More scribblin

diff --git a/theory.tex b/theory.tex
index c24fa801d43baf22e9659fc4c0b0c58914b9ac3a..8bde3d46d43fdd6836e648bef7b6d3253e84ea24 100644 (file)
--- a/theory.tex
+++ b/theory.tex
@@ -39,7 +39,46 @@ into our anisotropy tensor. Then there are some problems related to
 stability which we can discuss here, or maybe we should leave it for a
 later, more implementation-focused chapter.
 
-\section{Continuous formulation}
+Again, we define the structure tensor
+\begin{equation}
+    J_\rho(\nabla u_\sigma) := K_\rho * \left( \nabla u_\sigma \otimes
+    \nabla u_\sigma \right).
+\end{equation}
+This is then decomposed
+\begin{equation}
+    J_\rho(x) = U(x)^T \Sigma(x) U(x)
+\end{equation}
+and say something about the number of dimensions. The numerical problems
+should be discussed somewhere but maybe not here. The eigenvalues are
+extracted such that
+\begin{equation}
+    \Sigma(x) = \begin{bmatrix}
+        \sigma_1(x) & 0 \\
+        0 & \sigma_2(x)
+    \end{bmatrix}
+\end{equation}
+where $\sigma_1 \gg \sigma_2$. This is then inserted into a new matrix
+$A = U^T \Lambda U$ where
+\begin{equation}
+    \Lambda = \begin{bmatrix}
+        \lambda_1 & 0 \\
+        0 & \lambda_2
+    \end{bmatrix}
+\end{equation}
+and
+\begin{align}
+    \lambda_1 &= \frac{1}{1 + \frac{(\sigma_1 - \sigma_2)^2}{\gamma^2}},
+    \\
+    \lambda_2 &= 1.
+\end{align}
+So the rotation is kept, while the size of the eigenvalues are changed.
+This should be visualized, with a figure showing the length and
+direction of the eigenvalues in the area around an edge. But why is one
+of them always 1? Oh, it is not always the shortest, it varies?
+
+\section{Analysis}
+
+\fixme{Section heading!}
 
 In spirit of my project I have included a section with this title. It
 will be a theory heavy chapter, probably including the anisotropic
@@ -48,6 +87,38 @@ some proofs and derivations as it is not easy to find this in the
 literature. This is one of the most important parts of the theory
 chapter, I guess.
 
+Existence and uniqueness should probably be mentioned here as well? But
+how much do we say about it? Grasmair is an OK reference, but he has the
+square root in there, what does that change? OK, talked to Markus about
+it, it is very technical and does not give much to do it properly, but
+should be discussed. Did he say that it could be done in some smaller
+space $C^1$ or $C^2$ and then we could leave the extension to someone
+else?  For existence we need coercivity (that we can't go infinitely far
+to get a better solution) and weak lower semi-continuity, which means we
+consider the weak topology induced by the weak convergence (?). We do
+some kind of extension with $+\infty$ for functions outside our space,
+but that means we lose coercivity, which we have to fix again.
+Uniqueness is related to convexity. When the anisotropy tensor uses the
+smoothed initial value, this is trivial, but not so much if we use the
+output image in the anisotropy tensor. For us though, this will not be
+the case.
+
+Earlier we had
+\begin{equation}
+    \abs{Du} = \int_{-\infty}^{+\infty} P(\{u \geq t\}) \, dt
+\end{equation}
+(or something like that) but now we have to change it up a little bit.
+Grasmair has a very general one in his coarea formula paper, which in
+the and gives
+\begin{equation}
+    \int_\Omega \alpha(Du) = \int_{-\infty}^{+\infty} P(\{u \geq
+    t\};\alpha;\Omega) \, dt = \int_{\partial^* U \cap \Omega}
+    \alpha\big(x,\nu_U(x)\big) \, d\mathcal{H}^{n-1}.
+\end{equation}
+This might be where we discuss the Ciaccoppolini sets and all that
+stuff. We must also clarify what a perimeter is, and how it relates to a
+contour later.
+
 \chapter{Discrete formulation}
 
 This is where we discretize! It will also be an important chapter, as
@@ -58,20 +129,39 @@ tested later). Then the stencil shape can be adjusted as well.
 
 \section{Discretization}
 
+It would probably be rewarding to look thorougly into the
+Cauchy--Crofton formula, especially in the Riemann metric case. We have
+to verify that all the approximations work, and that it indeed converges
+also in this case.
+
 \section{Graph cut formulation}
 
+Maybe this is more tightly connected with the previous section, but the
+main point is that we describe how the edge weights are computed.
+
 \subsection{Networks}
 
+But first we have to say what a graph is. And also what a cut is!
+
 \fixme{Maybe just call them graphs this time?}
 
 \subsection{Network representable energy functions}
 
+Then, how we can represent an energy function as a graph, and how
+finding the minimum cut will give a minimum energy value.
+
 \subsection{Network construction}
 
+Then we describe how we actually create the graph.
+
 \chapter{Maximum flow approach}
 
+Next up is the maximum flow. Here we can get a lot from the project.
+
 \section{Flow networks}
 
+What is flow. Max--flow--min--cut theorem.
+
 \section{Augmenting flow algorithms}
 
 \subsection{Ford--Fulkerson}