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## Systematic regularization of linear inverse solutions of the EEG source localization problem

Venue: | NeuroImage |

Citations: | 23 - 3 self |

### Citations

1366 |
Solutions of Ill-Posed Problems”,
- Tikhonov, Arsenin
- 1977
(Show Context)
Citation Context ...mate a unique solution. These constraints are generally based on a priori knowledge about the sources. One common approach to solve this problem is the WMN solution or Tikhonov regularization method (=-=Tikhonov and Arsenin, 1977-=-), in which the a priori constraint can be interpreted in a Bayesian sense. The WMN solution constrains the reconstructed source distribution by minimizing a linear mixture of some weighted norm �Hj� ... |

370 |
Analysis of discrete ill-posed problems by means of the L-curve.
- HANSEN
- 1992
(Show Context)
Citation Context ...ifferent values of �. The curve obtained usually has an L shape (in ordinary or double logarithmic scale), hence its name “L-curve.” A satisfactory � would lie close to the inflection of the L-curve (=-=Hansen, 1992-=-). But Engl and Grever (1994) showed that hyperparameter choice strategies, based on this plot alone, e.g., on the corner of the L-curve, are independent of the norm of the difference between the true... |

312 |
Electric Fields of the Brain: The Neurophysics of EEG.
- Nuñez
- 1981
(Show Context)
Citation Context ...constraints using anatomical and physiological information derived from other imaging modalities. This approach has already been introduced in Phillips et al. (2002) and rests upon three assumptions (=-=Nunez, 1981-=-; Hämäläinen et al., 1993; Dale and Sereno, 1993): sources are located in gray matter, they are oriented orthogonal to the cortical sheet, and, for a sufficiently dense dipole distribution, they are l... |

307 |
Recovery of inter-block information when block sizes are unequal.
- Patterson, Thompson
- 1971
(Show Context)
Citation Context ...A� � A t C �1 b, (24) where, for any matrix B, B � denotes an arbitrary generalized inverse of B, i.e., any solution to BB � B � B. This is equivalent to maximizing the objective likelihood function (=-=Patterson and Thompson, 1971-=-) p�b�x���C� �1/2 e �1/2�b�Ax� t C �1 �b�Ax� , (25) which is equivalent to minimizing the function Maximizing Eq. (27) is equivalent to minimizing F�b; x, �� � �2 log�p�b�x, ��� � log�A tC��� �1A� � l... |

306 |
Maximum likelihood approaches to variance component estimation and to related problems.
- Harville
- 1977
(Show Context)
Citation Context ...x and � but a linear parameterization of C, i.e., C(�) � ¥ � iG i where G i are n � n symmetric matrices whose elements are known, leads to a much simpler and less computationally demanding approach (=-=Harville, 1977-=-). Simple ReML Solution One such case of linear parameterization of C involves a diagonal covariance matrix with each element parameterized by only one element of �. With only two 2 2 hyperparameters ... |

278 |
Standardized low-resolution brain electromagnetic tomography (sLORETA): technical details. Methods Find Exp Clin Pharmacol.
- RD
- 2002
(Show Context)
Citation Context ...e hyperparameter � was still obtained through the ReML algorithm. In the literature (Pascual-Marqui et al., 1994; Pascual-Marqui, 1999), the MS solution is usually implemented in the LORETA software (=-=Pascual-Marqui, 1998-=-) for the ideal noise-free case, using294 PHILLIPS, RUGG, AND FRISTON FIG. 2. Localization error (LE), left, and log 10 �, right, for the four solutions (IBF10, IBF5, WMN, and MS with and without noi... |

236 |
Basic mathematical and electromagnetic concepts of the biomagnetic problem.
- Sarvas
- 1987
(Show Context)
Citation Context ...g and Ebersole, 1994; Scherg et al., 1999; Aine et al., 2000), and the “linear distributed” approach, in which “all” possible source locations are considered simultaneously (Backus and Gilbert, 1970; =-=Sarvas, 1987-=-; Hämäläinen and Ilmoniemi, 1994; Grave de Peralta Menendez and Gonzalez Andino, 1999; Pascual-Marqui, 1999; Uutela et al., 1999). Here we present a linear distributed approach, based on the weighted ... |

231 |
Low resolution electromagnetic tomography: A new method for localizing electrical activity in the brain,”
- Pascual-Marqui, Michel, et al.
- 1994
(Show Context)
Citation Context ... ratios (SNRs): SNR � 1.5 (middle) and SNR � 6 (bottom). basis functions (spatial or temporal) were employed but the hyperparameter � was still obtained through the ReML algorithm. In the literature (=-=Pascual-Marqui et al., 1994-=-; Pascual-Marqui, 1999), the MS solution is usually implemented in the LORETA software (Pascual-Marqui, 1998) for the ideal noise-free case, using294 PHILLIPS, RUGG, AND FRISTON FIG. 2. Localization ... |

187 |
Magnetoencephalography—theory, instrumentation, and applications to noninvasive studies of the working human brain,”
- Hamalainen, Hari, et al.
- 1993
(Show Context)
Citation Context ...sing anatomical and physiological information derived from other imaging modalities. This approach has already been introduced in Phillips et al. (2002) and rests upon three assumptions (Nunez, 1981; =-=Hämäläinen et al., 1993-=-; Dale and Sereno, 1993): sources are located in gray matter, they are oriented orthogonal to the cortical sheet, and, for a sufficiently dense dipole distribution, they are locally coherent (i.e., th... |

157 |
Interpreting magnetic fields of the brain: minimum norm estimates.
- Hämäläinen, Ilmoniemi
- 1994
(Show Context)
Citation Context ..., 1994; Scherg et al., 1999; Aine et al., 2000), and the “linear distributed” approach, in which “all” possible source locations are considered simultaneously (Backus and Gilbert, 1970; Sarvas, 1987; =-=Hämäläinen and Ilmoniemi, 1994-=-; Grave de Peralta Menendez and Gonzalez Andino, 1999; Pascual-Marqui, 1999; Uutela et al., 1999). Here we present a linear distributed approach, based on the weighted minimum L2-norm (WMN) method, th... |

156 |
Dynamic statistical parametric mapping: combining fMRI and MEG for highresolution imaging of cortical activity.
- Dale, Liu, et al.
- 2000
(Show Context)
Citation Context ...N method. Further “soft” constraints can be introduced in the weighting of the WMN solution using, for example, hemodynamic measures of brain activity as spatial priors (Rugg, 1998; Liu et al., 1998; =-=Dale et al., 2000-=-). The focus of this paper is the relative weighting of these priors in the context of observation noise. In the context of noisy data, regularization is necessary to control the relative influence of... |

119 |
Uniqueness in the inversion of inaccurate gross earth data,
- Backus, Gilbert
- 1970
(Show Context)
Citation Context ...iltner et al., 1994; Scherg and Ebersole, 1994; Scherg et al., 1999; Aine et al., 2000), and the “linear distributed” approach, in which “all” possible source locations are considered simultaneously (=-=Backus and Gilbert, 1970-=-; Sarvas, 1987; Hämäläinen and Ilmoniemi, 1994; Grave de Peralta Menendez and Gonzalez Andino, 1999; Pascual-Marqui, 1999; Uutela et al., 1999). Here we present a linear distributed approach, based on... |

114 |
Multimodal image coregistration and partitioning—a unified framework.
- Ashburner, KJ
- 1997
(Show Context)
Citation Context ...owitz, 1967; Hämäläinen and Sarvas, 1989; Meijs et al., 1989; de Munck, 1992; Schlitt et al., 1995; Ferguson and Stroink, 1997; Buchner et al., 1997). A structural MR image of the head was segmented (=-=Ashburner and Friston, 1997-=-) and divided into three volumes with homogeneous isotropic conductivity: the brain, the skull, and the scalp volume. Then the potential at the electrode sites on the scalp surface was computed for an... |

108 |
Visualization of magnetoencephalographic data using minimum current estimates,”
- Uutela, Hämäläinen, et al.
- 1999
(Show Context)
Citation Context ...ible source locations are considered simultaneously (Backus and Gilbert, 1970; Sarvas, 1987; Hämäläinen and Ilmoniemi, 1994; Grave de Peralta Menendez and Gonzalez Andino, 1999; Pascual-Marqui, 1999; =-=Uutela et al., 1999-=-). Here we present a linear distributed approach, based on the weighted minimum L2-norm (WMN) method, that employs constraints using anatomical and physiological information derived from other imaging... |

106 |
Realistic Conductivity Geometry Model of the Human Head for Interpretation of Neuromagnetic Data,”
- H€am€al€ainen, Sarvas
- 1989
(Show Context)
Citation Context ...oblem is written as Ni v � � ��r� i, j� i� � �. (2) i�1 2.1. Head and Source Model � was calculated using the “boundary element method” (BEM) for a three-shell realistic head model (Geselowitz, 1967; =-=Hämäläinen and Sarvas, 1989-=-; Meijs et al., 1989; de Munck, 1992; Schlitt et al., 1995; Ferguson and Stroink, 1997; Buchner et al., 1997). A structural MR image of the head was segmented (Ashburner and Friston, 1997) and divided... |

99 | Bayesian inference for variance components using only error contrasts. - Harville - 1974 |

92 | Review of methods for solving the EEG inverse problem,”
- Pascual-Marqui
- 1999
(Show Context)
Citation Context ...h, in which “all” possible source locations are considered simultaneously (Backus and Gilbert, 1970; Sarvas, 1987; Hämäläinen and Ilmoniemi, 1994; Grave de Peralta Menendez and Gonzalez Andino, 1999; =-=Pascual-Marqui, 1999-=-; Uutela et al., 1999). Here we present a linear distributed approach, based on the weighted minimum L2-norm (WMN) method, that employs constraints using anatomical and physiological information deriv... |

69 |
Spatiotemporal imaging of human brain activity using functional MRI constrainedmagnetoencephalography data: Monte Carlo simulations.
- Liu, Belliveau, et al.
- 1998
(Show Context)
Citation Context ...le, a classical WMN method. Further “soft” constraints can be introduced in the weighting of the WMN solution using, for example, hemodynamic measures of brain activity as spatial priors (Rugg, 1998; =-=Liu et al., 1998-=-; Dale et al., 2000). The focus of this paper is the relative weighting of these priors in the context of observation noise. In the context of noisy data, regularization is necessary to control the re... |

42 | Anatomically informed basis functions for EEG source localization: combining functional and anatomical constraints
- Phillips, Rugg, et al.
(Show Context)
Citation Context ...ocation priors. Two criteria were used to assess and compare the performance of the different methods (IBF10, IBF5, WMN, and MS): “locational error” (LE) and “root mean square error” (RMSE) (see also =-=Phillips et al., 2002-=-). The LE is defined as the distance between the location r� r of the maximum (absolute value) of the reconstructed source jr and the location r� o of the original source set jo. In order to obtain a ... |

39 | Continuous probabilistic solutions to the biomagnetic inverse problem. - Ioannides, Bolton, et al. - 1990 |

35 | On the numerical accuracy of the boundary element method - Meijs, Weier, et al. - 1989 |

32 |
On bioelectric potentials in an inhomogeneous volume conductor
- Geselowitz
- 1967
(Show Context)
Citation Context ...ce localization problem is written as Ni v � � ��r� i, j� i� � �. (2) i�1 2.1. Head and Source Model � was calculated using the “boundary element method” (BEM) for a three-shell realistic head model (=-=Geselowitz, 1967-=-; Hämäläinen and Sarvas, 1989; Meijs et al., 1989; de Munck, 1992; Schlitt et al., 1995; Ferguson and Stroink, 1997; Buchner et al., 1997). A structural MR image of the head was segmented (Ashburner a... |

28 |
Inverse localization of electric dipole current sources in finite element models of the human head. Electroencephalogr Clin Neurophysiol 102:267–278.
- Buchner, Knoll, et al.
- 1997
(Show Context)
Citation Context ...dary element method” (BEM) for a three-shell realistic head model (Geselowitz, 1967; Hämäläinen and Sarvas, 1989; Meijs et al., 1989; de Munck, 1992; Schlitt et al., 1995; Ferguson and Stroink, 1997; =-=Buchner et al., 1997-=-). A structural MR image of the head was segmented (Ashburner and Friston, 1997) and divided into three volumes with homogeneous isotropic conductivity: the brain, the skull, and the scalp volume. The... |

27 |
Multiple source analysis of interictal spikes: goals, requirements, and clinical value,”
- Scherg, Bast, et al.
- 1999
(Show Context)
Citation Context ...ories: “equivalent current dipole” approaches, in which the EEG/MEG signals are assumed to be generated by a relatively small number of focal sources (Miltner et al., 1994; Scherg and Ebersole, 1994; =-=Scherg et al., 1999-=-; Aine et al., 2000), and the “linear distributed” approach, in which “all” possible source locations are considered simultaneously (Backus and Gilbert, 1970; Sarvas, 1987; Hämäläinen and Ilmoniemi, 1... |

24 | Using the L-curve for determining optimal regularization parameters,’’ - Engl, Grever - 1994 |

21 | A critical analysis of linear inverse solutions to the neuroelectromagnetic inverse problem,” - Menendez, Andino - 1998 |

21 |
A test of brain electrical source analysis (BESA): a simulation study.
- Miltner, Braun, et al.
- 1994
(Show Context)
Citation Context ...success. These can be divided into two main categories: “equivalent current dipole” approaches, in which the EEG/MEG signals are assumed to be generated by a relatively small number of focal sources (=-=Miltner et al., 1994-=-; Scherg and Ebersole, 1994; Scherg et al., 1999; Aine et al., 2000), and the “linear distributed” approach, in which “all” possible source locations are considered simultaneously (Backus and Gilbert,... |

20 |
Factors affecting the accuracy of the boundary element method in the forward problem. I. Calculating surface potentials
- Ferguson, Stroink
- 1997
(Show Context)
Citation Context ...s calculated using the “boundary element method” (BEM) for a three-shell realistic head model (Geselowitz, 1967; Hämäläinen and Sarvas, 1989; Meijs et al., 1989; de Munck, 1992; Schlitt et al., 1995; =-=Ferguson and Stroink, 1997-=-; Buchner et al., 1997). A structural MR image of the head was segmented (Ashburner and Friston, 1997) and divided into three volumes with homogeneous isotropic conductivity: the brain, the skull, and... |

16 | A linear discretization of the volume conductor boundary integral equation using analytically integrated elements - Munck - 1992 |

12 |
Convergent approaches to electrophysiological and hemodynamic investigations of memory,
- Rugg
- 1998
(Show Context)
Citation Context ...g, for example, a classical WMN method. Further “soft” constraints can be introduced in the weighting of the WMN solution using, for example, hemodynamic measures of brain activity as spatial priors (=-=Rugg, 1998-=-; Liu et al., 1998; Dale et al., 2000). The focus of this paper is the relative weighting of these priors in the context of observation noise. In the context of noisy data, regularization is necessary... |

10 |
Multistart algorithms for MEG empirical data analysis reliably characterize locations and time courses of multiple sources
- Aine, Huang, et al.
- 2000
(Show Context)
Citation Context ...rrent dipole” approaches, in which the EEG/MEG signals are assumed to be generated by a relatively small number of focal sources (Miltner et al., 1994; Scherg and Ebersole, 1994; Scherg et al., 1999; =-=Aine et al., 2000-=-), and the “linear distributed” approach, in which “all” possible source locations are considered simultaneously (Backus and Gilbert, 1970; Sarvas, 1987; Hämäläinen and Ilmoniemi, 1994; Grave de Peral... |

10 |
Ranken, “Evaluation of boundary element methods for the EEG forward problem: Effect of linear interpolation
- Schlitt, Heller, et al.
- 1995
(Show Context)
Citation Context ... and Source Model � was calculated using the “boundary element method” (BEM) for a three-shell realistic head model (Geselowitz, 1967; Hämäläinen and Sarvas, 1989; Meijs et al., 1989; de Munck, 1992; =-=Schlitt et al., 1995-=-; Ferguson and Stroink, 1997; Buchner et al., 1997). A structural MR image of the head was segmented (Ashburner and Friston, 1997) and divided into three volumes with homogeneous isotropic conductivit... |

8 |
Brain source imaging of focal and multifocal epileptiform EEG activity
- Scherg, Ebersole
- 1994
(Show Context)
Citation Context ...divided into two main categories: “equivalent current dipole” approaches, in which the EEG/MEG signals are assumed to be generated by a relatively small number of focal sources (Miltner et al., 1994; =-=Scherg and Ebersole, 1994-=-; Scherg et al., 1999; Aine et al., 2000), and the “linear distributed” approach, in which “all” possible source locations are considered simultaneously (Backus and Gilbert, 1970; Sarvas, 1987; Hämälä... |

5 | Backus and Gilbert method for vector fields - Menendez, Andino - 1999 |

5 | Reply to comments by - Pascual-Marqui - 1995 |

2 |
Improved localisation of cortical activity by combining EEG and MEG with MRI cortical surface reconstruction: A linear approach
- Dale, Sereno
- 1993
(Show Context)
Citation Context ...ological information derived from other imaging modalities. This approach has already been introduced in Phillips et al. (2002) and rests upon three assumptions (Nunez, 1981; Hämäläinen et al., 1993; =-=Dale and Sereno, 1993-=-): sources are located in gray matter, they are oriented orthogonal to the cortical sheet, and, for a sufficiently dense dipole distribution, they are locally coherent (i.e., their activity changes sm... |