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| -rw-r--r-- | buch/papers/kugel/applications.tex | 40 |
1 files changed, 29 insertions, 11 deletions
diff --git a/buch/papers/kugel/applications.tex b/buch/papers/kugel/applications.tex index 32095c4..1af0018 100644 --- a/buch/papers/kugel/applications.tex +++ b/buch/papers/kugel/applications.tex @@ -93,7 +93,7 @@ will assume that the potential distribution on the head can be written as a finite linear combination of spherical harmonics: \begin{equation*} V(\vartheta, \varphi) - = \sum_{n=0}^N \sum_{m=-n}^n a_{m,n} Y^m_n(\vartheta, \varphi), + = \sum_{n=1}^N \sum_{m=-n}^n a_{m,n} Y^m_n(\vartheta, \varphi), \end{equation*} where the values $a_{m,n}$ are the unknowns of our interpolation problem. Now to the measurements: we let $\phi_1, \phi_2, \ldots, p_M$ be the measured voltages @@ -125,20 +125,20 @@ world! Thus, to solve this problem, we will use the suspiciously convenient fact that (hint: eigenvalues) \begin{equation*} \surflaplacian V(\vartheta, \varphi) - = \sum_{n=0}^N \sum_{m=-n}^n a_{m,n} + = \sum_{n=1}^N \sum_{m=-n}^n a_{m,n} \surflaplacian Y^m_n(\vartheta, \varphi) - = \sum_{n=0}^N \sum_{m=-n}^n a_{m,n} + = \sum_{n=1}^N \sum_{m=-n}^n a_{m,n} n(n+1) Y^m_n(\vartheta, \varphi). \end{equation*} So that when substituted into \eqref{kugel:eqn:eeg-min} results in \begin{align*} - \int_{\partial S} \left[ - \sum_{n=0}^N \sum_{m=-n}^n n(n+1) |a_{m,n}| + \int_{\partial S} \left| + \sum_{n=1}^N \sum_{m=-n}^n n(n+1) a_{m,n} Y^m_n(\vartheta, \varphi) - \right]^2 ds - = \sum_{m, m'} \sum_{n, n'} |a_{m',n'}| |a_{m,n}| + \right|^2 ds + = \sum_{m, m'} \sum_{n, n'} a_{m',n'} \overline{a_{m,n}} n'(n'+1) n(n+1) - \underbrace{\int_{\partial S} Y^{m'}_{n'} Y^m_n \, ds}_{ + \underbrace{\int_{\partial S} Y^{m'}_{n'} \overline{Y^m_n} \, ds}_{ \langle Y^{m'}_{n'}, Y^m_n \rangle }, \end{align*} @@ -151,14 +151,32 @@ dropped and \eqref{kugel:eqn:eeg-min} simplifies down to \nocite{pascual-marqui_current_1988} \begin{equation} a^*_{m,n} = \arg \min_{a_{m,n}} - \sum_{n=0}^N \sum_{m=-n}^n n^2 (n+1)^2 |a_{m,n}|^2. + \sum_{n=1}^N \sum_{m=-n}^n n^2 (n+1)^2 |a_{m,n}|^2. \end{equation} At this point, we could continue solving for an analytical solution to the minimization problem, for example by differentiating with respect to some $a_{j,k}$, setting that to zero and so forth, but the job of the spherical -harmonics ends here. So, we will not pursue this further, and instead discuss a -few interesting problems that come with this approach. +harmonics ends here. So, we will not pursue this further, and instead briefly +discuss a few interesting implications and problems. + +\subsubsection{Sampling, smoothness and problems} + +The most interesting perhaps unforeseen fact is that with this method we are +getting a free (!) spectral analysis, since the coefficients $a_{m,n}$ are the +spectrum of the interpolated electric field $V(\vartheta, \varphi)$. However, +like in the non spherical Fourier transformation, we only get a \emph{finite} +resolution since our measurement are spatially discrete. In fact, if we know the +mean angular inter-electrode distance $\gamma$ we can actually formulate a +Nyquist frequency limit just like in the usual Fourier theory: +\begin{equation*} + f_N = \frac{\pi}{2T} + \iff + n_N = \left\lfloor \frac{\pi}{2\gamma} \right\rfloor. +\end{equation*} + +\nocite{wingeier_spherical_2001} + \subsection{Measuring Gravitational Fields} |