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-rw-r--r--buch/papers/kugel/applications.tex40
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}