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diff --git a/buch/papers/kugel/applications.tex b/buch/papers/kugel/applications.tex index 1af0018..15a57d5 100644 --- a/buch/papers/kugel/applications.tex +++ b/buch/papers/kugel/applications.tex @@ -78,13 +78,13 @@ of the brain? The details of how EEG actually works gets very complicated very quickly, but we will try our best to give an broad overview of the mathematical machinery that -makes it possible to measure brain waves. See, the problem neither the physicist -nor the mathematician considered is that we cannot measure the electric field in -its entirety. As show in figure \ref{kugel:fig:eeg-electrodes} the electrodes -give measurements that are only available at discrete locations, and we are thus -quite a lot of missing data. In other words, we have an interpolation problem. -And (at this point not so surprisingly) we will show that it can be solved using -the spherical harmonics. +makes it possible to measure brain waves. The problem neither the physicist nor +the mathematician considered is that we cannot measure the electric field in its +entirety. As show in figure \ref{kugel:fig:eeg-electrodes} the electrodes give +measurements that are only available at discrete locations, and we are thus +missing quite a lot of data. Or in other words, we have an interpolation +problem, which (at this point not so surprisingly) we will show can be solved +using the spherical harmonics. To solve this new interpolation problem, we will start with a blatantly engineering assumption: the human head is a sphere of radius $R$, with the value @@ -160,7 +160,8 @@ $a_{j,k}$, 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 briefly discuss a few interesting implications and problems. -\subsubsection{Sampling, smoothness and problems} +\subsubsection{Sampling, Smoothness and Problems} +\nocite{wingeier_spherical_2001, ruffini_spherical_2002} 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 @@ -168,15 +169,37 @@ 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*} +Nyquist frequency 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} +\end{equation} +Before concluding this overview of EEG, we should point out that in practice +there are about a million problems with this oversimplified approach. We do not +intend to give an in depth explanation (since the authors themselves are not +experts in any of these fields), but there are a few problems that are too big +to ignore, so we will very briefly discuss them now. The first important +real-world problem is that the electrodes are not necessarily at a reasonable +location, so the constraint \eqref{kugel:eqn:eeg-min-constraints} is a bit too +strong, and may end up fitting some noise or disturbances in the measurement. A +simple solution may for example be to introduce a smoothness factor $\lambda > +0$ as follows: +\begin{equation} + V(\vartheta, \varphi) = \sum_{n=1}^N \sum_{m=-n}^n + \frac{a_{m,n}}{1 + \lambda n^2(n+1)^2} Y^m_n(\vartheta, \varphi). +\end{equation} +To find proper smoothness factor $\lambda$, is another problem of its own, thus +we will not discuss it here, since this is getting too long already. Another +important issue is that in the real world, we cannot ``evenly distribute'' the +electrodes on our head. As shown in the image, most of the electrodes are on a +cap, and then there are just a few on the face, and almost none near the jawline +and chin. This not something that can be ignored, and in fact, makes the +analysis much more difficult. Finally, the most obvious problem is that human +heads are not perfect spheres. Here too, it is possible to account for this fact +and model the head with a more complex shape at the cost of making the math +quite unwieldy. \subsection{Measuring Gravitational Fields} |