MEG beamforming using Bayesian PCA for adaptive data covariance matrix regularization

doi:10.1016/j.neuroimage.2011.04.041

NeuroImage

Volume 57, Issue 4, 15 August 2011, Pages 1466-1479

https://doi.org/10.1016/j.neuroimage.2011.04.041 Get rights and content

Abstract

Beamformers are a commonly used method for doing source localization from magnetoencephalography (MEG) data. A key ingredient in a beamformer is the estimation of the data covariance matrix. When the noise levels are high, or when there is only a small amount of data available, the data covariance matrix is estimated poorly and the signal-to-noise ratio (SNR) of the beamformer output degrades. One solution to this is to use regularization whereby the diagonal of the covariance matrix is amplified by a pre-specified amount. However, this provides improvements at the expense of a loss in spatial resolution, and the parameter controlling the amount of regularization must be chosen subjectively. In this paper, we introduce a method that provides an adaptive solution to this problem by using a Bayesian Principle Component Analysis (PCA). This provides an estimate of the data covariance matrix to give a data-driven, non-arbitrary solution to the trade-off between the spatial resolution and the SNR of the beamformer output. This also provides a method for determining when the quality of the data covariance estimate maybe under question. We apply the approach to simulated and real MEG data, and demonstrate the way in which it can automatically adapt the regularization to give good performance over a range of noise and signal levels.

Research highlights

► New beamforming method that adapts to the information available using Bayesian PCA ► Provides a non-arbitrary trade‐off between spatial resolution and accuracy ► Automatically adapts to give good performance over a range of noise and signal levels. ► Improves spatial localization of high temporal resolution information from MEG data

Introduction

The MEG inverse problem involves the estimation of current distributions inside the head that give rise to the magnetic fields measured outside. Beamforming is a commonly used method for solving this problem (Veen et al., 1997, Vrba and Robinson, 2000), and corresponds to using an adaptive spatial filter that is designed to extract the origins of a signal from some pre-specified spatial location. A beamformer can be scanned over the whole brain to create a three-dimensional image showing areas of localized brain activity.

The spatial filter that represents the beamformer for a given location is a set of weights that is to be applied to the MEG sensor data. These weights are determined from knowing the forward model, i.e. the lead field matrix, and an estimate of the MEG data covariance. The accuracy of the MEG data covariance matrix estimation is therefore crucial, and it must be estimated from temporal windows of finite length (Barnes and Hillebrand, 2003). However, it can often be beneficial to focus the beamformer on a specific time period and frequency band (Dalal et al., 2009, Fawcett et al., 2004). This introduces a trade-off between the need for large integration windows to give good data covariance estimation, and the desire to focus on a specific time-frequency window. It is therefore possible that we can be operating in regimes where the quality of the data covariance estimate may be under question.

When the amount of data used to estimate the data covariance matrix is unavoidably low, e.g. due to focused time-frequency windows, and/or relatively few trials of data, one solution is to use regularization. Regularization amplifies the diagonal of the covariance matrix by a pre-specified amount, and is sometimes referred to as diagonal loading (Vrba and Robinson, 2000). Although providing some robustness against erroneous covariance estimates, and hence improving the estimate of electrical activity at a specific location, this regularization removes the spatial selectivity of the beamformer, in the limit tending to that of a dipole fit (Hillebrand and Barnes, 2003). This lack of spatial selectivity gives rise to poor immunity to environmental noise (Litvak et al., 2010, Adjamian et al., 2009) and poorer spatial resolution of beamformer images (Brookes et al., 2008). As a result the amount of regularization can have a large influence on the final beamformer image, the choice to date has been largely subjective; two popular choices being to use the lowest eigenvalue of the covariance matrix (Robinson and Vrba, 1999), or simply to use zero (Barnes et al., 2004).

In this paper, we propose a method that provides an objective solution to the problem of how to choose the amount of regularization that is required. This uses a Bayesian Principle Component Analysis (PCA) of the data to determine the size of the subspace (dimensionality) of the data that can be reliably estimated given the amount of evidence available in the data (Bishop, 1999). The estimate of this dimensionality acts as a surrogate for estimating the amount of regularization of the data that is required to give a good estimation of the data covariance matrix. Probabilistic Bayesian PCA provides an estimate of the data covariance matrix to give a data-driven, non-arbitrary solution to the trade-off between the spatial resolution and the SNR of the beamformer output. The estimate of dimensionality itself can also be used as an indicator of when there is insufficient information to reliably estimate the data covariance.

Section snippets

LCMV beamforming

The N × T matrix of MEG signals, y, recorded at the N MEG sensors over T time points is modeled as $y = \sum_{i = 1}^{L} H (r_{i}) m (r_{i}) + e$ where H(r_i) is the N × 3 lead field matrix and m(r_i) is the 3 × T vector timecourse for a dipole at location r_i, i = 1 … L, and e ~ (0, C_e) is the noise with C_e = σ_e²I. Using this forward model we can use a beamformer to optimize the 3 × N spatial filter, W(r_i), that estimates the dipole timecourse at location r_i from the sensor data as $\hat{m} (r_{i}) = W (r_{i}) y$

Here we use a linearly constrained minimum variance

Methods

The simulated data were prepared in Matlab with the same positioning of the head with respect to the MEG scanner (and the same subsequent forward model) as a real MEG dataset acquired using the 306-channel whole head Elekta-Neuromag MEG system as described in Methods in Real MEG data with a sampling rate of 200 Hz. We consider two different simulations. First, a low dimensional setup with just three dipolar sources located in the occipital cortex as shown in Fig. 1(a). Second, a more realistic

Data acquisition

The recordings were performed on a single subject using a 306 channel Elekta Neuromag system comprising 102 magnetometers and 204 planar gradiometers. MEG data were recorded at a sampling rate of 1000 Hz with a 0.1 Hz high pass filter. Before acquisition of the MEG data, a three-dimensional digitizer (Polhemus Fastrack) was used to record the patient's head shape relative to the position of the headcoils, with respect to three anatomical landmarks which could be registered on the MRI scan (the

Discussion

We have proposed an adaptive solution to the problem of having insufficient data to reliably estimate the data covariance matrix in MEG beamforming. Bayesian PCA is used to provide a data-driven estimate of the data covariance matrix that automatically trades-off between the spatial resolution and the SNR of the beamformer output. The output from the Bayesian PCA can also be used to calculate the implied dimensionality of the data, i.e. the number of effectively retained components. This can

Acknowledgments

Funding for Mark Woolrich is from the UK EPSRC and the Wellcome Trust. Thanks to Susie Murphy for the use of her dataset.

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Citation Excerpt :
In the first one, one wants to know what signal would be generated on the head if the dipole information is given. In the second problem, the signal is given, and one is interested in knowing the dipole information by which such a signal has been generated [1–11]. In this paper, we focus on the second problem.
Data with finite samples results in accuracy and robustness reduction of data covariance matrix estimation, which in turn results in performance reduction of minimum variance beamformer (MVB) for brain source localization (BSL). General linear combination (GLC) and convex combination (CC) are methods of interest for data covariance matrix estimation and increasing its accuracy and robustness because their scalar coefficients are obtained automatically and adaptively. However, based on our best knowledge, the applicability of GLC and CC algorithms has not been investigated for BSL to inform us about their performance. In this paper, we have two goals: 1) Investigation of GLC and CC covariance matrices applications for BSL is carried out using various simulated MEG scenarios and real MEG and clinical epilepsy data; 2) Modified GLC and CC are developed for more accurate and robust estimation of data covariance matrix when data with finite samples is available. In the modified versions, the scalar coefficients are replaced by diagonal matrix form coefficients. These matrix form coefficients are computed using the Hadamard product and mean square error concept. The evaluations show that the CC and modified CC based MVBs are not robust for BSL due to very small values of coefficients. Based on the simulated, real, and clinical data results, it can be stated that the modified GLC is significantly superior to conventional GLC in terms of localization error, spatial resolution (all z < -2; all p-values < 0.001), and offering reliable results. Also, the proposed GLC offers fewer missed sources and less sensitivity to the depth biasing problem, particularly in a high signal-to-noise ratio. In conclusion, it can be said that the covariance matrix of modified GLC which is user-free and robust against the finite data samples can improve the MVB performance for BSL in terms of localization error and spatial resolution.

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MEG beamforming using Bayesian PCA for adaptive data covariance matrix regularization

Abstract

Research highlights

Introduction

Section snippets

LCMV beamforming

Methods

Data acquisition

Discussion

Acknowledgments

J. Neurosci. Methods

Neuroimage

Neuroimage

Neuroimage

Neuroimage

Neuroimage

Neuroimage

Neuroimage

Neuroimage

Neuroimage

Neuroimage

Neuroimage

Statistical flattening of meg beamformer images

Hum. Brain Mapp.

Realistic spatial sampling for meg beamformer images

Hum. Brain Mapp.