\chapter{Results}
-We show some basic results for different parameters.
+In the previous chapters we have carefully constructed a method always
+trying to argue how the choices we make can have a positive impact on
+the image restoration results. The anisotropic total variation is the
+main advance from my project work \cite{project}, and we would like to
+see what its effects are.
\section{Anisotropy}
-We have a few figures showing results with predescribed tensors, so that
-we know they actually have an effect.
-
-We show how the blurring and integration scale parameters have an
-effect, maybe the fingerprint is a good example. Maybe a depiction of
-the tensors is a good idea here.
-
-We show how it affects regularization, and contrast loss, using a 1D
-slice through a noisy fingerprint image.
-
\begin{figure}
\centering
- \begin{subfigure}[b]{0.40\textwidth}
+ \begin{subfigure}[b]{0.30\textwidth}
\centering
\includegraphics[width=\textwidth]{fig/circle100n.png}
\caption{
}
\end{subfigure}
~
- \begin{subfigure}[b]{0.40\textwidth}
+ \begin{subfigure}[b]{0.30\textwidth}
\centering
\includegraphics[width=\textwidth]{fig/anisotropy_circle.png}
\caption{
- Anisotropically restored.
+ Uniform tensor.
+ }
+ \label{fig:tensor_experiment_unif}
+ \end{subfigure}
+ ~
+ \begin{subfigure}[b]{0.30\textwidth}
+ \centering
+ \includegraphics[width=\textwidth]{fig/circle100a.png}
+ \caption{
+ Normal tensor.
+ }
+ \end{subfigure}
+ \caption{
+ Funny caption. Parameters.
+ }
+ \label{fig:tensor_experiment}
+\end{figure}
+
+The anisotropy was introduced into the total variation in order to
+lessen the regularization done across what we know, or at least are
+pretty sure to be edges in the image. Before considering the anisotropy
+tensor construction described earlier, we will look at how a simple
+predescribed tensor affects the regularization. Imagine a tensor which
+is a diagonal matrix $A = \operatorname{diag}(1, \epsilon)$, where
+$\epsilon \ll 1$ is small. This would result in us down-weighting the
+size of $\nabla \tilde{f}_\sigma$ in the $y$-direction, and thus
+regularizing mostly in the $x$-direction.
+
+The results of this experiment can be seen in Figure
+\ref{fig:tensor_experiment}, where a noisy picture of a circle has been
+restored in two different ways, first with a uniform anisotropy tensor
+$A = \operatorname{diag}(1, \epsilon)$, and then with the tensor
+described in Section \ref{sec:anisotropy_tensor}. We see in Figure
+\ref{fig:tensor_experiment_unif} that the uniform tensor gives a strong
+smoothing in the $x$-direction, while no apparent smoothing in the
+$y$-direction.
+
+In the construction of the tensor in Section
+\ref{sec:anisotropy_tensor}, we have several parameters. The \emph{noise
+scale} $\sigma$ controls the smoothing done before $\nabla
+\tilde{f}_\sigma$ is calculated, and thus controls how sensitive we are
+to noise in the input image $f$. All details of scale less than $\sigma$
+will be considered to be noise.
+\begin{figure}
+ \centering
+ \begin{subfigure}[t]{0.30\textwidth}
+ \centering
+ \includegraphics[width=\textwidth]{fig/gradlow.png}
+ \caption{
+ Low smoothing.
+ }
+ \end{subfigure}
+ ~
+ \begin{subfigure}[t]{0.30\textwidth}
+ \centering
+ \includegraphics[width=\textwidth]{fig/gradnscale.png}
+ \caption{
+ Higher noise scale.
+ }
+ \label{fig:scale_comp_high}
+ \end{subfigure}
+ ~
+ \begin{subfigure}[t]{0.30\textwidth}
+ \centering
+ \includegraphics[width=\textwidth]{fig/gradiscale.png}
+ \caption{
+ Low noise scale, but high integration scale.
+ }
+ \label{fig:scale_comp_int}
+ \end{subfigure}
+ \caption{
+ Funny caption. Parameters.
+ }
+ \label{fig:scale_comp}
+\end{figure}
+
+Figure \ref{fig:scale_comp} shows a constructed zebra pattern of
+increasing width. The anisotropy should in theory help reduce contrast
+loss in this situation, by reducing the regularization done in the
+$x$-direction, but there is the question of at which scale it considers
+the lines to be detail instead of noise.
+
+In Figure \ref{fig:scale_comp_high} the noise scale is increased and the
+tensor no longer consider the left-most part of the image to be details,
+but rather noise.
+
+The \emph{integration scale} $\rho$ of \eqref{eq:s_def} represents the
+size of the structures we want to affect our anisotropy tensor. Although
+all edges and details larger than the noise scale will affect the edge
+detector $\nabla \tilde{f}_\sigma$, but the smoothing after of scale
+$\rho$ may ignore them depending on their size. In Figure
+\ref{fig:scale_comp_int} a high value for $\rho$ means that the
+structure found in the inner square is almost completely ignored in
+favour of the larger, more coherent structure around.
+
+\begin{figure}
+ \centering
+ \begin{subfigure}[t]{0.30\textwidth}
+ \centering
+ \includegraphics[width=\textwidth]{fig/lenaedge.png}
+ \caption{
+ Largest structure tensor eigenvalue.
+ }
+ \label{fig:lena_process_edge}
+ \end{subfigure}
+ ~
+ \begin{subfigure}[t]{0.30\textwidth}
+ \centering
+ \includegraphics[width=\textwidth]{fig/lenatensor.png}
+ \caption{
+ The anisotropy tensor eigenvector and eigenvalues.
+ }
+ \label{fig:lena_process_tensor}
+ \end{subfigure}
+ ~
+ \begin{subfigure}[t]{0.30\textwidth}
+ \centering
+ \includegraphics[width=\textwidth]{fig/lenarest.png}
+ \caption{
+ Restored Lena.
}
+ \label{fig:lena_process_restored}
\end{subfigure}
\caption{
Funny caption. Parameters.
}
+ \label{fig:lena_process}
\end{figure}
+It is often useful to look at intermediate results to better understand
+what happens and how the parameters affect our end result. In Figure
+\ref{fig:lena_process} we show three stages of our image restoration
+algorithm. Figure \ref{fig:lena_process_edge} shows the larges
+eigenvalue of our structure tensor, and is thus an indicator of where
+the edge structures are.
+
+Figure \ref{fig:lena_process_tensor} is a depiction of the anisotropy
+tensor in some of the points in the image, shown as eigenvectors with
+their corresponding eigenvalues as their length. We see that across
+edges, \fixme{what!?}
+
+Finally, Figure \ref{fig:lena_process_restored} shows the final restored
+image, and we see that heavy modifications have been made.
+
\begin{figure}
\centering
\begin{subfigure}[t]{0.30\textwidth}
\caption{
Restored without much anisotropy.
}
+ \label{fig:finger_contrast_tv}
\end{subfigure}
~
\begin{subfigure}[t]{0.30\textwidth}
\caption{
Restored with much anisotropy.
}
+ \label{fig:finger_contrast_atv}
\end{subfigure}
\caption{
Funny caption. Parameters.
}
+ \label{fig:finger_contrast}
\end{figure}
+We have claimed that the anisotropic total variation will reduce some of
+the contrast loss one can encounter with regular total variation
+regularization. An example of this can be seen in Figure
+\ref{fig:finger_contrast} where a portion of a fingerprint with
+added Gaussian noise has been restored in two different ways. We see
+that when the anisotropy is increased, more of the contrast between dark
+and light portions of the fingerprint is retained. This can also be seen
+in Figure \ref{fig:contrast_plot} where a one-dimensional slice has been
+taken through Figure \ref{fig:finger_contrast_tv} and
+\ref{fig:finger_contrast_atv}. Although much of the same structure is
+found for this particular regularization parameter $\beta$, the peaks
+are much higher for the anisotropic total variation.
+
\begin{figure}
\centering
\input{fig/contrast_plot}
\caption{Showing some contrast loss.}
+ \label{fig:contrast_plot}
\end{figure}
+So far we have just inspected the restored image visually to judge the
+results of the method. Depending on the application, we might have
+different wishes for the results. In many cases we just want to remove the
+noise $\delta$ in \fixme{ref} where the captured image $f$ is assumed to
+consist of an actual image $u$ and an additional noise term $\delta$
+such that $f = u + \delta$. The noise removed by the restoration method,
+$\hat{\delta} = f - u^*$ is called the method noise. Any assumptions of
+properties we have on $\delta$, we optimally want $\hat{\delta}$ to
+fulfill as well. Thus if we assumed independently distributed Gaussian
+noise, we want to remove that kind of noise. An indication of problems
+can in that case be if the method noise contains a lot of the details of
+the image.
+
+Figure \ref{fig:lena_method_noise} shows the results after restoring the
+noisy Lena picture in Figure \fixme{noisylena}, with different
+parameters, and their method noise. We note that the method noise of the
+regular total variation method shows and outline of Lena, which
+indicates some detail loss. Further, we see that the method noise of the
+anisotropically restored image, has less details.
+
+\begin{figure}
+ \centering
+ \begin{subfigure}[t]{0.30\textwidth}
+ \centering
+ \includegraphics[width=\textwidth]{fig/lena_tv.png}
+ \end{subfigure}
+ ~
+ \begin{subfigure}[t]{0.30\textwidth}
+ \centering
+ \includegraphics[width=\textwidth]{fig/lena_atv.png}
+ \end{subfigure}
+ ~
+ \begin{subfigure}[t]{0.30\textwidth}
+ \centering
+ \includegraphics[width=\textwidth]{fig/lena_atv72.png}
+ \end{subfigure}
+
+ \centering
+ \begin{subfigure}[t]{0.30\textwidth}
+ \centering
+ \includegraphics[width=\textwidth]{fig/lena_method_tv.png}
+ \caption{
+ Regular total variation.
+ }
+ \end{subfigure}
+ ~
+ \begin{subfigure}[t]{0.30\textwidth}
+ \centering
+ \includegraphics[width=\textwidth]{fig/lena_method_atv.png}
+ \caption{
+ Anisotropic total variation.
+ }
+ \end{subfigure}
+ ~
+ \begin{subfigure}[t]{0.30\textwidth}
+ \centering
+ \includegraphics[width=\textwidth]{fig/lena_method_atv72.png}
+ \caption{
+ Size 72 neighborhood.
+ }
+ \end{subfigure}
+ \caption{
+ Funny caption. Parameters.
+ }
+ \label{fig:lena_method_noise}
+\end{figure}
+
+\begin{figure}
+ \centering
+ \input{fig/factory/lena_gamma_seq/corr.tex}
+ \caption{
+ The correlation between the method noise and the noisy image for
+ different anisotropy parameters $\gamma$.
+ }
+\end{figure}
+
\section{Neighborhood}
We show how the different neighborhoods work. Here we might make some
nice, very contrived examples, maybe. To show that both too small and
too big is bad.
+The neighborhoods were introduced as a way to describe the discrete set
+of lines $\mathcal{L}_D$ in the discretization of the regularization
+term in Section \ref{sec:disc_regularization}. And as discussed, we want
+the stencil to have many short edges, such that the angular
+differences $\Delta \phi$, the inter-line distances $\Delta \rho$ and the
+edge lengths $e$ are ``small.''
+
+Figure \ref{fig:circle_arti} shows a circle that has been strongly
+regularized using a 8-point stencil. The circle has clearly been
+deformed to look more like an octagon. This is due to the fact that the
+8-point stencil measures horizontal, vertical and 45\textdegree{} lines
+exactly in the Cauhcy--Crofton formula, while the length of other lines
+overestimated as the sum of their three components.
+
\begin{figure}
\centering
\begin{subfigure}[b]{0.40\textwidth}
\caption{
Not so funny caption. Include some parameters.
}
+ \label{fig:circle_arti}
\end{figure}
-\section{Comparison}
-
-Maybe? Compare with the regular total variation method, and anisotropic
-diffusion. Question is, where do we get our anisotropic diffusion
-results from? Code our own? Or use some popular implementation?
-
projects / master / commitdiff