From ddfa6f1258287cbbe4ece43869e6431a655f664b Mon Sep 17 00:00:00 2001
From: Nao Pross <np@0hm.ch>
Date: Tue, 30 Aug 2022 13:19:13 +0200
Subject: kugel: Start EEG interpolation problem

---
 buch/papers/kugel/applications.tex | 42 ++++++++++++++++++++++++++++++++------
 1 file changed, 36 insertions(+), 6 deletions(-)

(limited to 'buch/papers/kugel/applications.tex')

diff --git a/buch/papers/kugel/applications.tex b/buch/papers/kugel/applications.tex
index 39e4cfc..eb6e02f 100644
--- a/buch/papers/kugel/applications.tex
+++ b/buch/papers/kugel/applications.tex
@@ -35,15 +35,15 @@ 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, since variables and functions represent physical quantities. So,
-we will quickly remind that the Laplacian operator does the following to an
-electric potential $\phi$:
+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
-  \iff
+  = \rho / \varepsilon,
+  \quad \text{or} \quad
   \iiint_\Omega \nabla \cdot \mathbf{E} \, dv
   = \iint_{\partial \Omega} \mathbf{E} \cdot d\mathbf{s}
   = \Phi / \varepsilon.
@@ -58,7 +58,37 @@ 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}) is the net electric charge.
+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?
+
+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.
+
+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 $R$ begin the radius of the average human head. So, we will assume that the
+potential distribution on the head can be written as a finite linear combination
+of spherical harmonics:
+\begin{equation*}
+  \phi(\vartheta, \varphi)
+    = \sum_{n=0}^N \sum_{m=-n}^n a_{m,n} Y^m_n(\vartheta, \varphi)
+\end{equation*}
 
 \subsection{Measuring Gravitational Fields}
 
-- 
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