\chapter{Results and Conclusions}
\label{chap:results}

This chapter outlines the results of the experiments performed using the methods outlined in the preceding chapter. For each section in this chapter an brief comparison of the similarity exhibited by each feature is given followed by lower dimensional representations of the feature space. Finally, plots examining the quality of the visualisation are given.

Each of the experiments in this section used a single image from each patient in the full set of 360 images of both left, right and mediolateral oblique and craniocaudal views chosen at random. This meant was so as not to overrepresent a single patient for the real dataset. Therefore, 90 real mammograms in total were used for each experiment. The set of phantom mammograms used in this project consisted of 60 images but were divided into 6 groups corresponding to only a single ``case". Because of this limitation a single breast phantom from each case was selected at random to be included in the data used for the experiments, bringing the total number of images used in all experiments to 96.

Features extracted from both sets of images are compared using a two sample Kolmogorov-Smirnov (KS) test and are reported in the relevant section. The three dimensionality reduction algorithms were tested with each of the different feature sets and visualisations of the feature space in both 2 and 3 dimensions are shown for each.

Two different kinds of quality measure derived from the co-ranking matrix were computed for each of the visualisations shown in this section. The quality metrics used were trustworthiness \& continuity and the local continuity meta-criteria up to a neighbourhood of size $k=50$.

\section{Blob features}
The results for the two sample KS test between the real and synthetic datasets for blob features detected over 10 scales are shown in table \ref{table:blob_features_ks}. An explanation of what each type of feature means can be found in section \ref{subsubsec:implementation-blob-features}. 2D projections of the blob feature space as produced by t-SNE, Isomap and locally linear embedding are shown in figures \ref{fig:blob_SNE_mapping}, \ref{fig:blob_iso_mapping}, and \ref{fig:blob_LLE_mapping} respectively.

In each of the figures the real mammograms are shown as coloured dots and the breast phantoms are shown as coloured triangles. The labels corresponding to the colours used in each of the visualisations are the image's BIRADS class (in the case of real mammograms) or the volumetric breast density (VBD) (in the case of breast phantoms).

Generally speaking the phantom mammograms are shown to be very different from the real mammograms features used. The size of blobs detected in the phantom mammograms are generally larger and contain far fewer small scale blobs when compared to real mammograms. This is reflected in the projections for this feature space.

In the t-SNE projections it can be seen that all of the phantom blobs, regardless of VBD, are grouped towards the bottom of the distribution which contains a relatively high proportion of BIRADS risk class 3 \& 4 images. The projections produced by Isomap and LLE generally separate the breast phantoms for the original data more than with t-SNE. Several of the real mammograms with features that match those of the breast phantoms are also interspersed with the phantom mammograms. This can be seen best in the 2D projection produced by Isomap in figure \ref{fig:blob_iso_mapping}. These real mammograms are ones which tend to have a higher number of larger blobs detected than real mammograms in general.

\begin{table}[H]
\label{table:blob_features_ks}
\centering
\primitiveinput{tables/blob_features_ks}
\caption{Comparison of the Kolmogorov-Smirnov test results for each feature generated from the radii of blobs detected in an image between real and phantom mammograms.}
\end{table}

The quality evaluation for each of the visualisations are shown in figures \ref{fig:TC_2d_blobs}, \ref{fig:TC_3d_blobs}, and \ref{fig:LCMC_blobs}. The quality metrics shown in these figures suggest that the 2D visualisation produced by Isomap is the most trustworthy, but t-SNE has better continuity. In the case of the 3D projections Isomap provides a better visualisation with it topping both criteria. For both projections Isomap show better performance in terms of the LCMC, but is overtaken for larger values of $k$ in 2D projections by t-SNE and in 3D projections by LLE.

\clearpage

\begin{figure}[H]
	\centering
	\subfigure{\includegraphics[width=0.4\textwidth]{figures/mappings/blob_SNE_mapping_2d.png}}
	\subfigure{\includegraphics[width=0.49\textwidth]{figures/mappings/blob_SNE_mapping_3d.png}}
	\caption{2D \& 3D projections of the blob feature space produced by the t-SNE algorithm with a learning rate of 200 and perplexity of 20.}\label{fig:blob_SNE_mapping}
\end{figure}

\begin{figure}[H]
	\centering
	\subfigure{\includegraphics[width=0.4\textwidth]{figures/mappings/blob_iso_mapping_2d.png}}
	\subfigure{\includegraphics[width=0.49\textwidth]{figures/mappings/blob_iso_mapping_3d.png}}
	\caption{2D \& 3D projections of the blob feature space  produced by the Isomap algorithm with 4 neighbours.}\label{fig:blob_iso_mapping}
\end{figure}

\begin{figure}[H]
	\centering
	\subfigure{\includegraphics[width=0.4\textwidth]{figures/mappings/blob_lle_mapping_2d.png}}
	\subfigure{\includegraphics[width=0.49\textwidth]{figures/mappings/blob_lle_mapping_3d.png}}
	\caption{2D \& 3D projections of the blob feature space produced by the LLE algorithm with 4 neighbours.}\label{fig:blob_LLE_mapping}
\end{figure}
\clearpage

\clearpage
\begin{figure}[H]
	\centering
	\subfigure{\includegraphics[width=0.49\textwidth]{figures/quality_measures/blob_trustworthiness_2d.png}}
	\subfigure{\includegraphics[width=0.49\textwidth]{figures/quality_measures/blob_continuity_2d.png}}
	\caption{Trustworthiness (left) and continuity (right) of the 2D projections produced from blob features.}\label{fig:TC_2d_blobs}
\end{figure}

\begin{figure}[H]
	\centering
	\subfigure{\includegraphics[width=0.49\textwidth]{figures/quality_measures/blob_trustworthiness_3d.png}}
	\subfigure{\includegraphics[width=0.49\textwidth]{figures/quality_measures/blob_continuity_3d.png}}
	\caption{Trustworthiness (left) and continuity (right) of the 3D projections produced from blob features.}\label{fig:TC_3d_blobs}
\end{figure}

\begin{figure}[H]
	\centering
	\subfigure{\includegraphics[width=0.49\textwidth]{figures/quality_measures/blob_lcmc_2d.png}}
	\subfigure{\includegraphics[width=0.49\textwidth]{figures/quality_measures/blob_lcmc_3d.png}}
	\caption{LCMC of both the 2D projection (left) and 3D projection (right) of the feature space for blobs.}\label{fig:LCMC_blobs}
\end{figure}
\clearpage

\subsection{Line features}
Results for the KS two sample test between the two feature spaces can be seen in table \ref{table:line_features_ks}. Again, as with blobs, the distributions for real and synthetic mammograms are not in particularly good agreement according to the test. 2D and 3D projections for the line feature space for each of the three different dimensionality reduction algorithms are shown in figures \ref{fig:line_SNE_mapping}, \ref{fig:line_iso_mapping}, and \ref{fig:line_LLE_mapping} respectively.

Note that in these results the value of the min area feature was removed. When using the min area feature all projections would produce two clusters. One smaller cluster would be created containing which contained blobs where the min area was much larger than the rest of the dataset. This is most likely caused by a thresholding issue causing some unwanted smaller lines to be retained for some of the images.

\begin{table}[H]
\label{table:line_features_ks}
\centering
\primitiveinput{tables/line_features_ks}
\caption{Comparison of the Kolmogorov-Smirnov test results for each feature generated from the area of lines detected in an image between real and phantom mammograms.}
\end{table}

The t-SNE projection shows that there is a general transition from images which contain a high mean area of linear structure on the bottom right towards images which have a lower mean area in the top left. In terms of BIRADS risk it can be seen that very low and very high risk classes are grouped towards the bottom right, reflecting that less liner structure is detected using the orientated bins method in low risk classes (because there's very little dense structure present) and high risk classes (because dense areas are mostly large, homogenous blobs not linear structure). As can be clearly seen from the visualisation the synthetic images are grouped towards the lower end reflecting the smaller size of the linear structures detected. The 3D version produced by t-SNE produced a far worse projection with less visible structure.

The projections produced by Isomap produce more clearly show the synthetics begin grouped in with the very low/high risk blobs towards the left of the mapping. For LLE the majority of the blobs are pushed away from the main body of the mapping and closer to some of the outliers of the real mammograms.

Examination of the three quality criteria show that Isomap produces better projections across both the 2D and 2D visualisations. t-SNE produced a 2D mapping which was comparable to Isomap in terms of trustworthiness and continuity, and LCMC. However in the case of the 3D case this was radically worse over all measures.

\begin{figure}[H]
	\centering
	\subfigure{\includegraphics[width=0.4\textwidth]{figures/mappings/line_SNE_mapping_2d.png}}
	\subfigure{\includegraphics[width=0.49\textwidth]{figures/mappings/line_SNE_mapping_3d.png}}
	\caption{2D \& 3D projections of the line feature space produced by the t-SNE algorithm with a learning rate of 200 and perplexity of 20.}\label{fig:line_SNE_mapping}
\end{figure}

\begin{figure}[H]
	\centering
	\subfigure{\includegraphics[width=0.4\textwidth]{figures/mappings/line_iso_mapping_2d.png}}
	\subfigure{\includegraphics[width=0.49\textwidth]{figures/mappings/line_iso_mapping_3d.png}}
	\caption{2D \& 3D projections of the line feature space  produced by the Isomap algorithm with 4 neighbours.}\label{fig:line_iso_mapping}
\end{figure}

\begin{figure}[H]
	\centering
	\subfigure{\includegraphics[width=0.4\textwidth]{figures/mappings/line_lle_mapping_2d.png}}
	\subfigure{\includegraphics[width=0.49\textwidth]{figures/mappings/line_lle_mapping_3d.png}}
	\caption{2D \& 3D projections of the line feature space produced by the LLE algorithm with 4 neighbours.}\label{fig:line_LLE_mapping}
\end{figure}
\clearpage

\clearpage
\begin{figure}[H]
	\centering
	\subfigure{\includegraphics[width=0.49\textwidth]{figures/quality_measures/line_trustworthiness_2d.png}}
	\subfigure{\includegraphics[width=0.49\textwidth]{figures/quality_measures/line_continuity_2d.png}}
	\caption{Trustworthiness (left) and continuity (right) of the 2D projections produced from line features.}\label{fig:TC_2d_lines}
\end{figure}

\begin{figure}[H]
	\centering
	\subfigure{\includegraphics[width=0.49\textwidth]{figures/quality_measures/line_trustworthiness_3d.png}}
	\subfigure{\includegraphics[width=0.49\textwidth]{figures/quality_measures/line_continuity_3d.png}}
	\caption{Trustworthiness (left) and continuity (right) of the 3D projections produced from line features.}\label{fig:TC_3d_lines}
\end{figure}

\begin{figure}[H]
	\centering
	\subfigure{\includegraphics[width=0.49\textwidth]{figures/quality_measures/line_lcmc_2d.png}}
	\subfigure{\includegraphics[width=0.49\textwidth]{figures/quality_measures/line_lcmc_3d.png}}
	\caption{LCMC of both the 2D projection (left) and 3D projection (right) of the feature space for lines.}\label{fig:LCMC_lines}
\end{figure}
\clearpage

\begin{figure}[H]
	\label{fig:mammogram-histogram}
	\centering
	\includegraphics[width=0.8\textwidth]{Images/inverted_hist.png}
	\caption{Comparison of the histogram of a real mammogram (blue) against a synthetic mammogram (green). The distribution of intensities are radically different from one another.}
\end{figure}

\subsection{Intensity \& Texture Features}
\label{subsec:results-texture}
Intensity and texture features showed the lowest similarity between the real and phantom datasets. As can be seen in figure \ref{fig:mammogram-histogram} the intensity distribution of the phantom mammograms are nothing like a real mammogram. Because of this difference the results show that the they are clearly in different spaces. The results of the KS two sample test showed that the two distributions generated for both intensity and texture features are completely different. The results of the two sample Kolmogorov-Smirnov test are included in appendix \ref{appendix:ks-test} because the number of features for blobs is based on the number of scales used leading to the a large number (up to 100) of entries.

The feature spaces presented in this section were generated by taking ROIs defined by the blob and line features detected from each of the mammograms by computing intensity and texture features from each image patch. Because blobs are detected across a range of scales the average value for each scale is used leading to a feature space of size $nk$ where $n$ is the number of features and $k$ is the number of scales (e.g. for texture: $n = 4$ and $k = 10$ so it has $40$ features).

The results of performing dimensionality reduction on both of these feature spaces is that in all almost all cases the synthetic mammograms are separated from the real mammograms into an isolated cluster on their own. This is because the values for both types of feature are on average much higher for the breast phantoms than the real mammograms.

For features derived from patches defined by blobs it can be seen that between the real mammograms there is a trend visible in both intensity and textural features which causes some transition between high to low risk. This is particularly visible for intensity features in the 2D plot produced by t-SNE shown in figure \ref{fig:intensity_SNE_mapping} and in the 2D plot for Isomap shown in figure \ref{fig:texture_iso_mapping} for texture features. For both intensity and texture feature LLE was shown to produce two clear clusters. One which contains the synthetics and one which contains the real mammograms. LLE still shows some variation between real in a similar way to what is seen in the projections produced by Isomap and LLE, but the resulting visualisation shows a much clear division.

In the case of intensity features this transition is caused by a higher detection of blobs which are of generally higher intensity. For texture features the trend is explained by a transition for blobs with low contrast and dissimilarity and high homogeneity and energy corresponding to high risk blobs transitioning through to low risk blobs where the reverse is true.

Intensity features derived from lines do show a degree of separation between high and low risk mammograms. High risk mammograms typically have a higher average intensity. This is most clearly seen in the 2D visualisation produced by t-SNE in \ref{fig:intensity_SNE_mapping_lines}. On the other hand the texture features do not appear to show any meaningful relationship between risk classes. LLE shows results that are also very similar to what is seen in blob features and the reasons behind this splitting are much the same.

The quality measures for intensity features derived from blobs show that these visualisations are generally less faithful in comparison to shape based features. Isomap is again a winner here with it producing better visualisations across all measures. Both t-SNE and LLE generally performed much worse than Isomap. The embedding by texture features are much more faithful. The 2D projections show a close call between Isomap and t-SNE. For 3D projections t-SNE is the clear loser, being dramatically worse compared with t-SNE and Isomap.

Quality measures taken from the visualisations for intensity from lines show that the mapping produced by Isomap is better for projections in both two and three dimensions, but that t-SNE is at least comparable to Isomap in 2D. The results for texture features derived from lines show that the t-SNE performance of t-SNE is better for 2D. In the three dimensions Isomap is again a clear winner.

%------------------------------------------------------------------------------------
% Blob intensity and texture features
%------------------------------------------------------------------------------------

\clearpage
\begin{figure}[H]
	\centering
	\subfigure{\includegraphics[width=0.4\textwidth]{figures/mappings/intensity_SNE_mapping_2d.png}}
	\subfigure{\includegraphics[width=0.49\textwidth]{figures/mappings/intensity_SNE_mapping_3d.png}}
	\caption{2D \& 3D projections of the intensity feature space produced by the t-SNE algorithm with a learning rate of 200 and perplexity of 20.}\label{fig:intensity_SNE_mapping}
\end{figure}

\begin{figure}[H]
	\centering
	\subfigure{\includegraphics[width=0.4\textwidth]{figures/mappings/intensity_iso_mapping_2d.png}}
	\subfigure{\includegraphics[width=0.49\textwidth]{figures/mappings/intensity_iso_mapping_3d.png}}
	\caption{2D \& 3D projections of the intensity feature space generated from blobs produced by the Isomap algorithm with 4 neighbours.}\label{fig:intensity_iso_mapping}
\end{figure}

\begin{figure}[H]
	\centering
	\subfigure{\includegraphics[width=0.4\textwidth]{figures/mappings/intensity_lle_mapping_2d.png}}
	\subfigure{\includegraphics[width=0.49\textwidth]{figures/mappings/intensity_lle_mapping_3d.png}}
	\caption{2D \& 3D projections of the intensity feature space generated from blobs produced by the LLE algorithm with 4 neighbours.}\label{fig:intensity_LLE_mapping}
\end{figure}
\clearpage

% Quality for blob intensity features
%------------------------------------------------------------------------------------
\clearpage
\begin{figure}[H]
	\centering
	\subfigure{\includegraphics[width=0.49\textwidth]{figures/quality_measures/intensity_trustworthiness_2d.png}}
	\subfigure{\includegraphics[width=0.49\textwidth]{figures/quality_measures/intensity_continuity_2d.png}}
	\caption{Trustworthiness (left) and continuity (right) of the 2D projections produced from intensity features from blobs.}\label{fig:TC_2d_intensity}
\end{figure}

\begin{figure}[H]
	\centering
	\subfigure{\includegraphics[width=0.49\textwidth]{figures/quality_measures/intensity_trustworthiness_3d.png}}
	\subfigure{\includegraphics[width=0.49\textwidth]{figures/quality_measures/intensity_continuity_3d.png}}
	\caption{Trustworthiness (left) and continuity (right) of the 3D projections produced from intensity features from blobs.}\label{fig:TC_3d_intensity}
\end{figure}

\begin{figure}[H]
	\centering
	\subfigure{\includegraphics[width=0.49\textwidth]{figures/quality_measures/intensity_lcmc_2d.png}}
	\subfigure{\includegraphics[width=0.49\textwidth]{figures/quality_measures/intensity_lcmc_3d.png}}
	\caption{LCMC of both the 2D projection (left) and 3D projection (right) of the feature space for intensity from blobs.}\label{fig:LCMC_intensity}
\end{figure}
\clearpage

\clearpage
\begin{figure}[H]
	\centering
	\subfigure{\includegraphics[width=0.4\textwidth]{figures/mappings/texture_SNE_mapping_2d.png}}
	\subfigure{\includegraphics[width=0.49\textwidth]{figures/mappings/texture_SNE_mapping_3d.png}}
	\caption{2D \& 3D projections of the texture feature space generated from blobs generated from blobs produced by the t-SNE algorithm with a learning rate of 200 and perplexity of 20.}\label{fig:texture_SNE_mapping}
\end{figure}

\begin{figure}[H]
	\centering
	\subfigure{\includegraphics[width=0.4\textwidth]{figures/mappings/texture_iso_mapping_2d.png}}
	\subfigure{\includegraphics[width=0.49\textwidth]{figures/mappings/texture_iso_mapping_3d.png}}
	\caption{2D \& 3D projections of the texture feature space generated from blobs produced by the Isomap algorithm with 4 neighbours.}\label{fig:texture_iso_mapping}
\end{figure}

\begin{figure}[H]
	\centering
	\subfigure{\includegraphics[width=0.4\textwidth]{figures/mappings/texture_lle_mapping_2d.png}}
	\subfigure{\includegraphics[width=0.49\textwidth]{figures/mappings/texture_lle_mapping_3d.png}}
	\caption{2D \& 3D projections of the texture feature space generated from blobs produced by the LLE algorithm with 4 neighbours.}\label{fig:texture_LLE_mapping}
\end{figure}
\clearpage

% Quality for blob texture features
%------------------------------------------------------------------------------------

\clearpage
\begin{figure}[H]
	\centering
	\subfigure{\includegraphics[width=0.49\textwidth]{figures/quality_measures/texture_trustworthiness_2d.png}}
	\subfigure{\includegraphics[width=0.49\textwidth]{figures/quality_measures/texture_continuity_2d.png}}
	\caption{Trustworthiness (left) and continuity (right) of the 2D projections produced from texture features from blobs.}\label{fig:TC_2d_texture}
\end{figure}

\begin{figure}[H]
	\centering
	\subfigure{\includegraphics[width=0.49\textwidth]{figures/quality_measures/texture_trustworthiness_3d.png}}
	\subfigure{\includegraphics[width=0.49\textwidth]{figures/quality_measures/texture_continuity_3d.png}}
	\caption{Trustworthiness (left) and continuity (right) of the 3D projections produced from texture features from blobs.}\label{fig:TC_3d_texture}
\end{figure}

\begin{figure}[H]
	\centering
	\subfigure{\includegraphics[width=0.49\textwidth]{figures/quality_measures/texture_lcmc_2d.png}}
	\subfigure{\includegraphics[width=0.49\textwidth]{figures/quality_measures/texture_lcmc_3d.png}}
	\caption{LCMC of both the 2D projection (left) and 3D projection (right) of the feature space for texture.}\label{fig:LCMC_texture}
\end{figure}
\clearpage

%------------------------------------------------------------------------------------
% Line intensity and texture features
%------------------------------------------------------------------------------------

\clearpage
\begin{figure}[H]
	\centering
	\subfigure{\includegraphics[width=0.4\textwidth]{figures/mappings/line_intensity_SNE_mapping_2d.png}}
	\subfigure{\includegraphics[width=0.49\textwidth]{figures/mappings/line_intensity_SNE_mapping_3d.png}}
	\caption{2D \& 3D projections of the intensity feature space for lines produced by the t-SNE algorithm with a learning rate of 200 and perplexity of 20.}\label{fig:intensity_SNE_mapping_lines}
\end{figure}

\begin{figure}[H]
	\centering
	\subfigure{\includegraphics[width=0.4\textwidth]{figures/mappings/line_intensity_iso_mapping_2d.png}}
	\subfigure{\includegraphics[width=0.49\textwidth]{figures/mappings/line_intensity_iso_mapping_3d.png}}
	\caption{2D \& 3D projections of the intensity feature space generated from lines produced by the Isomap algorithm with 4 neighbours.}\label{fig:intensity_iso_mapping_lines}
\end{figure}

\begin{figure}[H]
	\centering
	\subfigure{\includegraphics[width=0.4\textwidth]{figures/mappings/line_intensity_lle_mapping_2d.png}}
	\subfigure{\includegraphics[width=0.49\textwidth]{figures/mappings/line_intensity_lle_mapping_3d.png}}
	\caption{2D \& 3D projections of the intensity feature space for lines generated from blobs produced by the LLE algorithm with 4 neighbours.}\label{fig:intensity_LLE_mapping_lines}
\end{figure}
\clearpage

% Quality for line intensity features
%------------------------------------------------------------------------------------
\clearpage
\begin{figure}[H]
	\centering
	\subfigure{\includegraphics[width=0.49\textwidth]{figures/quality_measures/line_intensity_trustworthiness_2d.png}}
	\subfigure{\includegraphics[width=0.49\textwidth]{figures/quality_measures/line_intensity_continuity_2d.png}}
	\caption{Trustworthiness (left) and continuity (right) of the 2D projections produced from intensity features from lines.}\label{fig:TC_2d_intensity}
\end{figure}

\begin{figure}[H]
	\centering
	\subfigure{\includegraphics[width=0.49\textwidth]{figures/quality_measures/line_intensity_trustworthiness_3d.png}}
	\subfigure{\includegraphics[width=0.49\textwidth]{figures/quality_measures/line_intensity_continuity_3d.png}}
	\caption{Trustworthiness (left) and continuity (right) of the 3D projections produced from intensity features from lines.}\label{fig:TC_3d_intensity}
\end{figure}

\begin{figure}[H]
	\centering
	\subfigure{\includegraphics[width=0.49\textwidth]{figures/quality_measures/line_intensity_lcmc_2d.png}}
	\subfigure{\includegraphics[width=0.49\textwidth]{figures/quality_measures/line_intensity_lcmc_3d.png}}
	\caption{LCMC of both the 2D projection (left) and 3D projection (right) of the feature space for intensity from lines.}\label{fig:LCMC_intensity}
\end{figure}
\clearpage

\clearpage
\begin{figure}[H]
	\centering
	\subfigure{\includegraphics[width=0.4\textwidth]{figures/mappings/lines_texture_SNE_mapping_2d.png}}
	\subfigure{\includegraphics[width=0.49\textwidth]{figures/mappings/lines_texture_SNE_mapping_3d.png}}
	\caption{2D \& 3D projections of the texture feature space generated generated from lines produced by the t-SNE algorithm with a learning rate of 200 and perplexity of 20.}\label{fig:texture_SNE_mapping_lines}
\end{figure}

\begin{figure}[H]
	\centering
	\subfigure{\includegraphics[width=0.4\textwidth]{figures/mappings/lines_texture_iso_mapping_2d.png}}
	\subfigure{\includegraphics[width=0.49\textwidth]{figures/mappings/lines_texture_iso_mapping_3d.png}}
	\caption{2D \& 3D projections of the texture feature space generated from lines produced by the Isomap algorithm with 4 neighbours.}\label{fig:texture_iso_mapping_lines}
\end{figure}

\begin{figure}[H]
	\centering
	\subfigure{\includegraphics[width=0.4\textwidth]{figures/mappings/lines_texture_lle_mapping_2d.png}}
	\subfigure{\includegraphics[width=0.49\textwidth]{figures/mappings/lines_texture_lle_mapping_3d.png}}
	\caption{2D \& 3D projections of the texture feature space generated from lines produced by the LLE algorithm with 4 neighbours.}\label{fig:texture_LLE_mapping_lines}
\end{figure}
\clearpage

% Quality for line texture features
%------------------------------------------------------------------------------------

\clearpage
\begin{figure}[H]
	\centering
	\subfigure{\includegraphics[width=0.49\textwidth]{figures/quality_measures/lines_texture_trustworthiness_2d.png}}
	\subfigure{\includegraphics[width=0.49\textwidth]{figures/quality_measures/lines_texture_continuity_2d.png}}
	\caption{Trustworthiness (left) and continuity (right) of the 2D projections produced from texture features from lines.}\label{fig:TC_2d_texture}
\end{figure}

\begin{figure}[H]
	\centering
	\subfigure{\includegraphics[width=0.49\textwidth]{figures/quality_measures/lines_texture_trustworthiness_3d.png}}
	\subfigure{\includegraphics[width=0.49\textwidth]{figures/quality_measures/lines_texture_continuity_3d.png}}
	\caption{Trustworthiness (left) and continuity (right) of the 3D projections produced from texture features from lines.}\label{fig:TC_3d_texture}
\end{figure}

\begin{figure}[H]
	\centering
	\subfigure{\includegraphics[width=0.49\textwidth]{figures/quality_measures/lines_texture_lcmc_2d.png}}
	\subfigure{\includegraphics[width=0.49\textwidth]{figures/quality_measures/lines_texture_lcmc_3d.png}}
	\caption{LCMC of both the 2D projection (left) and 3D projection (right) of the feature space from lines for texture.}\label{fig:LCMC_texture}
\end{figure}
\clearpage

\section{Conclusions}
In summary, it can be concluded that the synthetic mammograms used as part of this experiment are not significantly related to real mammograms in terms of the features derived during this study. The synthetic mammograms appear to be closest to real mammograms in terms of the shape of the structures present in the image. The best results from this experiment were derived from the feature space of multi-scale blobs. Line features also seemed to positively show the the real and synthetic mammograms are at least in the same space in terms of shape. However, texture and intensity features derived from either of the features clearly show that the two datasets are not in the same intensity space. The two sample KS test confirms that the features created from both datasets are statistically not drawn from the same distribution and therefore we must conclude that they are, for all features presented here, effectively different.

This conclusion roughly correlates with the limitations discussed by the authors of the synthetic data \cite{bakic2002mammogram1, bakic2002mammogram2, bakic2003mammogram3}. They state that the synthetic mammograms are closest to real mammograms in terms of shape but are not so close in terms of intensity and texture. In the experiments with shape features presented here the major difference between the real and synthetic mammograms was caused by a lack of small scale structure being detected in the synthetic mammograms. This causes them to be grouped in with mammograms which are deemed to be of higher risk, regardless of their effective risk, because of the lack of small blob counts detected. The same can be said for line features where the number and size of lines is generally smaller, regardless of the risk.

The quality evaluation of the visualisations produced suggests that Isomap produces visualisations which best capture the local neighbourhood structure of mappings. t-SNE appears to often show similar levels of faithfulness in compared to Isomap in two dimensions, but was shown to be much worse for three dimensions.


\vsssub
\subsubsection{Track output post-processor} \label{sec:ww3trck}
\vsssub

\proddefH{ww3\_trck}{w3trck}{ww3\_trck.ftn}
\proddeff{Input}{ww3\_trck.inp}{Traditional configuration file.}{10} (App.~\ref{sec:config191})
\proddefa{track\_o.ww3}{Raw track output data.}{11}
\proddeff{Output}{standard out}{Formatted output of program.}{6}
\proddefa{track.ww3}{Formatted data file.}{51}

\vspace{\baselineskip}
\noindent
This post-processor will convert the raw track output data to an integer
compressed formatted file. The file contains the following header records :

\begin{list}{$\bullet$}{\itemsep 0mm \parsep 0mm}
\item File identifier (character string of length 34).
\item Number of frequencies and directions, first direction and directional
      increment (radians, oceanographic convention).
\item Radian frequencies of each frequency bin.
\item Corresponding directional bin size times frequency bin size to obtain
      discrete energy per bin.
\end{list}

\noindent
For each output point varying in time and position, the following records are printed :
\begin{list}{$\bullet$}{\itemsep 0mm \parsep 0mm}
\item Date and time in {\tt yyyymmdd hhmmss} format, longitude and latitude in
      degrees, and a status identifier `{\F ice}', `{\F lnd}' or `{\F
      sea}'. The following two records are written only for sea points.
\item Water depth in meters, current and wind u and v components in meters per
      second, friction velocity in meters per second, air-sea temperature
      difference in degrees centigrade and scale factor for spectrum.
\item The entire spectrum in integer packed format (can be read using free
      format).
\end{list}

\pb