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I - General Notions of Topology

    I.A -  A Continuous Theory

Topology is a branch of mathematics that studies the properties of geometric figures that are preserved through deformations, twistings and stretchings, without regard to size and absolute position.  In topology, the important mathematical notions are those of continuity and of continuous transformations; tearing, which would generate discontinuities, is prohibited.

Topology studies the properties of spatial objects by abstracting their inherent connectivity while ignoring their detailed form. One of the central ideas is that geometric objects, such as circles, curves, surfaces, can be treated as objects in their own right, independently of how they are represented or how they are embedded in  space. The exact geometry of the objects, their location and the details of their shape are irrelevant to the study of their topological properties.  In essence, this amounts to characterizing the topology of an object by its number of disconnected components, holes and cavities, but not by their position. For instance, a circle is topologically equivalent to any closed loop, no matter how different these two curves may appear. Similarly, the surface of a coffee mug with a handle is topologically the same as the surface of a doughnut (this type of surface is called a one-handled torus).

However, by ignoring the embedding space, it then becomes impossible to distinguish a torus from a knotted torus (see Figure below). This has lead mathematicians to define several levels of topological equivalence depending on the chosen set of continuous transformations. Given a specific set of transformations, two geometric figures represent the same topological object, or the same equivalence class, if both of them can be deformed into a third one by using continuous transformations from the considered set only.  

Intrinsic topology, homotopy type and Euler-characteristic.a-b) Two toruses that are homeomorphically equivalent. They share the same intrinsic topology. However, they do not share the same homotopy type as one can not be continuously transformed into the other. c) A geometric object with a spherical topology; its Euler-characteristic is X=v-e+f=8-12+6=2. d) A geometric object with a toroidal topology and an Euler-characteristic of X=v-e+f=16-32+16=0.

    1.B - Notions of Topological Equivalence

In this work, we are interested in locating anatomical structures from medical images. These structures are geometric entities, which are often referred to as topological spaces in the mathematics community. These shapes can be represented equivalently as surfaces or as volumes. Depending on the context, a topological space X might refer to the volume or surface representation. Using these two dual representations, two distinct levels of topological equivalence are usually considered. But we first need to introduce some mathematical definitions.   

An homeomorphism M from a space X into a space Y is a continuous, one-to-one transformation with a continuous inverse M^{-1}.

Homeomorphisms have some important properties that we will use later in this dissertation (particularly in Chapter~\ref{chap:genetic}). The Jacobian J_{\mathscr M}=|\frac{\partial \mathscr M}{\partial \mathbf x}|$ of the transformation\footnote{The fact that the Jacobian exists might seem confusing at first. Indeed, we have only assumed that the mapping $\mathscr M$ was an homeomorphism, and we did not specify that the mapping $\mathscr M$ was differentiable. However, in dimension 1, 2, and 3, any pair of homeomorphic smooth manifolds are diffeomorphic! This surprising property does not hold in higher dimension.} is non-singular (i.e. strictly positive or strictly negative). This is, of course, the multidimensional analog of monoticity. Another important property of the Jacobian is the fact that it relates the $n$-dimensional volumes of $X$ and $Y$: $d^n\mathbf y=J_{\mathscr M}d^n\mathbf x$, where $\mathbf y=J(\mathbf x)$.
\begin{definition}{\bf Homotopy}\\
An homotopy is a continuous transformation from one function into another. An homotopy between two functions $f$ and $g$ from a space $X$ into a space $Y$ is a continuous map $G:X\times [0,1]\to Y$ with $G(\mathbf x,0)=f(\mathbf x)$ and $G(\mathbf x,1)=g(\mathbf x)$, where $\times$ denotes set pairings. One says that two maps  $f_0$ and $f_1$ are homotopic if there exists a homotopy connecting them, and one writes $f_0\simeq f_1$.

In simpler terms, two objects are said to be homotopic if one can be continuously deformed into the other. For instance, a line segment is homotopic to a point, and a circle is homotopic to a solid torus. We note that, contrary to homeomorphism, homotopy does not consider the dimension of the topological objects. For instance, the unit ball in $\mathbb R^n$, $\{ \mathbf x\in \mathbb R^n /\  \| \mathbf x\| \leq 1 \}$, is homotopy equivalent to the point $\{\mathbf x=0\}$. Other levels of topological equivalence can be defined by considering the dimensionality of the topological objects. Homotopy is one of the main concepts of Algebraic Topology. For more details on homotopies and algebraic topology, we refer the reader to an excellent book on algebraic topology~\cite{hatcher:02}. \\

Using these two set of continuous transformations, two levels of topological equivalence are usually considered:
  \emph{Intrinsic Topology} : the intrinsic topology of an object   is defined by the set of properties that are preserved by  \emph{homeomorphic} transformations defined on the surface of the considered object. Under this set of equivalence, the embedding space is ignored : a knotted solid torus has the same intrinsic topology as a simple torus; and a hollow sphere is of the same topology as two spheres.
  \emph{Homotopy type} : the homotopy type of an object is the set of properties that are preserved by \emph{homotopic} transformations. Formally, we define two spaces $X$ and $Y$ to share the same homotopy   type, or to be homotopy equivalent, if there are maps $f:X\to Y$ and $g:Y\to X$, such that the composition $f\circ g$ is homotopic to the identity map of $Y$ ($f\circ g\simeq \mathbb I_Y$), and the composition $g\circ f$ is homotopic to the identity map of $X$ ($g\circ f\simeq \mathbb I_X$). Homotopy, which was first formulated by  Poincarr\'e around 1900, provides a measure of an object's topology by considering the embedding space. At this level of topological  equivalence, a torus is topologically different from a knotted torus, since one cannot be continuously transformed into the other (see Fig.~\ref{fig:topology}-a,b).

In this dissertation, the required level of topological equivalence is provided by homotopy. The anatomical structures to be segmented define smooth 2D compact (i.e. closed) orientable manifolds\footnote{A  manifold is a topological space such that each of its points has a neighborhood  that is homeomorphic to an open planar disk.} embedded in the real 3D Euclidean space. For such ``simple'' surfaces, the study of their  differential properties provides deep insights about their topology, as  the topology of such surfaces have profound connections with differential geometry. 

    1.C - Topology and Differential Geometry

Differential geometry is the study of Riemannian manifolds. Differential geometry, which deals with metricable notions on manifolds, has some surprising and fundamental links with topology. The connections arise from a set of theorems of elementary geometry (we refer the reader to the book on elementary differential geometry of  O'Neill for a proof of these theorems~\cite{oneill:97}). We first introduce a few notations and definitions.

\begin{definition}{\bf Rectangle and 2-segment}\\
A rectangle $R$ is a region of the 2D plane $R:a\leq u \leq b\ ,\ c\leq v
\leq d$, with $(u,v)\in \mathbb R^2$.  The interior $R^\circ $ of the
rectangle $R$ is the open set $a< u < b\ ,\  c< v< d$.
A 2-segment is a transformation from a rectangle $R$ into $\mathbb R^3$ that is a one-to-one regular mapping from the interior $R^\circ $ of the rectangle $R$ into $\mathbb R^3$.

\begin{definition}{\bf Rectangular decomposition of a surface $\mathscr{C}$}\\
A rectangular decomposition of a surface $\mathscr{C}$ is a finite collection of one-to-one 2-segments whose images cover $\mathscr{C}$ in such a way that if any two intersect, they do so in either a single common vertex or single common edge.

\begin{theorem}{\bf Rectangular decomposition}\\
Every compact surface $\mathscr{C}$ has a rectangular decomposition $\mathscr{D}(\mathscr{C})$. \end{theorem}

\begin{theorem}{\bf Euler-characteristic of a rectangular decomposition}\\
If $\mathscr{D}(\mathscr{C})$ is a rectangular decomposition of a compact surface $\mathscr{C}$, let $v$, $e$, and $f$ be the number of vertices,  edges, and faces in $\mathscr{D}(\mathscr{C})$. Then the integer ($v-e+f$) is the same for every rectangular decomposition of $\mathscr{C}$. This integer $\chi (\mathscr{C})$ is called the Euler-characteristic of $\mathscr{C}$.

The fact that a rectangular decomposition is used to compute the Euler-characteristic of the surface is merely a convenience for the proof of the theorem. Arbitrary polygons could as well have been used to decomposed $\mathscr{C}$. In the resulting polygonal decomposition, the different polygons would still be required to fit neatly, but they would not have the same number of sides. An arbitrary polygonal decomposition is called a tessellation, while, when only triangles are used, the decomposition is called a triangulation of $\mathscr{C}$.

\begin{theorem}{\bf Topological invariance of the Euler-characteristic}\\
{If  $\mathscr{C_M}$ and $\mathscr{C_N}$ are two compact orientable surfaces, $\chi(\mathscr{C_M}) = \chi(\mathscr{C_N})$ if and only if $\mathscr{C_M}$ and $\mathscr{C_N}$ are homeomorphic.}

%\begin{theorem}{\bf Gauss-Bonnet Theorem}\\
%The total Gaussian curvature of a compact orientable geometric surface $\mathscr{C}$ is $2\pi \chi(\mathscr{C})$.

%The Gauss-Bonnet theorem relates the Euler-characteristic of a surface to its total Gaussian curvature.
%The total Gaussian curvature of a compact orientable manifold is completely
%determined by its Euler-characteristic. It does not depend on the
%local shape of the surface but only on the surface connectivity.

Thm.~\ref{thm:invariance} is of central importance. It states that \emph{the Euler-characteristic of a surface is a topological invariant}. Two surfaces that have the same Euler-characteristic  share the same \emph{intrinsic} topology. However, we note that the Euler-characteristic does not define the homotopy type of a surface, since the embedding space is being ignored. Particularly, this implies that a discrete representation of a surface using a polygonal  decomposition with the desired Euler-characteristic might be self-intersecting in the 3D embedding space. We will discuss this important point later. \\    

The Euler-characteristic is of great practical interest because it can be calculated from any polyhedral decomposition $\mathscr{D}$ of the surface by $\chi = v - e + f$, where $v$, $e$ and $f$ denote respectively the
number of vertices,  edges and faces of the polyhedron $\mathscr{D}$. The Euler-characteristic of a sphere $\mathscr{S}$ is $\chi(\mathscr{S})=2$ (see Fig.~\ref{fig:topology}-c). This implies that any surface $\mathscr{C}$ with $\chi(\mathscr{C})=2$ is topologically equivalent (i.e. homeomorphic) to a sphere and therefore does not contain any handles. Surfaces with an Euler-characteristic $\chi(\mathscr{C})\neq 2$ have a topology that is different from that of a sphere. However, the Euler-characteristic does not provide any information about the localization of the topological differences.

Also, Thm.~\ref{thm:eulercharacteristic} states that the way a surface  is decomposed (i.e. tessellated) does not influence its topology. Any polyhedral decomposition of a surface will encode for the same intrinsic topology.\\

In fact, any compact, connected, and orientable surface is homeomorphic to a sphere with some number of handles. This number of handles is a topological invariant called the \emph{genus}. For example, a sphere is of genus $0$ and a torus is of  genus $1$. The genus $g$ is directly related to the Euler-characteristic $\chi$ by the formula $\chi=2-2g$.  In the case of multiple surfaces involving $K$  connected components, the total genus is related to the total Euler-characteristic by the formula: $\chi=2(K-g)$.

    1.D - On Topological Defects

We have already mentioned that an anatomical structure can be either represented by a volumic representation or by a surface representation, the two descriptions being dual representations. In this work, we call a topological defect any deviation from the spherical topology. Since we are considering 2D, smooth, orientable, and compact surfaces that are embedded in the 3D Euclidean space, 3 types of topological defects can be encountered:
\item{Disconnected components: in the presence of image artifacts, segmentations often contain several connected components, which might either constitute parts of the same structure or erroneous pieces of a   segmentation. }
\item{Cavities: cavities could be either the result of unexpected anatomical structures that are located inside the volume of interest, such as tumors, or, most frequently, the result of of image artifacts. Cavities are usually easy to detect and correct retrospectively if interpreted as connected background components. }
\item{Handles or holes: a handle or hole in a volume or a surface is identified whenever there exists a continuous loop that cannot be homotopically deformed onto a point within the manifold itself. These loops are called non-separating loops and constitute a fundamental concept of algebraic topology~\cite{hatcher:02}. Particularly, these are used to define the fundamental group of an object~\cite{hatcher:02,mangin-etal:95}.  

Finally, we note that for each defect present in an object (i.e. the foreground object) there exists a corresponding
defect in the background: a disconnected foreground component can be interpreted as a background cavity; a foreground cavity is a disconnected background component; and a handle in a foreground component defines another handle in the background component.

This foreground/background duality provides a methodology to  correct a topological defect~\cite{kriegeskorte-goeble:01,han-xu-etal:01}  (i.e. any deviation from the spherical topology).  For instance, the presence of a handle in an object could be corrected by either cutting the handle in the foreground object, or cutting the corresponding handle in the background object. Cutting the background handle can be interpreted as filling the corresponding hole. We will make use of this dual representation in Chapter~\ref{chap:digital_topology}.

Topology in Medical Imaging

We know that most macroscopic structures of the brain have the topology of a sphere. For instance, the highly folded cerebral cortex has the simple spherical topology, which means that it can be smoothly unfolded onto a sphere ( movie ).

Accurately locating (i.e. segmenting) specific structures from a medical image (MRI, CT, ...) can be a challenging task. Segmenting under topological constraints, i.e. achieving accurate location while ensuring that the topology of the object is correct, becomes even more difficult. However, it should be obvious that being able to achieve accurate and topologically correct representation of different structures is certainly of interest in medical imaging (Intersubject Registration, Spherical Coordinate System, Shape Analysis, Visualization...).

There are two ways of achieving accurate segmentation under topological constrainsts for medical images. Topological constraints can be directly incorporated into the segmentation process or retrospectively applied to an already segmented structure.  Both approaches use the same techniques, and we will now detail them.