Leweke, Sarah: The inverse magneto-electroencephalography problem for the spherical multiple-shell model : theoretical investigations and numerical aspects. 2018
Inhalt
- Zusammenfassung
- Abstract
- Contents
- Introduction
- Basics
- Preliminaries
- Notation
- Jacobi and Legendre Polynomials
- Vector Calculus in Spherical Geometries
- Scalar Spherical Harmonics
- Theory of Distributions
- Modelling the Magnetoencephalography Problem
- Modelling the Electroencephalography Problem
- Solving the Direct Problem
- Preliminaries
- Orthonormal Systems on the Interval
- Vector Spherical Harmonics
- Definition of Vector Spherical Harmonics
- Orthogonality and Completeness of Vector Spherical Harmonics
- Harmonicity of Vector Spherical Harmonics
- Decomposition of via Vector Spherical Harmonics
- Vector Legendre Polynomials
- Vector Outer Harmonics
- Orthonormal Systems on the Ball
- Vector Legendre-type Integral Kernels
- Vector Legendre-type Integral Operators
- Definition of the Integral Operators
- Continuity and Differentiability of the Potential
- Solution of the Direct Problem
- A Harmonic Vector Legendre-type Integral Operator
- Direct Magnetoencephalography Problem
- Direct Electroencephalography Problem
- Solving the Inverse Problem
- Introduction to Inverse Problems
- Ill-Posedness of the VLI Problem
- Continuous and Star-shaped Problem
- Harmonic VLI Problem
- Uniqueness Constraints for the Continuous VLI Problem
- Inverse Magneto-electroencephalography Problem
- Scalar General Integral Problem
- Regularization
- Preliminaries
- Regularized Functional Matching Pursuit Algorithm
- Numerical Solution of the MEG and EEG Problem
- Synthetic Test Case
- Foundation for Implementation
- Other Reconstruction Methods
- Regularized Ritz Method
- Scalar Spline Method
- Scalar Splines
- Scalar Splines for the MEG Problem
- Scalar Synthetic Test Current
- Corresponding Vector-Valued Neuronal Current
- Scalar Splines for the EEG Problem
- Vector Spline Method
- Numerical Results
- Final Remarks
- Appendix