Norm Minimization Algorithm Research Articles

An efficient matrix bi-factorization alternative optimization method for low-rank matrix recovery and completion

In recent years, matrix rank minimization problems have aroused considerable interests from machine learning, data mining and computer vision communities. All of these problems can be solved via their convex relaxations which minimize the trace norm instead of the rank of the matrix, and have to be solved iteratively and involve singular value decomposition (SVD) at each iteration. Therefore, those algorithms for trace norm minimization problems suffer from high computation cost of multiple SVDs. In this paper, we propose an efficient Matrix Bi-Factorization (MBF) method to approximate the original trace norm minimization problem and mitigate the computation cost of performing SVDs. The proposed MBF method can be used to address a wide range of low-rank matrix recovery and completion problems such as low-rank and sparse matrix decomposition (LRSD), low-rank representation (LRR) and low-rank matrix completion (MC). We also present three small scale matrix trace norm models for LRSD, LRR and MC problems, respectively. Moreover, we develop two concrete linearized proximal alternative optimization algorithms for solving the above three problems. Experimental results on a variety of synthetic and real-world data sets validate the efficiency, robustness and effectiveness of our MBF method comparing with the state-of-the-art trace norm minimization algorithms.

Fast seismic data recovery by combined minimum norm algorithm in DTCW domain

With the development of high-density sampling technology and high-precision seismology, the computational cost that dramatically increases with the sampling rate and resolution requirement becomes very challenging in seismic data recovery. A dual tree complex wavelet (DTCW)-based sparsity-preserving minimization model for seismic data recovery is studied in this paper. Different to other orthogonal transforms, the multi-dimensional DTCW transform (DTCWT) is shift invariant and direction sensitive, which ensures the sparse representation and stable recovery of seismic reflections. Compared with other over-complete transforms, DTCWT is superior in less redundancy and computation complexity. Based on these advantages, a modified split Bregman iterative algorithm in DTCW domain for solving the combined norm minimization model is proposed and its convergence is then established. As a crucial step of the algorithm, how to choose model parameters optimally for over-complete transforms is specifically discussed in this paper. We apply this method to recovery seismic data from severe noisy background. Numerical results show that the proposed DTCWT-based algorithm appears to give significant improvements over the fully decimated orthogonal wavelet transform (DWT) or non-decimated wavelet transform (DyDWT) based-ones both in SNR and visual quality of the results. Its memory and computation cost is much lower than DyDWT and discrete curvelet transform (DCurT)-based methods. Meanwhile, the artificial reflections in the results of the proposed method are much less than some other redundant transform-based ones. So, the DTCWT-based iteration algorithm is applicable in real seismic applications.

Dysfunctional and compensatory duality in mild cognitive impairment during a continuous recognition memory task

One of the current issues of debate in the study of mild cognitive impairment (MCI) is deviations of oscillatory brain responses from normal brain states and its dynamics. This work aims to characterize the differences of power in brain oscillations during the execution of a recognition memory task in MCI subjects in comparison with elderly controls. Magnetoencephalographic (MEG) signals were recorded during a continuous recognition memory task performance. Oscillatory brain activity during the recognition phase of the task was analyzed by wavelet transform in the source space by means of minimum norm algorithm. Both groups obtained a 77% hit ratio. In comparison with healthy controls, MCI subjects showed increased theta (p<0.001), lower beta reduction (p<0.001) and decreased alpha and gamma power (p<0.002 and p<0.001 respectively) in frontal, temporal and parietal areas during early and late latencies. Our results point towards a dual pattern of activity (increase and decrease) which is indicative of MCI and specific to certain time windows, frequency bands and brain regions. These results could represent two neurophysiological sides of MCI. Characterizing these opposing processes may contribute to the understanding of the disorder.

Harmony: EEG/MEG linear inverse source reconstruction in the anatomical basis of spherical harmonics.

EEG/MEG source localization based on a “distributed solution” is severely underdetermined, because the number of sources is much larger than the number of measurements. In particular, this makes the solution strongly affected by sensor noise. A new way to constrain the problem is presented. By using the anatomical basis of spherical harmonics (or spherical splines) instead of single dipoles the dimensionality of the inverse solution is greatly reduced without sacrificing the quality of the data fit. The smoothness of the resulting solution reduces the surface bias and scatter of the sources (incoherency) compared to the popular minimum-norm algorithms where single-dipole basis is used (MNE, depth-weighted MNE, dSPM, sLORETA, LORETA, IBF) and allows to efficiently reduce the effect of sensor noise. This approach, termed Harmony, performed well when applied to experimental data (two exemplars of early evoked potentials) and showed better localization precision and solution coherence than the other tested algorithms when applied to realistically simulated data.

Time-constrained functional connectivity analysis of cortical networks underlying phonological decoding in typically developing school-aged children: A magnetoencephalography study

The study investigated functional associations between left hemisphere occipitotemporal, temporoparietal, and inferior frontal regions during oral pseudoword reading in 58 school-aged children with typical reading skills (aged 10.4±1.6, range 7.5–12.5years). Event-related neuromagnetic data were used to compute source-current density waveforms using a minimum norm algorithm. Temporally-constrained contributions were established for four ROIs (STG, SMG, ANG, and IFG) by controlling for the autoregressive effects of activity in each ROI. Unique contributions made by activity in one ROI to subsequent activity in a second ROI were assessed through multiple regression analyses. Forward associations between lateral (LOC) and ventral occipitotemporal cortices (fusiform gyrus) to each of the four main ROIs were also examined. The earliest significant contributions to SMG and ANG activation (at 200–250ms) were made by preceding activity in the fusiform gyrus. The degree of activity in IFG appeared to be determined by earlier activity in ANG and STG.

A fast tri-factorization method for low-rank matrix recovery and completion

In recent years, matrix rank minimization problems have received a significant amount of attention in machine learning, data mining and computer vision communities. And these problems can be solved by a convex relaxation of the rank minimization problem which minimizes the nuclear norm instead of the rank of the matrix, and has to be solved iteratively and involves singular value decomposition (SVD) at each iteration. Therefore, those algorithms for nuclear norm minimization problems suffer from high computation cost of multiple SVDs. In this paper, we propose a Fast Tri-Factorization (FTF) method to approximate the nuclear norm minimization problem and mitigate the computation cost of performing SVDs. The proposed FTF method can be used to reliably solve a wide range of low-rank matrix recovery and completion problems such as robust principal component analysis (RPCA), low-rank representation (LRR) and low-rank matrix completion (MC). We also present three specific models for RPCA, LRR and MC problems, respectively. Moreover, we develop two alternating direction method (ADM) based iterative algorithms for solving the above three problems. Experimental results on a variety of synthetic and real-world data sets validate the efficiency, robustness and effectiveness of our FTF method comparing with the state-of-the-art nuclear norm minimization algorithms.

Improving resolution of miniature spectrometers by exploiting sparse nature of signals

In this paper, we present a signal processing approach to improve the resolution of a spectrometer with a fixed number of low-cost, non-ideal filters. We aim to show that the resolution can be improved beyond the limit set by the number of filters by exploiting the sparse nature of a signal spectrum. We consider an underdetermined system of linear equations as a model for signal spectrum estimation. We design a non-negative L1 norm minimization algorithm for solving the system of equations. We demonstrate that the resolution can be improved multiple times by using the proposed algorithm.

一种基于GTD 模型和状态空间处理的运动目标超宽带散射中心提取方法

Movements of the scattering center on the condition of ultra-wideband (UWB) will impact the extraction of parameters. This paper presents an approach for extracting the moving targets UWB scattering center parameters based on the geometrical theory of diffraction (GTD) model and the state space approach. Firstly, the UWB GTD scattering model is transformed into state-space equations in order to estimate the range and range rate through the singular value decomposition. Secondly, the type parameter in the whole bandwidth is in-dicated dimension-reducedly by orthonormalized basis and estimated by traversal algorithm and minimum-norm algorithm. Finally, the intensity of scattering center is worked out based on the least square method. The paper provides the Cramer-Rao bound (CRB) for parameter estimation. The effectiveness of the approach is verified by computer simulation. The approach can estimate range, range rate, reflection intensity and type parameter simultaneously in order to improve the track of scattering centers and the target identification.

A GTD model and state space approach based method for extracting the UWB scattering center of moving target

Movements of the scattering center on the condition of ultra-wideband (UWB) will impact the extraction of parameters. This paper presents an approach for extracting the moving targets’ UWB scattering center parameters based on the geometrical theory of diffraction (GTD) model and the state space approach. Firstly, the UWB GTD scattering model is transformed into state-space equations in order to estimate the range and range rate through the singular value decomposition. Secondly, the type parameter in the whole bandwidth is indicated dimension-reducedly by orthonormalized basis and estimated by traversal algorithm and minimumnorm algorithm. Finally, the intensity of scattering center is worked out based on the least square method. The paper provides the Cramer-Rao bound (CRB) for parameter estimation. The effectiveness of the approach is verified by computer simulation. The approach can estimate range, range rate, reflection intensity and type parameter simultaneously in order to improve the track of scattering centers and the target identification.

Multimodal functional imaging using fMRI-informed regional EEG/MEG source estimation

We propose a novel method, fMRI-Informed Regional Estimation (FIRE), which utilizes information from fMRI in E/MEG source reconstruction. FIRE takes advantage of the spatial alignment between the neural and the vascular activities, while allowing for substantial differences in their dynamics. Furthermore, with a region-based approach, FIRE estimates the model parameters for each region independently. Hence, it can be efficiently applied on a dense grid of source locations. The optimization procedure at the core of FIRE is related to the re-weighted minimum-norm algorithms. The weights in the proposed approach are computed from both the current source estimates and fMRI data, leading to robust estimates in the presence of silent sources in either fMRI or E/MEG measurements. We employ a Monte Carlo evaluation procedure to compare the proposed method to several other joint E/MEG-fMRI algorithms. Our results show that FIRE provides the best trade-off in estimation accuracy between the spatial and the temporal accuracy. Analysis using human E/MEG-fMRI data reveals that FIRE significantly reduces the ambiguities in source localization present in the minimum-norm estimates, and that it accurately captures activation timing in adjacent functional regions.

Multimodal Integration of fMRI and EEG Data for High Spatial and Temporal Resolution Analysis of Brain Networks

Two major non-invasive brain mapping techniques, electroencephalography (EEG) and functional magnetic resonance imaging (fMRI), have complementary advantages with regard to their spatial and temporal resolution. We propose an approach based on the integration of EEG and fMRI, enabling the EEG temporal dynamics of information processing to be characterized within spatially well-defined fMRI large-scale networks. First, the fMRI data are decomposed into networks by means of spatial independent component analysis (sICA), and those associated with intrinsic activity and/or responding to task performance are selected using information from the related time-courses. Next, the EEG data over all sensors are averaged with respect to event timing, thus calculating event-related potentials (ERPs). The ERPs are subjected to temporal ICA (tICA), and the resulting components are localized with the weighted minimum norm (WMNLS) algorithm using the task-related fMRI networks as priors. Finally, the temporal contribution of each ERP component in the areas belonging to the fMRI large-scale networks is estimated. The proposed approach has been evaluated on visual target detection data. Our results confirm that two different components, commonly observed in EEG when presenting novel and salient stimuli, respectively, are related to the neuronal activation in large-scale networks, operating at different latencies and associated with different functional processes.

Performance Evaluation of a Novel, Bayesian Source Reconstruction Technique for MEG

Event Abstract Back to Event Performance Evaluation of a Novel, Bayesian Source Reconstruction Technique for MEG Julia Owen1*, David Wipf1, Hagai Attias2, Kensuke Sekihara3 and Srikantan Nagarajan1 1 University of California, Biomagnetic Imaging Laboratory, United States 2 Golden Metalic Inc., United States 3 Tokyo Metropolitan University, Department of Systems Design and Engineering, Japan Magnetoencephalography (MEG) non-invasively detects brain activity from direct measurements of the magnetic field with an array of sensors. Determining the spatial distribution, orientation, and time courses of the underlying sources is an open inverse problem. We have developed a novel empirical Bayesian scheme that improves upon existing methods of source reconstruction in terms of reconstruction accuracy, robustness, and efficiency. The algorithm derived from this model, Champagne, is designed to estimate the number and location of a sparse set of flexible dipoles that adequately explain the observed sensor data. This method relies on having access to pre- and post-stimulus data, where the pre-stimulus data is thought to contain no stimulus-evoked brain activity. We have shown that Champagne reliably reconstructs a large number of correlated dipoles. We have extended our performance evaluations to include additional simulated data and real datasets. To evaluate Champagne’s performance of simulated data, with real brain noise, we have tested the algorithm with many configurations of sources to obtain aggregate performance metrics using signal detection theory. This was done for both deep sources, sources that are close together, and configurations where there are both deep and superficial sources. We compare the performance of our method to other widely-used methods such as beamforming and minimum-norm algorithms. We have also developed a statistical analysis routine that uses non-parametric permutation testing on real datasets. This method obtains a null distribution from actual brain data by switching the pre- and post-stimulus periods before averaging. With this technique, we can compare Champagne’s performance on real brain data to the performance of other popular methods listed above. Our performance evaluations on real and simulated data provide complimentary information. Conference: Biomag 2010 - 17th International Conference on Biomagnetism , Dubrovnik, Croatia, 28 Mar - 1 Apr, 2010. Presentation Type: Poster Presentation Topic: Signal proccessing Citation: Owen J, Wipf D, Attias H, Sekihara K and Nagarajan S (2010). Performance Evaluation of a Novel, Bayesian Source Reconstruction Technique for MEG. Front. Neurosci. Conference Abstract: Biomag 2010 - 17th International Conference on Biomagnetism . doi: 10.3389/conf.fnins.2010.06.00108 Copyright: The abstracts in this collection have not been subject to any Frontiers peer review or checks, and are not endorsed by Frontiers. They are made available through the Frontiers publishing platform as a service to conference organizers and presenters. The copyright in the individual abstracts is owned by the author of each abstract or his/her employer unless otherwise stated. Each abstract, as well as the collection of abstracts, are published under a Creative Commons CC-BY 4.0 (attribution) licence (https://creativecommons.org/licenses/by/4.0/) and may thus be reproduced, translated, adapted and be the subject of derivative works provided the authors and Frontiers are attributed. For Frontiers’ terms and conditions please see https://www.frontiersin.org/legal/terms-and-conditions. Received: 23 Mar 2010; Published Online: 23 Mar 2010. * Correspondence: Julia Owen, University of California, Biomagnetic Imaging Laboratory, San Francisco, United States, julia.owen@ucsf.edu Login Required This action requires you to be registered with Frontiers and logged in. To register or login click here. Abstract Info Abstract The Authors in Frontiers Julia Owen David Wipf Hagai Attias Kensuke Sekihara Srikantan Nagarajan Google Julia Owen David Wipf Hagai Attias Kensuke Sekihara Srikantan Nagarajan Google Scholar Julia Owen David Wipf Hagai Attias Kensuke Sekihara Srikantan Nagarajan PubMed Julia Owen David Wipf Hagai Attias Kensuke Sekihara Srikantan Nagarajan Related Article in Frontiers Google Scholar PubMed Abstract Close Back to top Javascript is disabled. Please enable Javascript in your browser settings in order to see all the content on this page.

Multimodal functional imaging using fMRI-informed regional EEG/MEG source estimation.

We propose a novel method, fMRI-Informed Regional Estimation (FIRE), which utilizes information from fMRI in E/MEG source reconstruction. FIRE takes advantage of the spatial alignment between the neural and the vascular activities, while allowing for substantial differences in their dynamics. Furthermore, with the regional approach, FIRE can be efficiently applied to a dense grid of sources. Inspection of our optimization procedure reveals that FIRE is related to the re-weighted minimum-norm algorithms, the difference being that the weights in the proposed approach are computed from both the current estimates and fMRI data. Analysis of both simulated and human fMRI-MEG data shows that FIRE reduces the ambiguities in source localization present in the minimum-norm estimates. Comparisons with several joint fMRI-E/MEG algorithms demonstrate robustness of FIRE in the presence of sources silent to either fMRI or E/MEG measurements.

Source reconstruction of brain electromagnetic fields — Source iteration of minimum norm (SIMN)

A recursive scheme aiming at obtaining sparse and focal brain electromagnetic source distribution is proposed based on the interpretation that the weighted minimum norm is the minimum norm estimates of amplitudes on grid points for the source distribution specified by the diagonal elements of the weight matrix. The source distribution is updated so that, at each grid point, the number of current dipoles equals the total source strength estimate of the pre-specified current dipoles. The source strength of a pre-specified current dipole is estimated by projecting the vector of minimum norm estimate to the space spanned by the three column vectors, corresponding to the three amplitudes of the current dipole, of the resolution matrix. The norm of the projected vector yields the source strength estimate of the current dipole. Exact inverse solutions are obtained by this source iteration of minimum norm (SIMN) algorithm for noiseless MEG signals from multi-point sources provided the sources are sufficiently sparse and there are no substantial cancellations among the signals of the sources. For noisy data, a set of “noise sources” is introduced. The diagonal matrix formed by the “noise source numbers” plays the role of regularization matrix and Tikhonov regularization is applied to initialize the “noise source numbers”. Application to the source localization of real EEG data is also presented.

DoA Estimation of Line of Sight Signal in Multipath Channel for GNSS Receiver

The GNSS receivers suffer from the multipath interference which is highly correlated with the line of sight (LoS) signal. Such interference results in tracking and ranging errors. In this paper, we propose a novel algorithm that can estimate the direction of arrival (DoA) of the LoS signal in the presence of highly correlated multipath interference. The proposed algorithm combines the matching pursuits algorithm for multipath estimation and the minimum norm algorithm for DoA estimation. An efficient combination of two algorithms yields reliable estimates of the DoA of LoS signal as demonstrated by computer simulations.

Random projections based feature-specific structured imaging

We present a feature-specific imaging system based on the use of structured illumination. The measurements are defined as inner products between the illumination patterns and the object reflectance function, measured on a single photodetector. The illumination patterns are defined using random binary patterns and thus do not employ prior knowledge about the object. Object estimates are generated using L(1)-norm minimization and gradient-projection sparse reconstruction algorithms. The experimental reconstructions show the feasibility of the proposed approach by using 42% fewer measurements than the object dimensionality.

Working memory processing of natural environmental scenes in the human brain revealed by MEG

Abstract Results of the previous studies on working memory processing of natural scenes in the human brain using fMRI have shown activation of a distributed network involving mainly medial temporal lobe and prefrontal sources. In spite of its relative high spatial accuracy in localizing the involved areas, possible interactions between the localized source activities in time are played in a finer temporal scale. In this study we used multichannel whole-head magnetoencephalography (MEG) which possesses both a relative good spatial and temporal resolution, to shed light on the possible mechanisms of interactions between the involved sources in the brain, after localizing them properly. 12 healthy human subjects were presented with a small set of natural scenes from the seaside. The memory task was a continuous 1-back comparison task, and the control was an automatic visuomotor response task involving the same sequence of trials (the order of task performance was random). For source estimation a weighted L2 minimum-norm algorithm was used. Focusing in a time interval between 300–600 ms when the impact of deep encoding of the presented stimuli during the 1-back task performance is more prominent and comparable, the interaction between the localized occipital, inferior medial temporal, posterior parietal, and prefrontal sources, was found to be important for deep (elaborative) encoding of these stimuli in working memory.

Estimation of multiple phases in holographic moiré in presence of harmonics and noise using minimum-norm algorithm

The paper proposes a novel approach for estimating multiple phases in holographic moiré. The need to design such an algorithm is necessitated by the development of optical configurations containing two phase stepping devices, e.g. PZTs, with a view to measure simultaneously two phase distributions. The approach consists of first applying minimum-norm algorithm to extract phase steps imparted to the PZTs. Salient feature of the algorithm lies in its ability to handle nonsinusoidal waveforms and noise. This approach also provides the flexibility of using arbitrary phase steps, a feature most commonly attributed to generalized phase shifting interferometry. Once the phase steps are estimated for each PZT, the Vandermonde system of equations is designed to estimate the phase distributions.

An equivalent current source model and laplacian weighted minimum norm current estimates of brain electrical activity.

We have developed a method for estimating the three-dimensional distribution of equivalent current sources inside the brain from scalp potentials. Laplacian weighted minimum norm algorithm has been used in the present study to estimate the inverse solutions. A three-concentric-sphere inhomogeneous head model was used to represent the head volume conductor. A closed-form solution of the electrical potential over the scalp and inside the brain due to a point current source was developed for the three-concentric-sphere inhomogeneous head model. Computer simulation studies were conducted to validate the proposed equivalent current source imaging. Assuming source configurations as either multiple dipoles or point current sources/sinks, in computer simulations we used our method to reconstruct these sources, and compared with the equivalent dipole source imaging. Human experimental studies were also conducted and the equivalent current source imaging was performed on the visual evoked potential data. These results highlight the advantages of the equivalent current source imaging and suggest that it may become an alternative approach to imaging spatially distributed current sources-sinks in the brain and other organ systems.

Imaging and visualization of 3-D cardiac electric activity.

Noninvasive imaging of cardiac electric activity is of importance for better understanding the underlying mechanisms and for aiding clinical diagnosis and intervention of cardiac abnormalities. We propose to image the three-dimensional (3-D) cardiac bioelectric source distribution from body-surface electrocardiograms. Cardiac electrical sources were modeled by a current dipole distribution throughout the entire myocardium, and estimated by using the Laplacian weighted minimum norm (LWMN) algorithm from body-surface potentials. The estimated inverse solution of the current distribution was further improved by using a recursive weighting strategy for localized sources, such as origins of cardiac arrhythmias. Computer simulations were conducted to test the feasibility of the proposed approach by using a 3-D ventricle model embedded in a realistically shaped torso model. The boundary element method was used to solve the forward problem from assumed cardiac sources to the body-surface potentials. Two testing dipoles were placed in the left and right ventricles, simulating the early activation associated with ventricular arrhythmias. The LWMN inverse solution showed an equivalent source distribution over the entity of both ventricles, with spread areas of activity overlying the positions of the testing dipoles. The sharpened inverse image provides well-localized focal sources near the testing dipole positions. In summary, the present computer simulation suggests that the proposed 3-D cardiac current source imaging and localization approach appears to be a promising candidate for localizing and imaging sites of origins of cardiac activation.