ICWLS) estimator for multiple-target joint localization using TDOA/FDOA measurements from both target sources and UAV calibration emitters. The newly proposed method consists of two stages. In the first phase, the sensor locations are refined based on the calibration measurements as well as the prior knowledge of sensor locations. The second step provides the estimate of multiple-target locations by combining the measurements of target signals as well as the estimated values in the first phase. An efficient ICWLS algorithm is presented at each stage. Both the two algorithms are implemented by using matrix singular value decomposition (SVD), which is able to provide a closed-form solution and update the weighting matrix at every iteration. Finally, the convergence behavior and estimation mean-square-error (MSE) of the new estimator are deduced. Both theoretical analysis and simulation results show that the developed method can improve the TDOA/FDOA localization accuracy obviously with the help of UAV calibration emitters.

Iterative constrained weighted least squares estimator for TDOA and FDOA positioning of multiple disjoint sources in the presence of sensor position and velocity uncertainties

Sensor position and velocity uncertainties are known to be able to degrade the source localization accuracy significantly. This paper focuses on the problem of locating multiple disjoint sources using time differences of arrival (TDOAs) and frequency differences of arrival (FDOAs) in the presence of sensor position and velocity errors. First, the explicit Cramér–Rao bound (CRB) expression for joint estimation of source and sensor positions and velocities is derived under the Gaussian noise assumption. Subsequently, we compare the localization accuracy when multiple-source positions and velocities are determined jointly and individually based on the obtained CRB results. The performance gain resulted from multiple-target cooperative positioning is also quantified using the orthogonal projection matrix. Next, the paper proposes a new estimator that formulates the localization problem as a quadratic programming with some indefinite quadratic equality constraints. Due to the non-convex nature of the optimization problem, an iterative constrained weighted least squares (ICWLS) method is developed based on matrix QR decomposition, which can be achieved through some simple and efficient numerical algorithms. The newly proposed iterative method uses a set of linear equality constraints instead of the quadratic constraints to produce a closed-form solution in each iteration. Theoretical analysis demonstrates that the proposed method, if converges, can provide the optimal solution of the formulated non-convex minimization problem. Moreover, its estimation mean-square-error (MSE) is able to reach the corresponding CRB under moderate noise level. Simulations are included to corroborate and support the theoretical development in this paper.

Disjoint path covers joining prescribed source and sink sets in interval graphs

A disjoint path cover of a graph is a set of internally vertex-disjoint paths that altogether cover every vertex of the graph. Given two disjoint source and sink sets, S and T, in a graph G, a k-disjoint path cover of G joining S and T is a disjoint path cover composed of k paths, each of which runs from a source in S to a sink in T. In this paper, we give two short proofs for the characterization of interval graphs that possess a k-disjoint path cover (of one-to-one type) joining S and T for any S and T with |S|=|T|=1 and for the characterization of interval graphs that possess a k-disjoint path cover (of one-to-many type) joining S and T for any S and T with |S|=1 and |T|=k. Also, some partial results on the many-to-many disjoint path coverability of an interval graph are addressed. In addition, we discuss interval graphs in which the disjoint-path-cover property is preserved after removal of arbitrary f or less vertices.

A two-step weighted least-squares method for joint estimation of source and sensor locations: A general framework

It is well known that the Two-step Weighted Least-Squares (TWLS) is a widely used method for source localization and sensor position refinement. For this reason, we propose a unified framework of the TWLS method for joint estimation of multiple disjoint sources and sensor locations in this paper. Unlike some existing works, the presented method is based on more general measurement model, and therefore it can be applied to many different localization scenarios. Besides, it does not have the initialization and local convergence problem. The closed-form expression for the covariance matrix of the proposed TWLS estimator is also derived by exploiting the first-order perturbation analysis. Moreover, the estimation accuracy of the TWLS method is shown analytically to achieve the Cramér-Rao Bound (CRB) before the threshold effect takes place. The theoretical analysis is also performed in a common mathematical framework, rather than aiming at some specific signal metrics. Finally, two numerical experiments are performed to support the theoretical development in this paper.

Inverse scattering with disjoint source and observation sets on asymptotically hyperbolic manifolds

In this note, the scattering operator of an asymptotically hyperbolic manifold is only allowed to act on functions supported on the source set where is an open subset of the boundary, and the resulting functions are then restricted to the observation set where Γ is open. When Γ is the complement of the closure of we call the corresponding operator the off-diagonal scattering operator with respect to We prove that for a non-empty open proper subset such that the intersection of its closure and its complement is not empty, the off-diagonal scattering operator with respect to determines the manifold modulo isometries which are equal to the identity at the boundary. We also prove there is no analog of the L2 boundary controllability from an open subset of the boundary for radiation fields, which presents a possible obstacle to extending the inverse result for arbitrary disjoint subsets and

Structural total least squares algorithm for locating multiple disjoint sources based on AOA/TOA/FOA in the presence of system error

Single-station passive localization technology avoids the complex time synchronization and information exchange between multiple observatories, and is increasingly important in electronic warfare. Based on a single moving station localization system, a new method with high localization precision and numerical stability is proposed when the measurements from multiple disjoint sources are subject to the same station position and velocity displacement. According to the available measurements including the angle-of-arrival (AOA), time-of-arrival (TOA), and frequency-of-arrival (FOA), the corresponding pseudo linear equations are deduced. Based on this, a structural total least squares (STLS) optimization model is developed and the inverse iteration algorithm is used to obtain the stationary target location. The localization performance of the STLS localization algorithm is derived, and it is strictly proved that the theoretical performance of the STLS method is consistent with that of the constrained total least squares method under first-order error analysis, both of which can achieve the Cramer-Rao lower bound accuracy. Simulation results show the validity of the theoretical derivation and superiority of the new algorithm.

TDOA localization of multiple disjoint sources based on a calibration emitter

TDOA localization of multiple disjoint sources based on a calibration emitter

Performance Analysis and Dimension-Reduction Taylor Series Algorithms for Locating Multiple Disjoint Sources Based on TDOA Under Synchronization Clock Bias

Time difference of arrival (TDOA) localization technology, which does not require the time stamp of the source signals, is playing an increasingly important role in passive localization. Besides TDOA measurement noise and receiver position errors, another factor affecting TDOA positioning accuracy is the synchronization clock bias between different receivers. In this paper, the problem of TDOA source localization is considered under the condition that the TDOA measurements from multiple disjoint sources are subject to the same receiver position displacements and synchronization clock bias. First, based on the orthogonal projection matrix, the Cramer–Rao lower bounds of unknown parameters are derived in the presence of synchronization clock bias. Subsequently, according to the maximum likelihood (ML) criterion and Taylor series (TS) positioning method, the estimates obtained by ignoring the clock bias are shown to be biased, and the mean square error is found to increase quadratically with the 2-norm of the clock bias vector. To suppress the effect of synchronization clock bias, two dimension-reduction TS algorithms are proposed, and their asymptotically optimal performance is demonstrated under first-order error analysis. In addition, an ML-based closed-form solution for the clock bias vector is provided. Finally, simulation results verify the validity of the theoretical derivation and show the superiority of the proposed algorithms compared to the existing closed-form method.

A Constrained-Total-Least-Squares Method for Joint Estimation of Source and Sensor Locations: A General Framework

It is well known that sensor location uncertainties can significantly deteriorate the source positioning accuracy. Therefore, improving the sensor locations is necessary in order to achieve better localization performance. In this paper, a constrained-total-least-squares (CTLS) method for simultaneously locating multiple disjoint sources and refining the erroneous sensor positions is presented. Unlike conventional localization techniques, an important feature of the proposed method is that it establishes a general framework that is suitable for many different location measurements. First, a modified CTLS optimization problem is formulated after some algebraic manipulations and then the corresponding Newton iterative algorithm is derived to give the numerical solution. Subsequently, by using the first-order perturbation analysis, the explicit expression for the covariance matrix of the proposed CTLS estimator is deduced under the Gaussian assumption. Moreover, the estimation accuracy of the CTLS method is shown to achieve the Cramér-Rao bound (CRB) before the thresholding effect occurs by a rigorous proof. Finally, two kinds of numerical examples are given to corroborate the theoretical development in this paper. One uses the TDOAs/GROAs measurements and the other is based on the TOAs/FOAs parameters.

Distributed Rateless Codes with Unequal Error Protection Property for Space Information Networks

In this paper, we propose a novel distributed unequal error protection (UEP) rateless coding scheme (DURC) for space information networks (SIN). We consider the multimedia data transmissions in a dual-hop SIN communication scenario, where multiple disjoint source nodes need to transmit their UEP rateless coded data to a destination via a dynamic relay. We formulate the optimization problems to provide optimal degree distributions on the direct links and the dynamic relay links to satisfy the required error protection levels. The optimization methods are based on the And–Or tree analysis and can be solved by multi-objective programming. In addition, we evaluate the performance of the optimal DURC scheme, and simulation results show that the proposed DURC scheme can effectively provide UEP property under a variety of error requirements.

Blind separation of incoherent and spatially disjoint sound sources

Blind separation of sound sources aims at reconstructing the individual sources which contribute to the overall radiation of an acoustical field. The challenge is to reach this goal using distant measurements when all sources are operating concurrently. The working assumption is usually that the sources of interest are incoherent – i.e. statistically orthogonal – so that their separation can be approached by decorrelating a set of simultaneous measurements, which amounts to diagonalizing the cross-spectral matrix. Principal Component Analysis (PCA) is traditionally used to this end. This paper reports two new findings in this context. First, a sufficient condition is established under which “virtual” sources returned by PCA coincide with true sources; it stipulates that the sources of interest should be not only incoherent but also spatially orthogonal. A particular case of this instance is met by spatially disjoint sources – i.e. with non-overlapping support sets. Second, based on this finding, a criterion that enforces both statistical and spatial orthogonality is proposed to blindly separate incoherent sound sources which radiate from disjoint domains. This criterion can be easily incorporated into acoustic imaging algorithms such as beamforming or acoustical holography to identify sound sources of different origins. The proposed methodology is validated on laboratory experiments. In particular, the separation of aeroacoustic sources is demonstrated in a wind tunnel.

Sensor Gain-Phase Errors Estimation Using Disjoint Sources in Unknown Directions

Sensor gain-phase error estimation is necessary for equipment using sensor array, such as radar, sonar, and mobile communication before they come into service. Due to this requirement, we propose an offline calibration algorithm for sensor gain-phase errors using two auxiliary sources, which appear independently of both space and time, named disjoint sources. The significant superiority of this algorithm lies in the use of calibration sources in unknown directions. First, based on the data covariance matrix, the sensor phase errors are obtained, and the relation between phase error matrix and array manifold is established. Second, the directions of two disjoint sources are obtained by the way of 2D search based on eigen-structure subspace method. Third, we provide two methods to realize the algorithm. The proposed algorithm performs independently of phase errors. Moreover, the accurate direction measurement of calibration sources is not necessary. Computer simulations are shown to verify the efficacy of the proposed algorithm.