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diff --git a/buch/papers/kugel/applications.tex b/buch/papers/kugel/applications.tex index b2f227e..481a3a5 100644 --- a/buch/papers/kugel/applications.tex +++ b/buch/papers/kugel/applications.tex @@ -1,9 +1,206 @@ -% vim:ts=2 sw=2 et spell: +% vim:ts=2 sw=2 et spell tw=80: \section{Applications} -\subsection{Electroencephalography (EEG)} +As suggested in the previous section, the fact that it is possible to write a +Fourier style expansion of any function on the surface of the sphere is very +useful in many fields of physics and engineering. Here we will present a few of +the most interesting applications we came across during our research. + +\subsection{Electroencephalography} + +\begin{figure} + \centering + \subfigure[EEG Electrodes \label{kugel:fig:eeg-electrodes}]% + % {\kugelplaceholderfig{.4\linewidth}{5cm}} + {\includegraphics[width=.45\linewidth, frame]{papers/kugel/figures/electrodes}} + \qquad + \subfigure[Gauss' Law \label{kugel:fig:eeg-flux}]% + {\includegraphics[width=.4\linewidth]{papers/kugel/figures/flux}} + \caption{ + Electroencephalography. + \label{kugel:fig:eeg} + } +\end{figure} + +To start, we will look at an application that is from the field of medicine: +electroencephalography. The \emph{electroencephalogram} (EEG) is a measurement +of the electrical field on the scalp, which shows the brain's activity, and is +used in many fields of research such as neurology and cognitive psychology. The +measurement is done by wearing a cap that contains a number of evenly +distributed electrodes, each of which measures the electric potential (voltage) +at their location (figure \ref{kugel:fig:eeg-electrodes}). To see how this will +relate to the spherical harmonics, we will first quickly recap a bit of physics, +electrodynamics to be precise. + +\subsubsection{Electrodynamics} + +In section \ref{kugel:sec:construction:eigenvalue} we have shown that the +spherical harmonics arise from the surface spherical Laplacian operator, whose +origin we did not consider too much, which is how mathematicians do their work. +On the contrary, physicists usually do the opposite and start by discussing what +is happening in the real world, since variables represent physical quantities. +So, we will quickly remind that the Laplacian operator does the following to an +electric potential $\phi(x, y, z)$: +\begin{equation*} + \nabla^2 \phi + = \nabla \cdot \nabla \phi + = \nabla \cdot \mathbf{E} + = \rho / \varepsilon, + \quad \text{or} \quad + \iiint_\Omega \nabla \cdot \mathbf{E} \, dv + = \iint_{\partial \Omega} \mathbf{E} \cdot d\mathbf{s} + = \Phi / \varepsilon. +\end{equation*} +Put into words: on the left we have the differential form, where we recall that +the Laplacian (which is a second derivative) is the divergence of the gradient. +Unpacking the notation we first see that we have the gradient of the potential, +which is just the electric field $\mathbf{E}$, and then the divergence of said +electric field is proportional to the charge density $\rho$. So, the Laplacian +of the electric potential is the charge density! For those that are more +familiar with the integral form of Maxwell's equation, we have also included an +additional step using the divergence theorem, which brings us to the electric +Flux, which by Gauss' law (shown in the iconic\footnote{Every electrical +engineer has seen this picture so many times that is probably burnt in their +eyes.} figure \ref{kugel:fig:eeg-flux}) equals the net electric charge. + +Now, an important observation is that if we switch to spherical coordinates, the +physics does not change. So, the spherical Laplacian $\sphlaplacian$ of the +electric potential $\phi(r, \vartheta, \varphi)$ is still the charge density (in +spherical coordinates). And what about the surface spherical Laplacian +$\surflaplacian$? To that case the physics is also indifferent, the only change +is that the units result is a \emph{surface} charge density $\rho_s$. Thus, we +are done with physics and finally arrive at the engineers' perspective: how can +we use this fact to build something that reads the current flows on the surface +of the brain? + +\subsubsection{EEG as Interpolation Problem} + +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. 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 +of $R$ being the average radius of a human head (which is around 11 cm). So, we +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=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 +at points in space $p_1, p_2, \ldots, p_M$ (position of the electrodes). To +simplify, we will assume that the electrodes are reasonably evenly distributed, +which means that we have no points that are on top of each other or at wildly +different radii from the origin. With that out of the way, we can now write a +minimization problem: +\begin{subequations} + \begin{align} + a_{m,n}^* &= \arg \min_{a_{m,n}} + \int_{\partial S} | \surflaplacian V |^2 \, ds + = \int_0^{2\pi} \int_{0}^\pi | \surflaplacian V |^2 + \sin \vartheta \, d\vartheta d\varphi, + \label{kugel:eqn:eeg-min} \\ + &\text{under the constraints} \quad V(p_j) = \phi_j + \quad \text{ for } \quad 1 \leq j \leq M. + \label{kugel:eqn:eeg-min-constraints} + \end{align} +\end{subequations} +Essentially, with \eqref{kugel:eqn:eeg-min} we are are asking for the solution +to be smooth by minimizing the square of the total curvature (recall that the +surface spherical Laplacian $\surflaplacian$ is a measure of curvature), while +at the same time with \eqref{kugel:eqn:eeg-min-constraints}, we force the +solution to go through the measured points. The latter is the reason why we +needed to assumed that the measurements are at reasonable locations, something +that (as every engineer show know) is not necessarily the case in the real +world! Thus, to solve this problem, we will use the suspiciously convenient fact +that (hint: eigenvalues) +\begin{equation*} + \surflaplacian V(\vartheta, \varphi) + = \sum_{n=1}^N \sum_{m=-n}^n a_{m,n} + \surflaplacian Y^m_n(\vartheta, \varphi) + = \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=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'} \overline{a_{m,n}} + n'(n'+1) n(n+1) + \underbrace{\int_{\partial S} Y^{m'}_{n'} \overline{Y^m_n} \, ds}_{ + \langle Y^{m'}_{n'}, Y^m_n \rangle + }, +\end{align*} +where we used a ``sloppy'' double sum notation to indicate that we have a bunch +of terms of that form. We did not bother to properly expand the product of +double sums, because we can see that at the end we end up with an inner product +$\langle Y^{m'}_{n'}, Y^m_n \rangle$, which as we know equals $\delta_{m'm} +\delta_{n'n}$, so all of the terms where $n' \neq n$ or $m' \neq m$ can be +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=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 briefly +discuss a few interesting implications 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 +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 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} + +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} -\subsection{Quantisation of Angular Momentum} +% \subsection{Quantisation of Angular Momentum} |