It is challenging for generative models to learn a distribution over graphs because of the lack of permutation invariance: nodes may be ordered arbitrarily across graphs, and standard graph alignment is combinatorial and notoriously expensive. We propose AlignGraph, a group of generative models that combine fast and efficient graph alignment methods with a family of deep generative models that are invariant to node permutations. Our experiments demonstrate that our framework successfully learns graph distributions, outperforming competitors by 25% — 560% in relevant performance scores.

Many contemporary machine learning models require extensive tuning of hyperparameters to perform well. A variety of methods, such as Bayesian optimization, have been developed to automate and expedite this process. However, tuning remains extremely costly as it typically requires repeatedly fully training models. To address this issue, Bayesian optimization methods have been extended to use cheap, partially trained models to extrapolate to expensive complete models. While this approach enlarges the set of explored hyperparameters, including many low-fidelity observations adds to the intrinsic randomness of the procedure and makes extrapolation challenging. We propose to accelerate hyperparameter tuning for neural networks in a robust way by taking into account the relative amount of information contributed by each training example. To do so, we integrate importance sampling with Bayesian optimization, which significantly increases the quality of the black-box function evaluations and their runtime. To overcome the additional overhead cost of using importance sampling, we cast hyperparameter search as a multi-task Bayesian optimization problem over both hyperparameters and importance sampling design, which achieves the best of both worlds. Through learning a trade-off between training complexity and quality, our method improves upon validation error, in the average and worst-case. We show that this results in more reliable performance of our method in less wall-clock time across a variety of and datasets complex neural architectures.

Variational autoencoders (VAEs) optimize an objective that comprises a reconstruction loss (the distortion) and a KL term (the rate). The rate is an upper bound on the mutual information, which is often interpreted as a regularizer that controls the degree of compression. We here examine whether inclusion of the rate term also improves generalization. We perform rate-distortion analyses in which we control the strength of the rate term, the network capacity, and the difficulty of the generalization problem. Lowering the strength of the rate term paradoxically improves generalization in most settings, and reducing the mutual information typically leads to underfitting. Moreover, we show that generalization performance continues to improve even after the mutual information saturates, indicating that the gap on the bound (i.e. the KL divergence relative to the inference marginal) affects generalization. This suggests that the standard spherical Gaussian prior is not an inductive bias that typically improves generalization, prompting further work to understand what choices of priors improve generalization in VAEs.

Deep latent-variable models learn representations of high-dimensional data in an unsupervised manner. A number of recent efforts have focused on learning representations that disentangle statistically independent axes of variation by introducing modifications to the standard objective function. These approaches generally assume a simple diagonal Gaussian prior and as a result are not able to reliably disentangle discrete factors of variation. We propose a two-level hierarchical objective to control relative degree of statistical independence between blocks of variables and individual variables within blocks. We derive this objective as a generalization of the evidence lower bound, which allows us to explicitly represent the trade-offs between mutual information between data and representation, KL divergence between representation and prior, and coverage of the support of the empirical data distribution. Experiments on a variety of datasets demonstrate that our objective can not only disentangle discrete variables, but that doing so also improves disentanglement of other variables and, importantly, generalization even to unseen combinations of factors.

There exist many problems in science and engineering that involve optimization of an unknown or partially unknown objective function. Recently, Bayesian Optimization (BO) has emerged as a powerful tool for solving optimization problems whose objective functions are only available as a black box and are expensive to evaluate. Many practical problems, however, involve optimization of an unknown objective function subject to unknown constraints. This is an important yet challenging problem for which, unlike optimizing an unknown function, existing methods face several limitations. In this paper, we present a novel constrained Bayesian optimization framework to optimize an unknown objective function subject to unknown constraints. We introduce an equivalent optimization by augmenting the objective function with constraints, introducing auxiliary variables for each constraint, and forcing the new variables to be equal to the main variable. Building on the Alternating Direction Method of Multipliers (ADMM) algorithm, we propose ADMM-Bayesian Optimization (ADMMBO) to solve the problem in an iterative fashion. Our framework leads to multiple unconstrained subproblems with unknown objective functions, which we then solve via BO. Our method resolves several challenges of state-of-the-art techniques: it can start from infeasible points, is insensitive to initialization, can efficiently handle `decoupled problems' and has a concrete stopping criterion. Extensive experiments on a number of challenging BO benchmark problems show that our proposed approach outperforms the state-of-the-art methods in terms of the speed of obtaining a feasible solution and convergence to the global optimum as well as minimizing the number of total evaluations of unknown objective and constraints functions.

Segmenting objects of interest from 3D data sets is a common problem encountered in biological data. Small field of view and intrinsic biological variability combined with optically subtle changes of intensity, resolution, and low contrast in images make the task of segmentation difficult, especially for microscopy of unstained living or freshly excised thick tissues. Incorporating shape information in addition to the appearance of the object of interest can often help improve segmentation performance. However, the shapes of objects in tissue can be highly variable and design of a flexible shape model that encompasses these variations is challenging. To address such complex segmentation problems, we propose a unified probabilistic framework that can incorporate the uncertainty associated with complex shapes, variable appearance, and unknown locations. The driving application that inspired the development of this framework is a biologically important segmentation problem: the task of automatically detecting and segmenting the dermal-epidermal junction (DEJ) in 3D reflectance confocal microscopy (RCM) images of human skin. RCM imaging allows noninvasive observation of cellular, nuclear, and morphological detail. The DEJ is an important morphological feature as it is where disorder, disease, and cancer usually start. Detecting the DEJ is challenging, because it is a 2D surface in a 3D volume which has strong but highly variable number of irregularly spaced and variably shaped "peaks and valleys." In addition, RCM imaging resolution, contrast, and intensity vary with depth. Thus, a prior model needs to incorporate the intrinsic structure while allowing variability in essentially all its parameters. We propose a model which can incorporate objects of interest with complex shapes and variable appearance in an unsupervised setting by utilizing domain knowledge to build appropriate priors of the model. Our novel strategy to model this structure combines a spatial Poisson process with shape priors and performs inference using Gibbs sampling. Experimental results show that the proposed unsupervised model is able to automatically detect the DEJ with physiologically relevant accuracy in the range 10- 20 μm .

Algorithms for fully automatic segmentation of images are often not sufficiently generic with suitable accuracy, and fully manual segmentation is not practical in many settings. There is a need for semiautomatic algorithms, which are capable of interacting with the user and taking into account the collected feedback. Typically, such methods have simply incorporated user feedback directly. Here, we employ active learning of optimal queries to guide user interaction. Our work in this paper is based on constrained spectral clustering that iteratively incorporates user feedback by propagating it through the calculated affinities. The original framework does not scale well to large data sets, and hence is not straightforward to apply to interactive image segmentation. In order to address this issue, we adopt advanced numerical methods for eigen-decomposition implemented over a subsampling scheme. Our key innovation, however, is an active learning strategy that chooses pairwise queries to present to the user in order to increase the rate of learning from the feedback. Performance evaluation is carried out on the Berkeley segmentation and Graz-02 image data sets, confirming that convergence to high accuracy levels is realizable in relatively few iterations.

Cardiac electrical imaging from body surface potential measurements is increasingly being seen as a technology with the potential for use in the clinic, for example for pre-procedure planning or during-treatment guidance for ventricular arrhythmia ablation procedures. However several important impediments to widespread adoption of this technology remain to be effectively overcome. Here we address two of these impediments: the difficulty of reconstructing electric potentials on the inner (endocardial) as well as outer (epicardial) surfaces of the ventricles, and the need for full anatomical imaging of the subject's thorax to build an accurate subject-specific geometry. We introduce two new features in our reconstruction algorithm: a nonlinear low-order dynamic parameterization derived from the measured body surface signals, and a technique to jointly regularize both surfaces. With these methodological innovations in combination, it is possible to reconstruct endocardial activation from clinically acquired measurements with an imprecise thorax geometry. In particular we test the method using body surface potentials acquired from three subjects during clinical procedures where the subjects' hearts were paced on their endocardia using a catheter device. Our geometric models were constructed using a set of CT scans limited in axial extent to the immediate region near the heart. The catheter system provides a reference location to which we compare our results. We compare our estimates of pacing site localization, in terms of both accuracy and stability, to those reported in a recent clinical publication , where a full set of CT scans were available and only epicardial potentials were reconstructed.

Noninvasive imaging of cardiac electrical function has begun to move towards clinical adoption. Here, we consider one common formulation of the problem, in which the goal is to estimate the spatial distribution of electrical activation times during a cardiac cycle. We address the challenge of understanding the robustness and uncertainty of solutions to this formulation. This formulation poses a nonconvex, nonlinear least squares optimization problem. We show that it can be relaxed to be convex, at the cost of some degree of physiological realism of the solution set, and that this relaxation can be used as a framework to study model inaccuracy and solution uncertainty. We present two examples, one using data from a healthy human subject and the other synthesized with the ECGSIM software package. In the first case, we consider uncertainty in the initial guess and regularization parameter. In the second case, we mimic the presence of an ischemic zone in the heart in a way which violates a model assumption. We show that the convex relaxation allows understanding of spatial distribution of parameter sensitivity in the first case, and identification of model violation in the second.

One of the biggest challenges in averaging ECG or EEG signals is to overcome temporal misalignments and distortions, due to uncertain timing or complex non-stationary dynamics. Standard methods average individual leads over a collection of epochs on a time-sample by time-sample basis, even when multi-electrode signals are available. Here we propose a method that averages multi electrode recordings simultaneously by using spatial patterns and without relying on time or frequency.

The 12 papers in this special issue focus on medical imaging and image computing in computational physiology. Most of the papers are focused on various aspects of the cardiovascular system across various physiological processes and observational scales.

Reflectance confocal microscopy (RCM) is a non-invasive and in-vivo imaging modality, which can take images from different depths of the human skin. A challenging problem is to detect a clinically important subsurface section of the skin, the Dermis/Epidermis junction, in RCM images. This is a tough problem because of the huge variation of texture and intensity features across both intersubject and intrasubject tissues. On the other hand, there's almost no wrinkle-free part of the skin. This well-known phenomenon can be used as a histological clue for guessing the probability of being Dermis or Epidermis in the neighboring regions. In this paper, we develop a two-step wrinkle detector for RCM images. By analyzing the results on different RCM images, we conclude it has high sensitivity and specificity, but a relatively lower Jaccard index.

In this paper we propose a supervised 3D segmentation algorithm to locate the esophagus in thoracic CT scans using a variational framework. To address challenges due to low contrast, several priors are learned from a training set of segmented images. Our algorithm first estimates the centerline based on a spatial model learned at a few manually marked anatomical reference points. Then an implicit shape model is learned by subtracting the centerline and applying PCA to these shapes. To allow local variations in the shapes, we propose to use nonlinear smooth local deformations. Finally, the esophageal wall is located within a 3D level set framework by optimizing a cost function including terms for appearance, the shape model, smoothness constraints and an air/contrast model.

Fluorescence molecular tomography (FMT) allows in vivo localization and quantification of fluorescence biodistributions in whole animals. The ill-posed nature of the tomographic reconstruction problem, however, limits the attainable resolution. Improvements in resolution and overall imaging performance can be achieved by forming image priors from geometric information obtained by a secondary anatomical or functional high-resolution imaging modality such as X-ray computed tomography or magnetic resonance imaging. A particular challenge in using image priors is to avoid the use of assumptions that may bias the solution and reduced the accuracy of the inverse problem. This is particularly relevant in FMT inversions where there is not an evident link between secondary geometric information and the underlying fluorescence biodistribution. We present here a new, two step approach to incorporating structural priors into the FMT inverse problem. By using the anatomic information to define a low dimensional inverse problem, we obtain a solution which we then use to determine the parameters defining a spatially varying regularization matrix for the full resolution problem. The regularization term is thus customized for each data set and is guided by the data rather than depending only on user defined a priori assumptions. Results are presented for both simulated and experimental data sets, and show significant improvements in image quality as compared to traditional regularization techniques.

In this paper, we report new progress in developing the instrument and software platform of a combined X-ray mammography/diffuse optical breast imaging system. Particularly, we focus on system validation using a series of balloon phantom experiments and the optical image analysis of 49 healthy patients. Using the finite-element method for forward modeling and a regularized Gauss-Newton method for parameter reconstruction, we recovered the inclusions inside the phantom and the hemoglobin images of the human breasts. An enhanced coupling coefficient estimation scheme was also incorporated to improve the accuracy and robustness of the reconstructions. The recovered average total hemoglobin concentration (HbT) and oxygen saturation (SO2) from 68 breast measurements are 16.2 microm and 71%, respectively, where the HbT presents a linear trend with breast density. The low HbT value compared to literature is likely due to the associated mammographic compression. From the spatially co-registered optical/X-ray images, we can identify the chest-wall muscle, fatty tissue, and fibroglandular regions with an average HbT of 20.1+/-6.1 microm for fibroglandular tissue, 15.4+/-5.0 microm for adipose, and 22.2+/-7.3 microm for muscle tissue. The differences between fibroglandular tissue and the corresponding adipose tissue are significant (p < 0.0001). At the same time, we recognize that the optical images are influenced, to a certain extent, by mammographical compression. The optical images from a subset of patients show composite features from both tissue structure and pressure distribution. We present mechanical simulations which further confirm this hypothesis.

Voxel-based reconstructions in diffuse optical tomography (DOT) using a quadratic regularization functional tend to produce very smooth images due to the attenuation of high spatial frequencies. This then causes difficulty in estimating the spatial extent and contrast of anomalous regions such as tumors. Given an assumption that the target image is piecewise constant, we can employ a parametric model to estimate the boundaries and contrast of an inhomogeneity directly. In this paper, we describe a method to directly reconstruct such a shape boundary from diffuse optical measurements. We parameterized the object boundary using a spherical harmonic basis, and derived a method to compute sensitivities of measurements with respect to shape parameters. We introduced a centroid constraint to ensure uniqueness of the combined shape/center parameter estimate, and a projected Newton method was utilized to optimize the object center position and shape parameters simultaneously. Using the shape Jacobian, we also computed the Cramér-Rao lower bound on the theoretical estimator accuracy given a particular measurement configuration, object shape, and level of measurement noise. Knowledge of the shape sensitivity matrix and of the measurement noise variance allows us to visualize the shape uncertainty region in three dimensions, giving a confidence region for our shape estimate. We have implemented our shape reconstruction method, using a finite-difference-based forward model to compute the forward and adjoint fields. Reconstruction results are shown for a number of simulated target shapes, and we investigate the problem of model order selection using realistic levels of measurement noise. Assuming a signal-to-noise ratio in the amplitude measurements of 40 dB and a standard deviation in the phase measurements of 0.1 degrees , we are able to estimate an object represented with an eighth-order spherical harmonic model having an absorption contrast of 0.15 cm(-1) and a volume of 4.82 cm(3) with errors of less than 10% in object volume and absorption contrast. We also investigate the robustness of our shape-based reconstruction approach to a violation of the assumption that the medium is purely piecewise constant.

Shape-based solutions have recently received attention for certain ill-posed inverse problems. Their advantages include implicit imposition of relevant constraints and reduction in the number of unknowns, especially important for nonlinear ill-posed problems. We apply the shape-based approach to current-injection electrical impedance tomography (EIT) reconstructions. We employ a boundary element method (BEM) based solution for EIT. We introduce two shape models, one based on modified B-splines, and the other based on spherical harmonics, for BEM modeling of shapes. These methods allow us to parameterize the geometry of conductivity inhomogeneities in the interior of the volume. We assume the general shape of piecewise constant inhomogeneities is known but their conductivities and their exact location and shape is not. We also assume the internal conductivity profile is piecewise constant, meaning that each region has a constant conductivity. We propose and test three different regularization techniques to be used with either of the shape models. The performance of our methods is illustrated via both simulations in a digital torso model and phantom experiments when there is a single internal object. We observe that in the noisy environment, either simulated noise or real sources of noise in the experimental study, we get reasonable reconstructions. Since the signal-to-noise ratio (SNR) expected in modern EIT instruments is higher than that used in this study, these reconstruction methods may prove useful in practice.

We examine the problem of fluorescence molecular tomography using the normalized Born approximation, termed herein the Born ratio, from a statistical perspective. Experimentally verified noise models for received signals at the excitation and emission wavelengths are combined to generate a stochastic model for the Born ratio. This model is then utilized within a maximum likelihood framework to obtain an inverse solution based on a fixed point iteration. Results are presented for three experimental scenarios: phantom data with a homogeneous background, phantoms implanted within a small animal, and in vivo data using an exogenous probe.

The accuracy of the inverse solutions obtained by imaging modalities which use surface measurements can be significantly reduced by even small modeling errors. One important type of modeling error is inaccuracy in locating the sensors. In this paper, we first report on the sensitivity to sensor mislocation of a known-geometry inverse solution both through simulation and by calculating the Cramer-Rao lower bound on the estimate variance. We then introduce a method based on a likelihood ratio hypothesis test to use the statistical behavior of the residuals to identify potential sensor mislocation. Upon detection of sensor location error, we then propose an approach, based on principal component analysis of the residuals, to reduce the sensitivity of the reconstruction algorithm to this kind of modeling error. We show the simulation results using a method, introduced in our previous work, for electrical impedance tomography based on boundary element methods

In this paper, we present theoretical developments and experimental results for the problem of estimating the conductivity map inside a volume using electrical impedance tomography (EIT) when the boundary locations of any internal inhomogeneities are known. We describe boundary element method (BEM) implementations of advanced electrode models for the forward problem of EIT. We then use them in the inverse problem with known internal boundaries and derive the associated Jacobians. We report on the results of two EIT phantom studies, one using a homogeneous cubical tank, and one using a cylindrical tank with agar conductivity inhomogeneities. We test both the accuracy of our BEM forward model, including the electrode models, as well as our inverse solution, against the measured data. Results show good agreement between measured values and both forward-computed tank voltages and inverse-computed conductivities; for instance, in a phantom experiment, we reconstructed the conductivities of three agar objects inside a cylindrical tank with an error less than 2% of their true value.

Shape-based solutions have recently received attention for certain ill-posed inverse problems when the relevant constituent parameters can be modeled as isotropic and piecewise continuous or homogeneous. Their advantages include implicit imposition of relevant constraints and reduction in the number of unknowns, especially important for non-linear ill-posed problems. We introduce a method, based on modified B-splines and the boundary element method (BEM), for shape-based inverse problems and apply it to current-injection electrical impedance tomography (EIT) in the torso. We assume the general shape of piecewise constant inhomogeneities is known but their conductivities and their exact location and shape are not. We model the boundaries of the unknown inhomogeneities using bivariate spline basis shape functions and solve for the conductivities, locations, and boundaries. The performance of our method is illustrated via simulation in a realistic torso model.

New MR acquisition techniques are enabling fast acquisition of data from an entire 3D volume. Parallel MR imaging methods can provide additional acceleration to the data acquisition rate. However, the large computational memory requirements associated with 3D imaging requires new efficient reconstruction techniques. This manuscript presents an efficient implementation of SPACE-RIP for the rapid reconstruction of sub-sampled 3D MR data. Uniform sub-sampling effectively decouples the SPACE-RIP linear system of equations into a number of smaller systems which can each be solved independently, thus requiring fewer computational resources. We present a particular phase-encode sampling pattern to capitalize on this effect which allows SPACE-RIP to be computationally competitive with SENSE in 3D imaging, while providing the added benefits of self-calibrated coil sensitivity maps and improved artifact suppression through irregular sub-sampling.

Diffuse optical tomography (DOT) is an emerging medical imaging modality offering the possibility of recovering the distribution of optical absorption and scattering coefficients, and from them localize metabolic parameters such as oxygen saturation or neural activity, using nonionizing near-IR light. However it requires solving a badly ill-posed inverse problem. In this article we discuss the formulation of the DOT problem that uses an efficient interior-point-type optimization algorithm to find a DOT inverse solution in an admissible solution scenario. We present simulation results verifying the effectiveness of the approach.

Due to attenuation and spatial smoothing that occurs in the conducting media, the bioelectric inverse problem of estimating sources from remote measurements is ill-posed and solution requires regularization. Recent studies showed that employing Bayesian methods could help increase accuracy. The basic limitations are the availability of good a priori information about the solution, and the lack of a "good" error metric. In this paper, we employ Bayesian methods, and present the mathematical framework for incorporating additional information in the form of prior statistics, and extra measurements. We also use Bayesian error metrics to evaluate the reconstructions, and select prior models. We apply the methods to inverse electrocardiography problem. The results show that we can improve the reconstructions by including extra information, and Bayesian error metrics are useful in evaluating the results.

This paper deals with the problem of estimating the position of a user equipment operating in a wireless communication network. We present a new positioning method based on the angles of arrival (AOA) measured in several radio links between that user equipment and different base stations. The proposed AOA-based method leads us to a non-iterative closed-form solution of the positioning problem, and an statistical analysis of that solution is also included. The comparison between this method and the classical AOA-based positioning technique is discussed in terms of computational load, convergence of the solution and also in terms of the bias and variance of the position estimate.

Dynamic magnetic resonance imaging (MRI) refers to the acquisition of a sequence of MRI images to monitor temporal changes in tissue structure. We present a method for the estimation of dynamic MRI sequences based on two complimentary strategies: an adaptive framework for the estimation of the MRI images themselves, and an adaptive method to tailor the MRI system excitations for each data acquisition. We refer to this method as the doubly adaptive temporal update method (DATUM) for dynamic MRI. Analysis of the adaptive image estimate framework shows that calculating the optimal system excitations for each new image requires complete knowledge of the next image in the sequence. Since this is not realizable, we introduce a linear predictor to aid in determining appropriate excitations. Simulated examples using real MRI data are included to illustrate that the doubly adaptive strategy can provide estimates with lower steady state error than previously proposed methods and also the ability to recover from dramatic changes in the image sequence.

Motivated by work in the area of dynamic magnetic resonance imaging (MRI), we develop a new approach to the problem of reduced-order MRI acquisition. Efforts in this field have concentrated on the use of Fourier and singular value decomposition (SVD) methods to obtain low-order representations of an entire image plane. We augment this work to the case of imaging an arbitrarily-shaped region of interest (ROI) embedded within the full image. After developing a natural error metric for this problem, we show that determining the minimal order required to meet a prescribed error level is in general intractable, but can be solved under certain assumptions. We then develop an optimization approach to the related problem of minimizing the error for a given order. Finally, we demonstrate the utility of this approach and its advantages over existing Fourier and SVD methods on a number of MRI images.

This paper addresses the problem of designing an efficient and effective image sequence processing scheme that will successfully detect very small (point) targets in a cluttered background when both the target and clutter are moving through the image scene. The specific application area was detection of targets such as airplanes in infrared (IR) image sequences of a cloudy sky which have been taken by a stationary camera. In general we assume that targets are typically one to two pixels in extent and move only a fraction of a pixel per frame, are often low amplitude, and are found in scenes which also contain evolving clutter, e.g. clouds. Our algorithm is based on signal processing and detection theory, includes a perfect measurement performance analysis, and can be made computationally efficient compared to other approaches. Thus the algorithm could be applicable to other image sequence processing scenarios, using other acquisition systems besides IR, such as detection of small moving objects or structures in a biomedical or biological imaging scenario or the detection of satellites, meteors or other celestial bodies in night sky imagery acquired using a telescope. We present a GLRT solution, perfect measurement analysis including ROC curves, and results using real-world infrared data.

Restoring an original image from its blurred and noisy version is usually ill-posed. l/sub 2/ norm based regularized inverse solutions lead to restored images with the edges blurred. Some work has focused on using the l/sub 1/ norm of the gradient as a regularizer. Alternatively convex optimization approaches have been used to provide flexibility to impose multiple constraints on the solution. In this paper, we combine these methods using a convex optimization approach based on the ellipsoid algorithm to impose, among other constraints, an l/sub 1/ norm of the gradient constraint. This results in a restored image with good edge preservation capabilities.

Automatic segmentation of the electrocardiogram (ECG) is important in both clinical and research settings. Past algorithms have relied on incorporation of detailed heuristics. We avoid heuristics by employing a best-basis algorithm. As large variability of the local SNR causes the standard entropy criterion to produce an overly-fine segmentation, we introduce a novel optimality criterion which is based on a linear combination of the entropy measure and a function of a smoothness measure, and is quite general in form. We tested the algorithm on the MIT-BIH arrythmia database and body surface potential maps.

We describe an imaging modality whose goal is the display and interpretation of the electrical activity of the heart via measurements of electrical potentials made either through invasive procedures or non-invasively on the body surface. The goal of this imaging process is not to reconstruct anatomical geometry, but rather to display, analyze, and understand the normal and abnormal electrical activity of the heart in order to apply this knowledge to the diagnosis and monitoring of patients.>

The application of higher-order spectral methods to certain multichannel inverse problems where the observed time signals are linear combinations of unknown sources is proposed. The transfer matrix is static and known beforehand but is highly ill-conditioned, leading to the failure of standard least squares regularization techniques or low rank approximation when there is additive noise. The noise suppression properties of third order cumulants are used, along with the ability to obtain multiple estimates by extracting individual signals from their third order cross-correlations via cross-bicepstrum operations. Variations of this approach are presented, and normalization issues are discussed.>

Complex cepstrum techniques are applied to cross-bispectra in order to simultaneously reconstruct three independent finite-time signals, given the cross-bispectrum of the three signals only. The method is applied to simultaneously identify three systems via the cross-bispectrum of their outputs, knowing only the statistics of the input. A least-squares method is presented to estimate the cross-bicepstral parameters of the three signals simultaneously. The method is nonparametric and noniterative, assuming only that the signals (or impulse responses) are exponential, stable, and have no zeros on the unit circle. It takes full advantage of the two-dimensional nature of the bispectrum to reconstruct the three signals (or identify the three systems) simultaneously.>