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18
Sparse signal reconstruction from limited data using FOCUSS: A reweighted minimum norm algorithm
 IEEE Trans. Signal Processing
, 1997
"... Abstract—We present a nonparametric algorithm for finding localized energy solutions from limited data. The problem we address is underdetermined, and no prior knowledge of the shape of the region on which the solution is nonzero is assumed. Termed the FOcal Underdetermined System Solver (FOCUSS), t ..."
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Cited by 218 (12 self)
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Abstract—We present a nonparametric algorithm for finding localized energy solutions from limited data. The problem we address is underdetermined, and no prior knowledge of the shape of the region on which the solution is nonzero is assumed. Termed the FOcal Underdetermined System Solver (FOCUSS), the algorithm has two integral parts: a lowresolution initial estimate of the real signal and the iteration process that refines the initial estimate to the final localized energy solution. The iterations are based on weighted norm minimization of the dependent variable with the weights being a function of the preceding iterative solutions. The algorithm is presented as a general estimation tool usable across different applications. A detailed analysis laying the theoretical foundation for the algorithm is given and includes proofs of global and local convergence and a derivation of the rate of convergence. A view of the algorithm as a novel optimization method which combines desirable characteristics of both classical optimization and learningbased algorithms is provided. Mathematical results on conditions for uniqueness of sparse solutions are also given. Applications of the algorithm are illustrated on problems in directionofarrival (DOA) estimation and neuromagnetic imaging. I.
An empirical Bayesian solution to the source reconstruction problem in EEG
 NEUROIMAGE
, 2005
"... Distributed linear solutions of the EEG source localisation problem are used routinely. In contrast to discrete dipole equivalent models, distributed linear solutions do not assume a fixed number of active sources and rest on a discretised fully 3D representation of the electrical activity of the br ..."
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Cited by 29 (9 self)
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Distributed linear solutions of the EEG source localisation problem are used routinely. In contrast to discrete dipole equivalent models, distributed linear solutions do not assume a fixed number of active sources and rest on a discretised fully 3D representation of the electrical activity of the brain. The ensuing inverse problem is underdetermined and constraints or priors are required to ensure the uniqueness of the solution. In a Bayesian framework, the conditional expectation of the source distribution, given the data, is attained by carefully balancing the minimisation of the residuals induced by noise and the improbability of the estimates as determined by their priors. This balance is specified by hyperparameters that control the relative importance of fitting and conforming to various constraints. Here we formulate the conventional bWeighted Minimum NormQ (WMN) solution in terms of hierarchical linear models. An "ExpectationMaximisation" (EM) algorithm is used to obtain a
Comparison of Basis Selection Methods
, 1996
"... In this paper, we describe and evaluate three forward sequential basis selection methods: Basic Matching Pursuit (BMP), Order Recursive Matching Pursuit (ORMP) and Modified Matching Pursuit (MMP), and a parallel basis selection method: the FOCal Underdetermined System Solver (FOCUSS) algorithm. Com ..."
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Cited by 16 (3 self)
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In this paper, we describe and evaluate three forward sequential basis selection methods: Basic Matching Pursuit (BMP), Order Recursive Matching Pursuit (ORMP) and Modified Matching Pursuit (MMP), and a parallel basis selection method: the FOCal Underdetermined System Solver (FOCUSS) algorithm. Computer simulations show that the ORMP method is superior to the BMP method in terms of its ability to select a compact basis set. However, it is computationally more complex. The MMP algorithm is developed which is of intermediate computational complexity and has performance comparable to the ORMP method. All the sequential selection methods are shown to have difficulty in environments where the basis set contains highly correlated vectors. The drawback can be traced to the sequential nature of these methods suggesting the need for a parallel basis selection method like FOCUSS. Simulations demonstrate that the FOCUSS algorithm does indeed perform well in such correlated environments. However,...
Systematic regularization of linear inverse solutions of the EEG source localization problem
 NeuroImage
"... Distributed linear solutions of the EEG source localization problem are used routinely. Here we describe an approach based on the weighted minimum norm method that imposes constraints using anatomical and physiological information derived from other imaging modalities to regularize the solution. In ..."
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Cited by 12 (2 self)
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Distributed linear solutions of the EEG source localization problem are used routinely. Here we describe an approach based on the weighted minimum norm method that imposes constraints using anatomical and physiological information derived from other imaging modalities to regularize the solution. In this approach the hyperparameters controlling the degree of regularization are estimated using restricted maximum likelihood (ReML). EEG data are always contaminated by noise, e.g., exogenous noise and background brain activity. The conditional expectation of the source distribution, given the data, is attained by carefully balancing the minimization of the residuals induced by noise and the improbability of the estimates as determined by their priors.
A general approach to sparse basis selection: Majorization, concavity, and affine scaling
 IN PROCEEDINGS OF THE TWELFTH ANNUAL CONFERENCE ON COMPUTATIONAL LEARNING THEORY
, 1997
"... Measures for sparse best–basis selection are analyzed and shown to fit into a general framework based on majorization, Schurconcavity, and concavity. This framework facilitates the analysis of algorithm performance and clarifies the relationships between existing proposed concentration measures use ..."
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Cited by 6 (3 self)
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Measures for sparse best–basis selection are analyzed and shown to fit into a general framework based on majorization, Schurconcavity, and concavity. This framework facilitates the analysis of algorithm performance and clarifies the relationships between existing proposed concentration measures useful for sparse basis selection. It also allows one to define new concentration measures, and several general classes of measures are proposed and analyzed in this paper. Admissible measures are given by the Schurconcave functions, which are the class of functions consistent with the socalled Lorentz ordering (a partial ordering on vectors also known as majorization). In particular, concave functions form an important subclass of the Schurconcave functions which attain their minima at sparse solutions to the best basis selection problem. A general affine scaling optimization algorithm obtained from a special factorization of the gradient function is developed and proved to converge to a sparse solution for measures chosen from within this subclass.
Validation of SOBI components from highdensity EEG
, 2005
"... Secondorder blind identification (SOBI) is a blind source separation (BSS) algorithm that can be used to decompose mixtures of signals into a set of components or putative recovered sources. Previously, SOBI, as well as other BSS algorithms, has been applied to magnetoencephalography (MEG) and elec ..."
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Cited by 4 (0 self)
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Secondorder blind identification (SOBI) is a blind source separation (BSS) algorithm that can be used to decompose mixtures of signals into a set of components or putative recovered sources. Previously, SOBI, as well as other BSS algorithms, has been applied to magnetoencephalography (MEG) and electroencephalography (EEG) data. These BSS algorithms have been shown to recover components that appear to be physiologically and neuroanatomically interpretable. While some proponents of these algorithms suggest that fundamental discoveries about the human brain might be made through the application of these techniques, validation of BSS components has not yet received sufficient attention. Here we present two experiments for validating SOBIrecovered components. The first takes advantage of the fact that noise sources associated with individual sensors can be objectively validated independently from the SOBI process. The second utilizes the fact that the time course and location of primary somatosensory (SI) cortex activation by median nerve stimulation have been extensively characterized using converging imaging methods. In this paper, using both known noise sources and highly constrained and wellcharacterized neuronal sources, we provide validation for SOBI decomposition of highdensity EEG data. We show that SOBI is able to (1) recover known noise sources that were either spontaneously occurring or artificially induced; (2) recover neuronal sources activated by median nerve stimulation that were spatially and temporally consistent with estimates obtained from previous EEG, MEG, and fMRI studies; (3) improve the signaltonoise ratio (SNR) of somatosensoryevoked potentials (SEPs); and (4) reduce the level of subjectivity involved in the source localization process.
Mathematical analysis of lead field expansions
 IEEE Transactions on Medical Imaging
, 1999
"... Abstract—The solution to the bioelectromagnetic inverse problem is discussed in terms of a generalized lead field expansion, extended to weights depending polynomially on the current strength. The expansion coefficients are obtained from the resulting system of equations which relate the lead field ..."
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Cited by 4 (0 self)
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Abstract—The solution to the bioelectromagnetic inverse problem is discussed in terms of a generalized lead field expansion, extended to weights depending polynomially on the current strength. The expansion coefficients are obtained from the resulting system of equations which relate the lead field expansion to the data. The framework supports a family of algorithms which include the class of minimum norm solutions and those of weighted minimum norm, including FOCUSS (suitably modified to conform to requirements of rotational invariance). The weightedminimumnorm family is discussed in some detail, making explicit the dependence (or independence) of the weighting scheme on the modulus of the unknown current density vector. For all but the linear case, and with a single power in the weight, a highly nonlinear system of equations results. These are analyzed and their solution reduced to tractable problems for a finite number of degrees of freedom. In the simplest magnetic field tomography (MFT) case, this is shown to possess expected properties for localized distributed sources. A sensitivity analysis supports this conclusion. Index Terms—Biomagnetic inverse problem, magnetic field tomography, magnetoencephalography. I.
By Combining EEG and MEG with MRI Cortical Surface Reconstruction
"... We describe a comprehensive linear approach to the problem of imaging brain activity with high temporal as well as spatial resolution based on combining EEG and MEG data with anatomical constraints derived from MRI images. The “inverse problem ” of estimating the distribution of dipole strengths ove ..."
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Cited by 1 (1 self)
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We describe a comprehensive linear approach to the problem of imaging brain activity with high temporal as well as spatial resolution based on combining EEG and MEG data with anatomical constraints derived from MRI images. The “inverse problem ” of estimating the distribution of dipole strengths over the cortical surface is highly underdetermined, even given closely spaced EEG and MEG recordings. We have obtained much better solutions to this problem by explicitly incorporating both local cortical orientation as well as spatial covariance of sources and sensors into our formulation. An explicit polygonal model of the cortical manifold is first constructed as follows: (1) slice data in three orthogonal planes of section (needleshaped voxels) are combined with a linear deblurring technique to make a single highresolution 3D image (cubic voxels), (2) the image is recursively floodfilled to determine the topology of the graywhite matter border, and (3) the resulting continuous surface is refined by relaxing it against the original 3D grayscale image using a deformable template method, which is also used to computationally flatten the cortex for easier viewing. The explicit solution to an error minimization formulation of an optimal inverse linear operator (for a particular cortical manifold, sensor
Mixed EEG/MEG imaging: a way forward
"... this paper I will use variations of the probabilistic algorithm derived by Clark et al. [1, 3, 9]. One of the main disadvantages in working with EEG data is that anyaccurate model of the brain needs to know the conductivity profile of the head to a high degree of accuracy which is not so with MEG [8 ..."
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this paper I will use variations of the probabilistic algorithm derived by Clark et al. [1, 3, 9]. One of the main disadvantages in working with EEG data is that anyaccurate model of the brain needs to know the conductivity profile of the head to a high degree of accuracy which is not so with MEG [8, 19]. This is abig problem since nobody knows the conductivity profile of the head to sufficient accuracy. This does not preclude a study into the possible advantages of such an approach as long as the signal data is generated by a computer rather than measured in reality. In this paper I will assume that an accurate model of the conductivities of the head will eventually be available and when this occurs the results from this study should remain valid. Having said that a totally realistic model is impossible, it remains to choose an appropriate model for the head which (hopefully) still retains the essential properties of a realistic model. It has been suggested [8] that a simple conducting sphere model [19] is adequate for MEG signals from superficial sources from the visual and auditory cortices which seems to be the case from out analyses of real data [10, 12]. This is 1 the model which I will use for computing magnetic fields. It is known however that this is not the case for electric potentials# so I decided to use a layered anisotropic concentric sphere model [14,15]. The notation that I will use for both the MEG and EEG forward problems is as in figure 1 where r is the position of the measurement pointandr 0 is the position of the source dipole, of dipole moment M . There are N spheres of radii
MEG study of early cortical plasticity following multiple digit frequency discrimination training in humans
 CDHMMS for Maximum Relative Separation Margin,” Proc. ICASSP
, 2005
"... this paper, we will discuss results from Map I/II data. The ICAcleaned MEG signal was separated into trials of500 to 1500 ms relative to the stimulus onset. Average for each digit was constructed separately in each of the 5 runs. Magnetic field tomography (MFT) [8] was applied to the averaged data ..."
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this paper, we will discuss results from Map I/II data. The ICAcleaned MEG signal was separated into trials of500 to 1500 ms relative to the stimulus onset. Average for each digit was constructed separately in each of the 5 runs. Magnetic field tomography (MFT) [8] was applied to the averaged data to compute the 3D distribution of activity throughout the brain at each timeslice independently. The resulting brain images were superimposed on each subject's MRIs. Objective statistical measure KolmogorovSmirnov (KS) test was used to identify brain areas that showed significant changes between the post and prestimulus period by calculating the p values at 4913 voxels at each timeslice and selecting the areas with p<0.005. Cluster analysis was further applied to separate these areas into subareas with similar current flow direction. Regions of interest (ROI) were defined from these subareas as spherical regions with radius of 0.71.7 cm and labeled based on their anatomical locations. Activation curves for the modulus of the current density and along the main direction in these ROIs were calculated as a function of time