Adaptive Spatial Filters for Electromagnetic Brain Imaging (eBook)

Contents 7
Introduction 12
1.1 Functional brain mapping 12
1.2 Electromagnetic brain imaging 13
1.3 Spatial .lters 14
1.4 Book chapter organization 16
1.5 Acknowledgements 18
Sensor array outputs and spatial . lters 20
2.1 Neuromagnetic signals as sensor-array outputs 20
2.1.1 De.nitions 20
2.1.2 Sensor lead field 21
2.1.3 Linear independence of lead-field vectors 22
2.2 Bioelectromagnetic inverse problem 24
2.3 Expressions of data covariance matrices 26
2.3.1 Data and source covariance relationship 26
2.3.2 Formulation for uncorrelated sources 28
2.4 Low-rank signal modeling 29
2.4.1 Definition of noise and signal subspaces 29
2.4.2 Property of the data covariance matrix 30
2.5 Spatial filters 33
2.5.1 Source reconstruction using a spatial filter 33
2.5.2 Scalar and vector spatial filters 34
2.5.3 Resolution kernel, point-spread function, and beam response 36
Tomographic reconstruction and nonadaptive spatial filters 38
3.1 Minimum-norm method 38
3.1.1 Tomographic reconstruction formulation 38
3.1.2 Nonadaptive spatial-filter formulation 42
3.2 Variants of the minimum-norm filter 43
3.2.1 Weight-normalized minimum-norm filter 43
3.2.2 sLORETA filter 43
3.3 Spatial matched filter 45
3.4 Deriving the minimum-norm-based filters using leakage minimization 46
Adaptive spatial filters 48
4.1 Deriving weights for adaptive spatial filters 48
4.1.1 Minimum-variance spatial filter with the unit-gain constraint 48
4.1.2 Minimum-variance spatial filter with the array-gain constraint 50
4.1.3 Minimum-variance spatial filter with the unit-noisegain constraint 50
4.2 Prerequisites for the adaptive spatial-filter formulation 51
4.2.1 Uncorrelated source time courses 51
4.2.2 Low-rank signals 54
4.3 Scalar adaptive spatial filter: deriving the optimum source orientation 55
4.4 LCMV spatial filter 57
4.5 Vector adaptive spatial filter formulation 59
4.5.1 Unit-gain constraint spatial filter 59
4.5.2 Array-gain constraint spatial filter 60
4.5.3 Unit-noise-gain constraint spatial filter 62
4.5.4 Equivalence between the adaptive scalar and vector formulations 64
4.6 Frequency-domain implementation 65
4.7 Numerical examples 68
Location bias, spatial resolution, and beam response 76
5.1 Bias properties of various spatial filters 76
5.1.1 Definition of source location bias 76
5.1.2 Bias for the spatial matched filter 77
5.1.3 Bias for the minimum-norm filter 78
5.1.4 Bias for the weight-normalized minimum-norm filter 78
5.1.5 Bias for the sLORETA filter 79
5.1.6 Bias for the unit-gain minimum-variance spatial filter 79
5.1.7 Bias for the array-gain minimum-variance spatial filter 80
5.1.8 Bias for the unit-noise-gain minimum-variance spatial filter 80
5.2 Effects of noise on the location bias 81
5.3 Spatial resolution 82
5.4 Spatial-filter beam response 83
5.5 Numerical examples 85
Output SNR and array mismatch 94
6.1 Output SINR 94
6.2 Adaptive spatial filters that attain the maximum SINR 96
6.3 SNR transfer factor 98
6.4 Two types of SNR de.nitions for the vector minimum- variance spatial filter 100
6.5 Influence of array mismatch 103
6.6 Diagonal loading 104
6.7 Asymmetric diagonal loading 106
6.8 Eigenspace-projection spatial filter 108
6.8.1 Eigenspace projection 108
6.8.2 Extension to vector spatial-filter formulation 112
6.9 Numerical examples 114
Effects of low-rank interference 120
7.1 Influence of low-rank interference 120
7.1.1 Low-rank interference 120
7.1.2 Analysis when Rd is a rank-one matrix 122
7.1.3 Analysis when Rd is a rank-two matrix 124
7.2 Influence on output of the unit-noise-gain minimum- variance filter 125
7.3 Effects on the output of the eigenspaceprojected spatial filter 126
7.4 Numerical examples 127
Effects of high-rank interference 136
8.1 Influence of background brain activity 136
8.1.1 Point-spread function under background interference 136
8.1.2 Numerical examples 138
8.2 Prewhitening eigenspace-projection spatial filter 140
8.2.1 Prewhitening signal covariance estimation 140
8.2.2 Prewhitening eigenspace-projection spatial filter 143
8.3 Overestimation of signal-subspace dimensionality 144
8.4 Reconstruction of induced activity 146
8.4.1 General background 146
8.4.2 Prewhitening method 147
8.5 Numerical examples 149
Effects of source correlation 156
9.1 Performance of adaptive spatial filters in the presence of correlated sources 156
9.2 Signal cancellation and estimation of source correlation 158
9.3 Suppression of coherent interferences using the LCMV spatial filter 160
9.3.1 Weight-vector derivation 160
9.3.2 Extension to eigenspace-projected spatial filter 162
9.4 Imaging magnitude source coherence 163
9.5 Numerical examples 166
Effects of using the sample covariance matrix 174
10.1 Sample covariance matrix: the maximumlikelihood estimate of the true covariance matrix 174
10.2 Effects of using sample covariance matrices on the minimum- variance filters 175
10.3 Recovering from the sample covariance effects: Beamspace processing 177
10.4 Numerical examples 179
10.4.1 Effects of using sample covariance matrices 179
10.4.2 Recovering from the sample covariance effects 179
10.4.3 Effects of using sample covariance matrices on unit- noise- gain minimum- variance filter 180
Statistical evaluation of the spatial filter output 190
11.1 Problem with Gaussian-distribution-based methods 190
11.2 Evaluation of statistical significance using nonparametric statistics 191
11.2.1 Voxel-by-voxel statistical significance test 191
11.2.2 Multiple comparisons using maximum statistics 192
11.2.3 Modification for power image 193
11.2.4 Multiple comparisons using the false discovery rate 194
11.3 Deriving a voxel-wise empirical null distribution 196
11.3.1 Method when the signal is time-locked and the interference is non- time- locked to the stimulus 196
11.3.2 Method when both the signal and the interference are non- time- locked to the stimulus 197
11.4 Non-parametric method using reconstructed voxel time courses 198
11.5 Numerical examples 199
Methods related to adaptive spatial filters 204
12.1 Wiener filter 204
12.1.1 Minimum-mean-squared-error criterion 204
12.1.2 Derivation of the minimum-variance spatial filter 206
12.2 MUSIC algorithm 207
12.2.1 Single- and multi-dipole search 207
12.2.2 Making use of the noise subspace–the MUSIC algorithm 208
12.3 Scanning with the generalized-likelihoodratio test function 209
12.3.1 Data model 210
12.3.2 Deriving the scanning function 211
12.3.3 Numerical examples 214
Appendices 216
13.1 Maximum-likelihood estimation of noise and signal subspaces 216
13.2 Additional topics related to non-adaptive spatial filters 218
13.2.1 Determination of the optimum orientation for scalar non- adaptive spatial filters 218
13.2.2 Equivalence between the vector and scalar minimumnorm filters 219
13.3 Rayleigh-Ritz formula 220
13.4 Supplementary formulae when only one or two sources exist 222
13.5 Robustness of the prewhitening signal covariance estimation to the control- only- sources scenario 225
13.6 Derivation of GLRT scanning function in Eq. ( 12.45) 228
13.7 Bioelectromagnetic forward modeling 231
13.7.1 Quasi-static Maxwell’s equations 232
13.7.2 Magnetic field in an infinite homogeneous conductor 232
13.7.3 Electric potential in an infinite homogeneous conductor 234
13.7.4 Formulae in a bounded conductor with piecewise- constant conductivity 234
13.7.5 Magnetic field from a homogeneous spherical conductor 235
13.7.6 Magnetic field from a realistically-shaped conductor 237
13.7.7 Electric potential for a multiple-shell conductor 242
Bibliography 244
Index 254