1;Contents;7 2;Introduction;12 2.1;1.1 Functional brain mapping;12 2.2;1.2 Electromagnetic brain imaging;13 2.3;1.3 Spatial .lters;14 2.4;1.4 Book chapter organization;16 2.5;1.5 Acknowledgements;18 3;Sensor array outputs and spatial . lters;20 3.1;2.1 Neuromagnetic signals as sensor-array outputs;20 3.1.1;2.1.1 De.nitions;20 3.1.2;2.1.2 Sensor lead field;21 3.1.3;2.1.3 Linear independence of lead-field vectors;22 3.2;2.2 Bioelectromagnetic inverse problem;24 3.3;2.3 Expressions of data covariance matrices;26 3.3.1;2.3.1 Data and source covariance relationship;26 3.3.2;2.3.2 Formulation for uncorrelated sources;28 3.4;2.4 Low-rank signal modeling;29 3.4.1;2.4.1 Definition of noise and signal subspaces;29 3.4.2;2.4.2 Property of the data covariance matrix;30 3.5;2.5 Spatial filters;33 3.5.1;2.5.1 Source reconstruction using a spatial filter;33 3.5.2;2.5.2 Scalar and vector spatial filters;34 3.5.3;2.5.3 Resolution kernel, point-spread function, and beam response;36 4;Tomographic reconstruction and nonadaptive spatial filters;38 4.1;3.1 Minimum-norm method;38 4.1.1;3.1.1 Tomographic reconstruction formulation;38 4.1.2;3.1.2 Nonadaptive spatial-filter formulation;42 4.2;3.2 Variants of the minimum-norm filter;43 4.2.1;3.2.1 Weight-normalized minimum-norm filter;43 4.2.2;3.2.2 sLORETA filter;43 4.3;3.3 Spatial matched filter;45 4.4;3.4 Deriving the minimum-norm-based filters using leakage minimization;46 5;Adaptive spatial filters;48 5.1;4.1 Deriving weights for adaptive spatial filters;48 5.1.1;4.1.1 Minimum-variance spatial filter with the unit-gain constraint;48 5.1.2;4.1.2 Minimum-variance spatial filter with the array-gain constraint;50 5.1.3;4.1.3 Minimum-variance spatial filter with the unit-noisegain constraint;50 5.2;4.2 Prerequisites for the adaptive spatial-filter formulation;51 5.2.1;4.2.1 Uncorrelated source time courses;51 5.2.2;4.2.2 Low-rank signals;54 5.3;4.3 Scalar adaptive spatial filter: deriving the optimum source orientation;55 5.4;4.4 LCMV spatial filter;57 5.5;4.5 Vector adaptive spatial filter formulation;59 5.5.1;4.5.1 Unit-gain constraint spatial filter;59 5.5.2;4.5.2 Array-gain constraint spatial filter;60 5.5.3;4.5.3 Unit-noise-gain constraint spatial filter;62 5.5.4;4.5.4 Equivalence between the adaptive scalar and vector formulations;64 5.6;4.6 Frequency-domain implementation;65 5.7;4.7 Numerical examples;68 6;Location bias, spatial resolution, and beam response;76 6.1;5.1 Bias properties of various spatial filters;76 6.1.1;5.1.1 Definition of source location bias;76 6.1.2;5.1.2 Bias for the spatial matched filter;77 6.1.3;5.1.3 Bias for the minimum-norm filter;78 6.1.4;5.1.4 Bias for the weight-normalized minimum-norm filter;78 6.1.5;5.1.5 Bias for the sLORETA filter;79 6.1.6;5.1.6 Bias for the unit-gain minimum-variance spatial filter;79 6.1.7;5.1.7 Bias for the array-gain minimum-variance spatial filter;80 6.1.8;5.1.8 Bias for the unit-noise-gain minimum-variance spatial filter;80 6.2;5.2 Effects of noise on the location bias;81 6.3;5.3 Spatial resolution;82 6.4;5.4 Spatial-filter beam response;83 6.5;5.5 Numerical examples;85 7;Output SNR and array mismatch;94 7.1;6.1 Output SINR;94 7.2;6.2 Adaptive spatial filters that attain the maximum SINR;96 7.3;6.3 SNR transfer factor;98 7.4;6.4 Two types of SNR de.nitions for the vector minimum- variance spatial filter;100 7.5;6.5 Influence of array mismatch;103 7.6;6.6 Diagonal loading;104 7.7;6.7 Asymmetric diagonal loading;106 7.8;6.8 Eigenspace-projection spatial filter;108 7.8.1;6.8.1 Eigenspace projection;108 7.8.2;6.8.2 Extension to vector spatial-filter formulation;112 7.9;6.9 Numerical examples;114 8;Effects of low-rank interference;120 8.1;7.1 Influence of low-rank interference;120 8.1.1;7.1.1 Low-rank interference;120 8.1.2;7.1.2 Analysis when Rd is a rank-one matrix;122 8.1.3;7.1.3 Analysis when Rd is a rank-two matrix;124 8.2;7.2 Influence on output of the unit-noise-gain minimum- variance filter;125 8.3;7.3 Effects on the output of the eigens