
\chapter{What is OODA?\label{chap1A:What-is-OODA?}}

The fields of human endeavor currently known as \emph{statistics},
\emph{data science} and \emph{data analytics} have been radically
transformed over the recent past. These transformations have been
driven simultaneously by a massive increase in computational capabilities
coupled with a rapidly growing scientific appetite for ever deeper
understanding and insights. The notion of forming a \emph{\index{data matrix}data
matrix} provides a useful paradigm for understanding important aspects
of how these fields are evolving. In particular, the currently popular
context of \emph{Big Data} has several quite different facets, ranging
from \emph{low dimension high sample size} areas (the basis of classical
mathematical statistical thought, which is perhaps typified by sample
survey and census data), through both high dimension and sample sizes
(common for internet scale data sets of many types), and on to \emph{high
dimension low sample size\index{High Dimension Low Sample Size}}
contexts (frequently encountered in areas such as genetics, medical
imaging and other types of extremely rich but relatively expensive
measurements). The pressing need to analyze data in this wide array
of contexts has generated many exciting new ideas and approaches.

Yet a deeper look into these developments suggests that the organization
of data into a matrix may itself be imposing limitations. In particular,
there is a growing realization that the challenges presented by Big
Data are being eclipsed by the perhaps far greater challenges of \emph{Complex
Data}, which are typically not easily represented as an unconstrained
matrix of numbers. \emph{Object Oriented Data Analysis} (OODA) provides
a useful general framework for the consideration of many types of
Complex Data. It is deliberately intended to be particularly useful
in the analysis of data in complicated situations, diverse examples
of which are given in the first two chapters. The phrase \index{OODA terminology}OODA
in this context was coined by \citet{WangMarron2007}. An overview
of the area was given in \citet{MarronAlonso2014}. For more discussion
of Big Data and its relation to statistics see \citet{CarmichaelMarron2017DataSciStat},
and see many interesting viewpoints in the special issue edited by
\citet{Sangalli2018BigData}.

The OODA viewpoint is easily understood through taking \emph{data
objects\index{data objects|(}} to be the \emph{atoms} of a statistical
analysis, where atom is meant in the sense of elementary particle,
studied in several contexts of increasing complexity:
\begin{itemize}
\item In a first course in statistics atoms are numbers, and the goal is
to develop methods for understanding of variation in populations of
numbers.
\item A more advanced course, termed \emph{multivariate analysis} in the
statistical culture, generalizes the atoms, i.e. the data objects
from numbers to vectors and involves a host of methods for managing
uncertainty in that context. For example see \citet{MardiaKentBibby1979MultivarAnal},
\citet{Muirhead1982Multivar} and \citet{Koch2014Book} (for a more
up to date treatment). 
\item At the time of this writing a fashionable area in statistics is \emph{\index{Functional Data Analysis|(}Functional
Data Analysis} (FDA), where the goal is to analyze the variation in
a population of curves. A good introduction to this vibrant research
area, where functions are the data objects, can be found in \citet{RamsaySilverman2002,RamsaySilverman2005}
. A case study, illustrating many of the basic concepts of FDA which
are useful for understanding OODA is given in Section \ref{sec:Curves-as-Data}.
\item OODA provides the next step in terms of complexity of atoms of a statistical
analysis to a wide array of more complicated objects. The important
example of shapes as data objects is considered in the case study
of Section \ref{sec:Shapes-as-Data}. A wide variety of other examples,
which highlight the breadth of OODA, appears in Chapter \ref{chap1B:Breadth-of-OODA}. 
\end{itemize}
Note that each of the above areas can be thought of as containing
the preceding ones as special cases. E.g. multivariate analysis is
the case of FDA where the functions are discretely supported. Similarly
multivariate analysis and FDA are special cases of OODA. In later
chapters it is useful to recall that OODA includes these predecessors
as special cases. This is because often simple multivariate examples
are used for maximal clarity in the illustration of concepts and methods,
but the ideas are useful more generally for OODA. 

A good question is: What is the value\index{OODA terminology!value}
added to applied statistics and data science from the concept of OODA
and its attendant terminology? The terminology is based on very substantial
real world experience with a wide variety of complex data sets. A
fact that rapidly becomes clear in the course of interdisciplinary
research is that there frequently are substantial hurdles in terminology.
Especially at the beginning of such endeavors, it can feel like collaborators
are even speaking different languages, so often serious effort needs
to be devoted to the development of a common set of definitions just
to carry on a useful discussion. An added complication is that for
complex data contexts, it is frequently not obvious how to even ``get
a handle on the data''. Usually there are many options available,
which are most effectively decided upon through careful discussion
between domain scientists and statisticians. In such discussions,
the issue of \emph{what should be the data objects?} has proven to
frequently lead to useful choices, thus resulting in an effective
and insightful data analysis.

Real data examples demonstrating data object choices in a variety
of contexts are given in the following and Chapter \ref{chap1B:Breadth-of-OODA}.
In particular, Section \ref{sec:Curves-as-Data} introduces curves
as data objects. A more complex variant involves curves with interesting
variation in phase\index{phase!variation} in place of, or in addition
to, the usual FDA amplitude\index{amplitude!variation} variation
discussed in Section \ref{sec:Amplitude-and-Phase}. A mathematically
deeper case is considered in Section \ref{sec:Shapes-as-Data} where
\emph{shapes} are the data objects which require special treatment
as shapes are most naturally viewed as points on a curved manifold.
Section \ref{sec:Tree-Structured-Data} considers a perhaps even more
challenging data set of tree structured data\index{tree structured data}
objects. The data objects in Section \ref{sec:Sounds-as-Data} are
recordings of sounds, in particular human spoken words, which bring
special challenges in the choice of data objects. Finally, in Section
\ref{sec:Images-as-Data}, a fun example with images of faces as data
objects is considered. 

It is seen that the notion of data objects provides a particularly
useful format for discussing \emph{modes of variation\index{modes of variation|(}}
that give insights about population structure. This term is formally
defined in Section \ref{secBA.1:Foundations_Terminology}, but until
then the meaning should be intuitively clear from the context.

One more general aspect of OODA is that there are frequently three
major phases of this type of data analysis:
\begin{enumerate}
\item \emph{Object Definition}. This is the phase where the fundamental
issue of what should be the data objects is addressed. A number of
examples of this phase are provided in the rest of this chapter and
also in further examples in other sections.
\item \emph{Exploratory Analysis}\index{Exploratory Analysis}. Here the
goal is to find perhaps surprising population structure in data, often
using some type of visualization\index{visualization} method. A wide
variety of examples and methods for exploratory analysis are given
in the rest of this Chapter and in Chapters \ref{chap1B:Breadth-of-OODA},
\ref{chapBB:ExpConfAnal}, \ref{chapD:Preprocessing} - \ref{chapI:Tree-Structured-Data}.
Exploratory analysis frequently only appears sparingly in most classical
statistics courses, but is usually more prominent in machine learning\index{machine learning}.
However it has a strong statistical tradition, going back well before
the ideas nicely summarized in \citet{tukey1977exploratory}.
\item \emph{Confirmatory Analysis}\index{confirmatory analysis}. While
many great discoveries have been made using exploratory methods, it
is also very easy to make discoveries that are not real, in the sense
of being non-replicable sampling artifacts. For this reason it is
very important to validate such discoveries. This critical topic and
many variants of approaches to it are discussed in detail in the very
large classical statistical literature. Some less well known aspects,
that are particularly relevant to OODA are discussed in Chapter \ref{chapL:Confirmatory-Analysis}.
\end{enumerate}
A companion website to this book, containing links to available software,
the Matlab or R programs used to generate most of the figures in this
book, and additional graphics can be found at \citet{Marron2020OODAbookGitPage}.

Further discussion on other ideas and nomenclature related to OODA
can be found in Chapter \ref{chapC:BackgroundRelatedAreas}. Additional
big picture discussion of data science and statistics can be found
in \citet{Marron2017BigData} and \citet{CarmichaelMarron2017DataSciStat}.\index{data objects|)}

\section{Case Study: Curves as Data Objects \label{sec:Curves-as-Data}}

An interesting\index{data objects!curves|(} example of functional
data analysis (viewed here as an important special case of OODA) is
the \emph{\index{Spanish Mortality data|(}Spanish Mortality} data,
first studied from an FDA\index{Functional Data Analysis|)} viewpoint
in Section 2 of \citet{MarronAlonso2014}. Such data sets are available
at the Human Mortality Database of \citet{WilmothMortality2008}.
For a given population (e.g. citizens of one country) mortality data
are generally a matrix with rows and columns indexed by years and
ages. The matrix entries are the chance of a person of each age dying
in the given year, calculated as the number of deaths divided by the
number of people for that year - age pair. Here we study mortality
of males in Spain, mostly because there are interesting features in
the data, due to recent Spanish history. 

This data set provides good illustration of the issue of Object Definition,
because there are several data object choices to be made in the analysis
of this data. First, since these probabilities range over several
orders of magnitude, logarithms are useful to provide good visual
separation across a wide range of scale\index{scale!log}s. Particularly
strong interpretability comes from the choice of $\log_{10}$ of the
probability (e.g. $-2$ corresponds to a probability of $0.01$ as
opposed to about $0.135$ for the natural log). The utility of this
data object choice is demonstrated in Figure \ref{fig:Mortality-Raw},
where the raw probabilities are shown in the left panel (with much
interesting structure missed since this is very nearly 0 for the important
younger age groups) with $\log_{10}$ mortality in the right (highlighting
important contrasts among the younger ages). Second there are two
different ways to turn the matrix of data into functional data. One
is to consider data objects to be curves of mortality as a function
of age, with curves indexed by year. The other is (the matrix transpose)
where the mortality is viewed as a function of year, with data object
curves indexed by age. In this analysis, the former choice is used,
because it gives the best illustration of the usefulness of OODA concepts
and also gives an interesting narrative. The latter choice is considered
in Figure \ref{figP-C:SpanMortLoadCorr}. An analysis that also integrates
female mortality in an interesting way can be found in \citet{Feng2018JIVE}.
The choice here results in $n=95$ curves corresponding to the years
1908-2002. Ages considered here are 0 through 98, since larger ages
are problematic due to occasional small population sizes. The raw
data are shown as overlaid curves in Figure \ref{fig:Mortality-Raw}.
There the curves are distinguished using the standard graphical technique
of a rotating color palate (in this case the default 7 colors in Matlab). 

\begin{figure}[H]
\includegraphics[scale=0.75]{Chapter1/OODAbookChp1FigACMYK}

\caption{Spanish Mortality curves as a function of age. Raw male mortality
is in the left panel, with $\log_{10}$ mortality on the right. Years
are distinguished using a rotating color palette. Shows age effects
and large variation (factors of more than 10 for some age groups)
across years, as well as the data object choice of $\log_{10}$ mortality
being the more useful scaling of the data.\label{fig:Mortality-Raw}}
\end{figure}
This view already shows interesting aspects of the data. For example,
being born is a risky activity, with a high mortality rate. However,
the chance of dying falls off rapidly, up until the teen years when
risky behavior tends to begin. Then through adulthood the death rate
slowly increases, becoming quite high in old age. Also note the bundle
of curves is quite thick, with the axes indicating approximately a
10 fold change over the years, begging an investigation into how things
have changed over time. This is easily provided in Figure \ref{fig:Mortality-Colored},
by applying a different color scheme to the curves in the right panel
of Figure \ref{fig:Mortality-Raw}. Here time ordering of the curves
is highlighted through coloring with a rainbow scheme to indicate
years, starting with magenta ({[}1 0 1{]} in RGB coordinates) for
1908 and ranging through violet, blue, cyan, green, yellow and orange
to red ({[}1 0 0{]} in RGB) for 2002.

\begin{figure}[H]
\includegraphics[scale=0.75]{Chapter1/OODAbookChp1FigB2CMYK}

\caption{Spanish Mortality curves using a rainbow color scheme to indicate
progression in time (over years 1908-2002). Shows major improvements
in mortality over this time range.\label{fig:Mortality-Colored}}
\end{figure}
This shows a very clear overall improvement over the years in mortality,
due mostly to improvements in medicine and public health. Note also
that these improvements have benefited younger people more than the
old, as there is not yet much treatment available for aging. As happens
frequently with OODA data, additional visual insights come from careful
decomposition of the variation present in these curves, through a
\emph{\index{Principal Component Analysis|(}Principal Component Analysis}
(PCA). See Chapters \ref{chapBB:ExpConfAnal} and \ref{chapP:PCA-Details}
and \citet{Jolliffe2002} for background information concerning the
many ways this method is used. One important use of PCA is to gain
insight into how data objects relate to each other. Insight comes
from considering the data as lying in an abstract \emph{point cloud}
in $d=99$ dimensional space, where low dimensional projection\index{projection|(}s
frequently visually illustrate key relationships (e.g. clustering
of data objects). An often useful first step of a PCA is \emph{mean
centering}\index{mean!center}, which essentially moves the point
cloud so that it is centered at the origin. As seen in Figure \ref{fig:Mortality-Mean-Resid},
this centering operation itself can provide an informative decomposition
of the data into the mean and residuals about the mean. 

\begin{figure}[H]
\includegraphics[scale=0.75]{Chapter1/OODAbookChp1FigCCMYK}

\caption{Left panel is the mean mortality curve. Right panel contains the mean
residuals, where the mean is subtracted from each curve, using the
same color scheme. Shows that age effects are essentially common for
all (i.e. over time), in the sense of appearing in the mean. Improvements
over time appear in the residuals, with overall most improvement for
the young.\label{fig:Mortality-Mean-Resid}}
\end{figure}
The left panel of Figure \ref{fig:Mortality-Mean-Resid} shows the
mean curve, computed as the point-wise mean of the curves in Figure
\ref{fig:Mortality-Colored}. The right panel contains the mean residuals,
which are computed by subtracting the mean from each of the data curves,
while retaining the original year coloring. Note that the mean curve
contains many of the important features of the raw data, especially
those related to age. In particular, the danger of being born together
with low mortality for the young with increasingly higher mortality
for the old are all properties of the mean. These essentially do not
appear in the mean residuals, indicating that these are population
properties which have not changed much over time. A perhaps surprising
aspect of the mean is the occasional blips that appear. One might
think these are random noise, but note that they are quite periodic
and in fact appear at decades. This is a function of historically
poor record keeping. The early lack of birth certificates for the
full population led to some uncertainty of age at the time of death
for some, with subsequent rounding to decades which is clearly visible.
The mean residuals also reflect an important aspect of the population
structure, being driven by the changes over time. Most important are
the dramatic improvements in mortality that have been made over the
course of this study. This view also makes it clear that the young
have benefited the most with that benefit decreasing as a function
of age.

\begin{figure}[H]
\includegraphics[scale=0.75]{Chapter1/OODAbookChp1FigDCMYK}

\caption{PC1 mode of variation plot (left) and scores\index{scores!distribution plot|(}
distribution plot (right). The former shows that this dominant mode
of variation reflects most of the overall improvement in mortality.
Scores plot shows most of the improvements happened relatively rapidly,
plus highlights the 1918 Flu Pandemic (violet outlier on the right)
and the Spanish Civil War (light blue sharp trend to the right).\label{fig:Mortality-PC1}}
\end{figure}
PCA is usefully understood as decomposing the mean centered data in
the right panel of Figure \ref{fig:Mortality-Mean-Resid} into insightful
modes of variation (this concept is formally defined in Section \ref{subsecBA.1.1:FormalizingModes}).
One such mode is the variation revealed by the first principal component
as shown in Figure \ref{fig:Mortality-PC1}. Insight comes from thinking
of the above mentioned point cloud, where each data object (curve
in this case) is a point. The PCA modes of variation are developed
by seeking orthogonal directions of maximal variation within the point
cloud. The first PC \emph{direction} is the unit (i.e. norm 1) vector,
based at the sample mean, which maximizes the variance of the data
projected onto that vector. This direction is easily computed as the
first eigenvector of the sample covariance matrix (defined at (\ref{eqB.F:SampleCovariance})).
The entries of that vector (which indicate how it relates to the variables,
i.e. feature\index{feature}s, of the data set) are called the \emph{loadings\index{loadings|(}}.
Visual insight into these loadings comes from the \emph{mode of variation
plot\index{modes of variation!plot}} in the left panel of Figure
\ref{fig:Mortality-PC1}. The horizontal axis indexes the variables,
which are ages in this case, and the curves are all multiples of the
eigenvector. In particular the curves are projections of the data
curves onto the direction vector. These are the columns of the rank
1 matrix that is the product of the column vector of loadings times
the row vector of \emph{scores}\index{scores|(}, which are the projection
coefficients of each data object onto the eigenvector. In classical
multivariate analysis the scores are also called the principal components.
This matrix is the (least squares) best rank 1 approximation of the
mean residual matrix shown in the right panel of Figure \ref{fig:Mortality-Mean-Resid}.
This PC1 view highlights the \emph{dominant} mode of variation, which
nicely reflects the major overall improvement in mortality. In addition,
as life and death record keeping has improved over time, the decline
in age rounding effects is reflected in the decadal spikes pointing
upwards (early) and downwards (later). In particular, the rounding
was present earlier, not later, so it shows up partially in the mean
in Figure \ref{fig:Mortality-Mean-Resid}, and then as this contrast
in Figure \ref{fig:Mortality-PC1} (left panel). 

The right panel of Figure \ref{fig:Mortality-PC1} is the PC1 \emph{scores
distribution plot}, which will be used frequently in the following
to display detailed information as to how the data objects relate
to each other. Each circle represents one score, using the same color
scheme. Horizontal coordinates indicate the score and vertical coordinates
indicate order in the data set, in this case the year. The magenta
color of the top circle is the year 1908 and the red color of the
lowest circle is for the year 2002. The overall leftward trend again
shows the overall improvement in mortality over these years. The black
curve shows a \emph{\index{kernel density estimate}kernel density
estimate}, which can be thought of as a smooth histogram. The vertical
axis records the heights of this curve. Detailed discussion of kernel
density estimation\index{density estimation} is in Chapter \ref{chapN:Smoothing}.
See \citet{WandJones1995} for a more in depth overview. This type
of display of one dimensional distributions, which includes both the
actual data points and the smooth histogram, is used many other times
in the following. In this case it shows much higher density of scores
in the higher and lower regions, which is another way of seeing that
most of the overall transition from higher to lower mortality was
relatively rapid. A couple of smaller scale aspects are also clear
in this scores plot. The violet year, farthest to the right was the
year 1918, when many people around the world died during a flu pandemic,
which until recently was the largest ever well-documented epidemiological
event worldwide. Also notable is the shift towards higher mortality
(i.e. to the right) shown as light blue, which was the time of the
Spanish Civil War, just before World War II (in which Spain was not
a combatant).

\begin{figure}[H]
\includegraphics[scale=0.75]{Chapter1/OODAbookChp1FigECMYK}

\caption{PC2 mode of variation (left) and scores distribution (right), using
the same format as Figure \ref{fig:Mortality-PC1}. The loadings plot
shows this second mode of variation provides a contrast between the
20-45 year old with the rest. The scores plot shows the deep effects
of the flu pandemic, the Spanish civil war and automotive death rate.\label{fig:Mortality-PC2}}
\end{figure}
Figure \ref{fig:Mortality-PC1} showed the first mode of variation
in the mortality data called PC1. An interesting complementary mode
of variation is the second PC, as shown in Figure \ref{fig:Mortality-PC2}.
This represents the direction of second strongest variation (in the
sense of being orthogonal to the first direction) measured again in
terms of variance of projections. It is computed as the second eigen
direction of the sample covariance matrix. The PC2 mode of variation
plot (left panel) shows that this direction highlights differences
between the 20-45 year old cohort, with the union of the young and
the old. The color pattern is harder to interpret in this mode, but
is very clear in the scores distribution plot (right panel). Note
that the 20-45 year olds suffered even stronger effects from both
the pandemic and also the war, as they died at a substantially higher
rate than usual in those times. Another interesting feature is the
growing mortality for this cohort in the 1960s to 1980s (green to
orange). This period corresponds to growing access to automobiles,
and apparently the idea that young males are the group most prone
to risky automobile behavior. Note that in the final years, the direction
of this trend has fortunately reversed, which has been ascribed to
much improved car safety (such as seat belts) and also to major improvements
in roads.

The concept of modes of variation as determined by PCA loadings\index{loadings|)}
and scores is explored more deeply in Section \ref{secBA.1:Foundations_Terminology}.

\begin{figure}[H]
\includegraphics{Chapter1/OODAbookChp1FigFCMYK}

\caption{Scatterplot of PC1 vs. PC2 scores\index{scores!scatterplot}. This
shows many of the above historical trends in a single plot.\label{fig:Mortality-Scatterplot}}
\end{figure}
Figure \ref{fig:Mortality-Scatterplot} shows a scatterplot of the
bivariate distribution of the PC1 and PC2 scores, which provides a
useful and concise summary of both modes of variation, i.e. of much
of the structure in this data set. The one dimensional PC1 scores
distribution in the right panel of Figure \ref{fig:Mortality-PC1}
is on the horizontal axis, while the vertical axis has the corresponding
PC2 \index{scores|)}scores distribution from the right of Figure
\ref{fig:Mortality-PC2}. This is the two dimensional projection\index{projection|)}
of the data onto the plane with maximal variation. Note that the circles
representing the data objects (i.e. the mortality curves) are connected
with line segments in time order, which facilitates keeping the progression
of years in mind when interpreting the plot. The overall improvement
in mortality, with the exceptions of flu and war, are clear from the
main leftwards progression. Variation over time of the contrast between
the 20-45 year olds and the rest are also clear on the vertical axis,
nicely highlighting the flu, war and automobile effects.

For this data set, the most interesting views are in the first two
PC components. For others, more components can also be quite insightful.
A useful summary of several PC components is a matrix of such scatterplots,
with the axes carefully coordinated over both rows and columns. The
diagonal of such a display is most useful when it shows some sort
of 1-d distributional summary, e.g. the combination of jitter plot\index{jitter plot}s
(the colored circles) and kernel density estimates\index{kernel density estimate}
used to show the distribution of scores\index{scores!distribution plot|)}
in the right panels of Figures \ref{fig:Mortality-PC1} and \ref{fig:Mortality-PC2}.
Jitter plots are discussed in more detail in Section \ref{secB2:Data-Visualization}.
Further examples of such matrices of scatterplots can be found in
Figures \ref{fig:50d-ToyPCAScatMat}, \ref{fig:HiDimToyPCA }, \ref{fig:HiDimToyDWD}
and many other places in later chapters.

Mortality rates for other countries can be explored in a similar way.
For example mortality data from Switzerland (also available in \citet{WilmothMortality2008})
show similar flu pandemic and automobile effects as observed here,
but neither the data rounding (due to a longer period of good record
keeping) nor the war caused mortality effects are visible as expected.\index{data objects!curves|)}
\index{Spanish Mortality data|)}

\section{Case Study: Shapes as Data Objects\label{sec:Shapes-as-Data}}

A\index{data objects!shapes|(} particularly deep and important example
of shapes as data objects is the \emph{\index{Bladder-Prostate-Rectum data|(}Bladder-Prostate-Rectum}
data, motivated by the challenge of planning radiation treatment of
prostate cancer described in \citet{ChaneyPizer2004}.

\subsection{The Segmentation Challenge \label{subsec1A.B.A:Segmentation}}

Radiation treatment of cancer is quite effective, and administered
over the course of a number of days. The goal is to provide a maximal
radiation dose to the prostate while minimizing the impact on nearby
sensitive organs such as the attached bladder and the rectum, which
is adjacent. A major radiation treatment planning challenge is that
(even within the same person) the locations of all 3 organs vary widely
on the critical time scale of days. Computed Tomography (CT) images
are useful for visually locating these organs on a given day, with
CT preferred over Magnetic Resonance images due to its superior accuracy
of location. However \emph{segmentation}\index{segmentation|(}, i.e.
finding the set of \emph{voxels\index{voxel} }(three dimensional
analogs of pixel\index{pixel}s) inside each organ, was a challenging
problem because of poor contrast and noise, as shown in Figure \ref{fig:BladProsRect-Raw}.
That is one slice of a 3-d stack of images, showing a side view of
the hip region for one patient. The color scheme of CT is the same
as for x-rays, so dense objects such as bones show up as white. Thus
the upper right of Figure \ref{fig:BladProsRect-Raw} shows the tailbone,
and a hipbone passes through this slice in the lower center. Black
indicates the least dense regions which are gas bubbles in the rectum,
which is the curved lighter region containing the darkest spots starting
near the top center and curving down below and to the left of the
tail bone. The lighter gray region between the top of the rectum and
the small hip bone is the bladder. The prostate, which is the target
of the treatment, is a light gray region between the hip bone, the
bladder and the lowest visible section of the rectum.

\begin{figure}[H]
\includegraphics[scale=0.75]{Chapter1/OODAbookChp1FigN}

\caption{One slice of 3-d CT image in Bladder-Prostate-Rectum data. Bones are
white, black gas bubbles indicate the rectum. Bladder and prostate
are light gray near the center and lower center. This image shows
that automatic segmentation is very challenging.\label{fig:BladProsRect-Raw}}
\end{figure}
Segmentation of the prostate is quite challenging because of very
poor contrast with surrounding objects (it is essentially the same
shade of gray and has both lighter and darker regions nearby) and
because of the relatively high noise level. For these reasons, incorporation
of anatomical knowledge is essential to the segmentation process.
\emph{Manual segmentation} achieves this through an anatomically trained
technician drawing the boundary of an object on each slice of the
3-d image. The union of the interior voxels aggregated over slices
then gives a segmentation of the object. An example of that process
is in Figure \ref{fig:Segmented-Bladder}, which shows two views of
a manual segmentation of the bladder in Figure \ref{fig:BladProsRect-Raw}.
The left panel shows how voxels are aggregated across slices, using
a view orthogonal to that where the drawing was done. The right panel
is a rotated view of the highlighted collection of blue colored voxels
without the CT image, giving a clear impression of the 3-d object. 

\begin{figure}[H]
\includegraphics[scale=0.65]{Chapter1/OODAbookChp1FigO}

\caption{Left panel shows the results of a manual segmentation of the bladder,
performed sequentially on orthogonal slices. Right panel shows a rotated
view of the same bladder, to highlight the 3-d aspect of the segmentation.\label{fig:Segmented-Bladder}}
\end{figure}
While manual segmentation is quite effective at locating these organs
for planning radiation treatment, it is time consuming and hence it
is not practical to repeat this manual operation many times over the
course of radiation treatment (i.e. in a clinical setting). This has
motivated a lot of research on automatic segmentation of these organs,
much of which was developed in the references cited at the end of
this section. The key idea is to incorporate anatomical information
into the training process, using a Bayesian statistical model. The
starting point for this is a \emph{shape representation\index{shape representation|(}},
i.e. a parametric model for each organ. 

\subsection{General Shape Representations \label{subsec1A.B.B:ShapeRepresentations}}

In some contexts shape is conveniently represented by \emph{\index{data objects!landmark configurations}landmark\index{landmark|(}
configurations}, i.e. a set of points that correspond across members
of the data set, which can be readily found on each. The statistical
analysis of landmark configuration shape data objects was pioneered
by \citet{Kendall1984Shape} and \citet{Bookstein1986Shape}. For
introduction to the large literature on that, see \citet{DrydenMardia2016}.
The fundamental idea is illustrated by a toy data set of triangles
in $\mathbb{R}^{2}$ as data objects in Figure \ref{fig:ToyTriangles}.
An intuitive representation of each triangle is the configuration
of the $\mathbb{R}^{2}$ coordinates of the vertices (a 6-tuple) ,
which are natural landmarks. However many triangles with different
configurations have the same shape. In particular, the triangles to
the left of the dashed line are all translations, rotations and scalings
of each other, i.e. all have the same shape. Two other sets of common
shapes appear between the vertical lines, and to the right of the
dot-dashed line. The mathematical device of \emph{equivalence relation}
provides a convenient formulation of the notion of shape. Calling
two triangular configurations \emph{equivalent} when they are translations,
rotations and scalings of each other, results in \emph{equivalence
class\index{equivalence class}es}. These are the sets of all triangles
which can be translated, rotated and scaled into each other, i.e.
triangles of the same shape. These equivalence classes of identified
triangles then become the shape data objects. Spaces of equivalence
classes are widely studied in differential geometry, where they are
called \emph{quotient space\index{quotient space}s}. Common synonyms
for the equivalence classes are \emph{fiber\index{fiber}s} (frequently
used here in Chapter \ref{chapG:Manifold-Data-Analysis}) and \emph{orbit\index{orbit}s}
(appearing often here in Chapter \ref{chapH:FDA-Curve-Registration}).
As discussed in Section \ref{sec:Landmark-Shapes} and in Section
4.3.4 of  \citet{DrydenMardia2016}, the natural geometry of the quotient
space of triangle shapes is the sphere $S^{2}$ (see (\ref{eqB.C:Spheres})
for a formal definition). Sections \ref{secG.F:IntroManifolds} and
\ref{sec:Landmark-Shapes} contain a broader discussion of shape quotient
spaces, where it is seen that many of those are also curved. This
provides strong motivation for studying data objects lying on curved
manifold\index{manifold|(}s, as done below and in more depth in Section
\ref{secG-A:ShapeDataObjects}. 

\begin{figure}[H]
\includegraphics[scale=0.6]{Chapter1/OODAbookChp1FigWToyTris}

\caption{Toy data set of triangles in $\mathbb{R}^{2}$, to illustrate shapes
as data objects. Lines separate three equivalence classes (i.e. fibers
or orbits) with respect to translation, rotation and scaling. \label{fig:ToyTriangles} }
\end{figure}
While landmark approaches are useful for many tasks, they are typically
less useful in many medical imaging situations, such as soft tissues,
where landmark\index{landmark|)}s that correspond across cases can
be hard to find, with often very few obvious choices apparent. Hence,
there has been much research devoted to \emph{boundary representations}.
In the computer graphics world a very common boundary representation
is a triangular mesh, see e.g. \citet{Owen1998survey}. A major challenge
to the use of mesh representations in shape statistics is \emph{correspondence\index{correspondence}\index{}},
i.e. relating the mesh parameters (e.g. triangle vertices) across
instances of shape data objects. Two important approaches to this
are Active Shape Models, see \citet{CootesTaylor1994} for a good
introduction, and the entropy based ideas of \citet{CatesWhitaker2007}.
Another major formulation of boundary representations is through Fourier\index{Fourier!methods}
methods, e.g. as in \citet{KelemenGerig1999}. For sufficiently smooth
shapes, \citet{KurtekSrivastava2013} have shown that superior representation
comes from enhancing boundary representations by also including surface
normal vectors in the data objects.

\subsection{Skeletal Shape Representations \label{subsec1A.B.C:SkeletalShape}}

As discussed in \citet{SiddiqiPizer2008}, a \emph{medial representation\index{medial representation}}
can provide improvements for a number of imaging tasks. The key idea
is to base the representation on the more robust concept of 3-d solids,
instead of on 2-d boundary surfaces. For the reasons discussed in
Chapter 3 of \citet{SiddiqiPizer2008}, the concept of medial locus
has been generalized to give \emph{\index{skeletal representation|(}skeletal
representations}. As noted in \citet{pPzerJung2013} the enhanced
flexibility of skeletal representations allows for superior fits to
data. A skeletal representation of one bladder, prostate and rectum
instance is illustrated in Figure \ref{fig:BlaProsRect-Srep}. 

\begin{figure}[H]
\includegraphics[scale=0.75]{Chapter1/OODAbookChp1FigP}

\caption{Skeletal representation of a single bladder-prostate-rectum. Left
panel shows the central skeletal sheets, atoms and spokes for each
shape object. Center panel adds the implied boundaries as quad meshes,
using yellow for the bladder, green for the prostate, red for the
rectum. Right panel represents the implied boundaries using a light
source rendering. \label{fig:BlaProsRect-Srep}}
\end{figure}
The left panel of Figure \ref{fig:BlaProsRect-Srep} shows the interior
components of three skeletal representations, one for each organ.
Each has a set of yellow dots, called \emph{skeletal atoms}, connected
by green line segments, which are a discretization of the \emph{skeletal
sheet}, the 2-d surface which is approximately medial in the sense
of being equidistant from both boundaries. Each skeletal atom has
\emph{spokes}, shown as cyan and magenta line segments, extending
from the skeletal sheet to the boundary of the organ. Skeletal atoms
at the edge of the sheet each have one additional spoke shown in red,
extending to the edge of the organ. The central panel of Figure \ref{fig:BlaProsRect-Srep}
adds three colored meshes (yellow for the bladder, green for the prostate,
red for the rectum) which indicate the boundary of each that is implied
by the interior components as a quadrilateral mesh that connects the
ends of the spokes. The right panel shows the boundary more explicitly
by coloring the panels of the quad meshes and using a light source
shading in the same colors. The skeletal model is a parametric model
of shape, whose parameters are the 3-d locations of the yellow atoms,
the lengths of the spokes, and the angles of the spokes, each of which
is represented as a point on the sphere $S^{2}$. 

The data objects in this OODA case study are chosen to be the skeletal
models represented by the locations of $k$ atoms in $\mathbb{R}^{3}$,
$l$ positive spoke lengths in $\mathbb{R}_{+}$and $m$ directions
on $S^{2}$. For CT images where a manual segmentation has been performed,
the skeletal shape model can be fit to the binary image shown in blue
in Figure \ref{fig:Segmented-Bladder} (i.e. the various parameters
estimated), using direct methods such as least squares. However as
discussed above, for clinical applications such as radiation treatment
planning, with a need for a technician to perform this operation several
tens of times for one course of treatment, manual segmentation is
prohibitively expensive. This motivated the work cited at the end
of this section, on automating fitting of skeletal models (as shown
in Figure \ref{fig:BlaProsRect-Srep}) directly to raw CT images (as
shown in Figure \ref{fig:BladProsRect-Raw}). As discussed above,
this requires incorporation of something akin to anatomical information.
That is done using a Bayesian statistical approach. Essentially some
manual segmentations are used to train a prior distribution using
OODA, which is combined with a likelihood based on a new CT image,
to generate a posterior distribution which is maximized over the parameters
of the skeletal shape representation, to give an automatic segmentation. 

\subsection{Bayes Segmentation via Principal Geodesic Analysis \label{subsec1A.B.D:BayesSegmentation}}

The Bayes implementation employed in this type of application differs
somewhat from most modern Bayes applications. On one hand, the underlying
probability distributions are very basic, since only conjugate Gaussian
priors, likelihood and hence posteriors are used. This is a strong
contrast with the complicated models involving Monte Carlo Markov
Chain methods that are currently very prevalent in applications of
Bayes methods. On the other hand, this Bayes application is relatively
deep in two ways. First the number of parameters to fit is typically
much higher then the number of training instances, i.e. it lies in
the high dimensional OODA domain discussed in a general way in Chapter
\ref{chapM:HDasy}. The second complication is the non-Euclidean nature
of the reparameterizations, caused mostly by each spoke naturally
lying on the surface of the sphere $S^{2}$. As this research has
progressed, the high dimensionality has been handled by a variety
of methods related to PCA\index{Principal Component Analysis|)}.
More challenging is that skeletal parameterized data objects are naturally
elements of a space of the form $\mathbb{R}^{3k}\times\mathbb{R}_{+}^{l}\times(S^{2})^{m}$
(i.e. tuples of $k$ real numbers, $l$ positive reals, and $m$ points
on the sphere). Such spaces are called \emph{manifold\index{manifold|)}s}
in differential geometry (see Section \ref{secG.F:IntroManifolds}
for an introduction to aspects of this topic needed for OODA) and
are usefully thought of as curved surfaces (e.g. the surface of a
sphere).

The need to address the first complication (the high dimension) in
the bladder-prostate-rectum segmentation challenge described above
has led to a series of developments in terms of analogs of PCA for
data lying on the manifolds of skeletal representations. The \index{Principal Geodesic Analysis|(}Principal
Geodesic Analysis (PGA) of \citet{Fletcher2004PGA} represents an
important early advance in this work. The main idea of PGA is to consider
the Euclidean PCA basis as a set of orthogonal lines that (sequentially)
best fit the data. In PGA these best fitting lines are replaced by
best fitting \emph{geodesics} (e.g. great circles on $S^{2}$) which
are a natural analog of lines. The results of a PGA, based upon $n=17$
skeletal representations (collected over a sequence of days) from
a single patient are shown in Figure \ref{fig:BladProsRect-PGA}.\index{Bladder-Prostate-Rectum data|)}

Figure \ref{fig:BladProsRect-PGA} reveals clinically interesting
modes of variation\index{modes of variation|)} of these organs within
this person. The left column (first mode of variation) seems to reflect
vertical shift variation driven by the rectum. The second mode (middle
column) shows twisting, while the third (right column) is about emptying
and filling of the bladder. This input led to the Bayes segmentation
method giving very effective automatic segmentation. That was the
basis for the successful start-up company Morphormics, which was subsequently
purchased by the radiation treatment equipment manufacturer Accuray.

More recently there has been a series of improvements to PGA, motivated
by a succession of deeper and deeper integrations of statistical ideas
with differential geometry. Detailed discussion of this progression
appears in Section \ref{secG-A:ShapeDataObjects}. While this discussion
has focused mostly on segmentation using skeletal shape representations,\index{shape representation|)}
much important related work has been done on classification\index{classification!shapes}
as discussed in Chapter \ref{chapJ:Classification} and on confirmatory
analysis\index{confirmatory analysis!shapes} which appears here in
Chapter \ref{chapL:Confirmatory-Analysis}. A good overview of the
usefulness of \index{skeletal representation|)}skeletal representations,
especially in comparison to other types of representations can be
found in \citet{pPzerJung2013,Pizer2014SrPNSvPDM}, \citet{Schulz2016HypoTest}
and \citet{HongJP2016ClassSrep}. 

\begin{figure}[H]
\includegraphics{Chapter1/OODAbookChp1FigQ}

\caption{Principal Geodesic Analysis\index{Principal geodesic analysis|)}
of Bladder-Prostate-Rectum variation within one person. Columns give
visual impression of first 3 PGA modes of variation. All three plots
in the second row are the \index{Fréchet mean}Fréchet mean (notion
of center defined in (\ref{eq:ChpEEqBFrechetMean})). Top row shows
the three +2 standard deviation departures from the mean, and bottom
row shows the corresponding -2 standard deviation departures. This
gives three interpretable and sensible modes of variation. \label{fig:BladProsRect-PGA}}
\end{figure}
The bladder-prostate-rectum research\index{Bladder-Prostate-Rectum data}
that lies at the core of the discussion of this section was developed
in a series of papers. That includes \citet{ChaneyPizer2004}, \citet{BroadhurstStough2005},
\citet{davis2005automatic}, \citet{PizerJeong2005,PizerJoshi2005,PizerBroadhurst2006,PizerBroadhurst2007},
\citet{LuPizer2007}, \citet{StoughBroadhurst2007}, \citet{JeongStough2008},
\citet{MerckTracton2008} and \citet{FengFoskey2010}.\index{data objects!shapes|)}
