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MV-PURE Spatial Filters with Application to EEG/MEG Source Reconstruction

Tomasz Piotrowski, Jan Nikadon, David Gutierrez

Abstract

In this paper we propose spatial filters for a linear regression model which are based on the minimum-variance pseudo-unbiased reduced-rank estimation (MV-PURE) framework. As a sample application, we consider the problem of reconstruction of brain activity from electroencephalographic (EEG) or magnetoencephalographic (MEG) measurements. The proposed filters come in two versions depending on whether or not the EEG/MEG forward model explicitly considers interfering activity in the way of brain activity originating in regions different to those of main interest, but measured as correlated with signals of interest by the EEG/MEG sensor array. In both cases, the proposed filters are equipped with a rank-selection criterion minimizing the mean-square error (MSE) of the filter output. Therefore, we consider them as novel nontrivial generalizations of well-known linearly constrained minimum variance (LCMV) and nulling filters. In order to facilitate reproducibility of our research, we provide (jointly with this paper) comprehensive simulation framework that allows for estimation of error of signal reconstruction for a number of spatial filters applied to MEG or EEG signals. Based on this framework, chief properties of proposed filters are verified in a set of detailed simulations.

MV-PURE Spatial Filters with Application to EEG/MEG Source Reconstruction

Abstract

In this paper we propose spatial filters for a linear regression model which are based on the minimum-variance pseudo-unbiased reduced-rank estimation (MV-PURE) framework. As a sample application, we consider the problem of reconstruction of brain activity from electroencephalographic (EEG) or magnetoencephalographic (MEG) measurements. The proposed filters come in two versions depending on whether or not the EEG/MEG forward model explicitly considers interfering activity in the way of brain activity originating in regions different to those of main interest, but measured as correlated with signals of interest by the EEG/MEG sensor array. In both cases, the proposed filters are equipped with a rank-selection criterion minimizing the mean-square error (MSE) of the filter output. Therefore, we consider them as novel nontrivial generalizations of well-known linearly constrained minimum variance (LCMV) and nulling filters. In order to facilitate reproducibility of our research, we provide (jointly with this paper) comprehensive simulation framework that allows for estimation of error of signal reconstruction for a number of spatial filters applied to MEG or EEG signals. Based on this framework, chief properties of proposed filters are verified in a set of detailed simulations.

Paper Structure

This paper contains 38 sections, 5 theorems, 65 equations, 1 figure, 6 tables.

Table of Contents

  1. Introduction
  2. Notation
  3. EEG/MEG Measurement Model
  4. Interference-Free Measurement Model
  5. Spatial Filtering Under MSE Criterion
  6. LCMV Filter
  7. Conditioning of the Interference-Free Measurement Model
  8. Measurement Model in the Presence of Interference
  9. Spatial Filtering in the Presence of Interference
  10. Nulling Filter
  11. Conditioning of the Measurement Model in the Presence of Interference
  12. Challenging the Ill-Conditioning of the Measurement Model
  13. Proposed Spatial Filters in Presence of Interference
  14. Optimization Problems
  15. Closed Algebraic Forms
  16. ...and 23 more sections

Key Result

Theorem 1

For a given $r$ such that $1\leq r\leq l$, the solution to optimization problem (rr_nulling) is given by where Its corresponding MSE is given by where $c=tr(\bm{Q}).$

Figures (1)

  • Figure 1: Example of simulation setup showing ROIs and vertices representing sources of interest $\bm{q}$ (black), interference $\bm{q}_I$ (red), and background activity $\bm{q}_b$ (blue). Depending on the simulation setup, ROIs and vertices can be fixed or randomly selected. Perturbation of source location and direction can also be introduced. The perturbed sources are indicated using dashed line.

Theorems & Definitions (8)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Remark 1
  • Theorem 4
  • Theorem 5
  • Remark 2
  • Remark 3