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+\chapter{Introduction}
+\label{chap:introduction}
+
+Facial animation is an important discipline in computer animation,
+as faces convey a significant part of human expression or emotion.
+This makes good facial animation an essential requirement for character animation in film or video games.
+At the same time, animation efficiency can't be disregarded completely for high-quality results,
+so an approach combining both model quality and model usability is desired.
+\textit{Inverse kinematics} provides such an approach,
+as it enables direct manipulation of high-quality blend shape models or performance driven facial animation using motion capture.
+
+\section{Facial Animation Overview}
+\label{sec:facialanimationmethods}
+
+The simplest approach to facial animation could possibly be freeform deformation mixed with keyframe animation,
+which allows to recreate any emotion or expression at the cost of labor intensity and missing \textquote{guardrails}:
+This technique imposes no modeling restrictions, so unnatural facial deformations can be produced easily (see \autoref{fig:unnaturaldeformation}).
+
+\begin{figure}[h]
+  \centering
+  \begin{subfigure}[b]{0.3\textwidth}
+	\includegraphics[scale=0.3]{img/unnatural_deformation.png}
+  \end{subfigure}
+  \caption{Unnatural Deformation.~\autocite{directmanipulationblendshapes}}
+  \label{fig:unnaturaldeformation}
+\end{figure}
+
+To improve the model usability and prevent unnatural results,
+the facial model can also be controlled through a set of parameters,
+either chosen and implemented manually or automatically.
+
+The (probably) first parametric face model was produced by F.I.\ Parke in 1974~\autocite{parametricface}:
+It contains manually chosen parameters like \code{JAW ROTATION} (see \autoref{fig:parametricjawrotation}) or \code{MOUTH WIDTH} and applies different deformation operations like translation,
+rotation or scaling on certain facial regions to realize them.
+
+\begin{figure}[h]
+  \centering
+  \begin{subfigure}[b]{0.3\textwidth}
+	\includegraphics[scale=0.2]{img/parametric_model.png}
+  \end{subfigure}
+  \caption{Manually specified jaw rotation.~\autocite{parametricface}}
+  \label{fig:parametricjawrotation}
+\end{figure}
+
+A different method to parameterize a face model is the \textquote{blend shape} approach:
+Instead of using custom deformation operations on isolated parts of the mesh,
+a blend shape model interpolates between different pre-modeled expressions that have identical vertex amounts and triangulation.
+Multiple weighted expressions (\textquote{blend shape targets}) can be combined to generate the target expression.
+
+By generating the target expression through a weighted sum of complete face models,
+unwanted scale deformations can be introduced by choosing \textquote{wrong} weights.
+Thus, the blend shape method was refined to combine local deviations from a neutral face instead,
+called \textquote{delta blend shapes}.
+
+Instead of manually associating each parameter with a certain expression,
+model parameters can also be determined statistically,
+e.g.\ by doing a principal component analysis of a range of facial expressions.
+This generally results in non-interpretable model parameters,
+which makes this model harder to animate,
+as it is difficult to anticipate the resulting expression based on single parameter modifications.
+
+It is also possible to apply more general character animation techniques like skeletal animation to faces,
+as is often done in video games.
+This approach is efficient to animate,
+but weaknesses like unnatural skin and muscle movement become more significant in the case of faces,
+so skeletal face animation might not be suitable if quality requirements are high.
+
+Hybrid approaches,
+like supplying a bone-driven model with a physical skin and muscle simulation,
+can produce more realistic results,
+but are also highly dependent on the character,
+so the transfer of animations or \textquote{rigs} between characters might become difficult.
+Also, physical simulation usually comes with a large performance impact,
+which makes this technique unsuitable for real-time applications.
+
+Skeletal rigs can also be combined with blend shapes,
+as certain facial components that have a clear rotational axis (like eyes or jaw) are conveniently animated using a skeleton.
+Blend shapes can then be used to correct skin movement or add detail.
+
+\section{Kinematics}
+\label{sec:kinematics}
+
+Kinematics is a subfield of physics, dealing with the motion of bodies in a system while ignoring masses or forces.
+A kinematic system in computer animation, like a skeleton, is modeled using rigid \textquote{bones} and rotatable \textquote{joints}\footnote{
+  This differs for the blend shape approach,
+  which is further described in \autoref{chap:blendshapemodels}.
+}:
+
+\begin{figure}[h]
+  \centering
+  \begin{subfigure}[b]{0.3\textwidth}
+	\includegraphics[scale=0.1]{img/human_rig.png}
+  \end{subfigure}
+  \caption{Human skeleton rig.~\autocite{computeranimation}}
+  \label{fig:humanrig}
+\end{figure}
+
+The bones are ordered in a hierarchical relationship:
+Rotating a single \textquote{parent} bone around its joint also rotates all \textquote{children} bones accordingly along the same center of rotation.
+Controlling such a kinematic model can happen in two ways: Forward and inverse kinematics.
+
+\textbf{Forward kinematics} calculates the absolute position of a joint (also called \textquote{end effector}) based on given joint orientations/angles of the parent joints.
+In general, the forward kinematics problem can be expressed as \(x=f(\theta)\),
+where \(x=(x_1,\dots,x_n)^T\) are the effector positions and \(\theta=(\theta_1,\dots,\theta_n)^T\) are the joint angles.
+
+\textbf{Inverse kinematics} describes the opposite problem \(\theta=f^{-1}(x)\):
+The absolute effector positions are given and the angles that produce these positions are to be determined.
+Solving this problem for real-world models is difficult for multiple reasons:
+
+\begin{itemize}
+  \item High-quality kinematic models (e.g.\ for character animation) can have hundreds of degrees of freedom
+  \item Multiple solutions or no solutions at all are possible
+        (for a given \(x_2\) in \autoref{fig:2dforwardkinematics} there are two solutions for \(\theta\) already,
+         as \(\theta_1\) and \(\theta_2\) can be exchanged)
+  \item Different solutions can be ordered by criteria that are difficult to define analytically,
+        for example by \textquote{intuitiveness} or if a solution is appropriate in the context of an animation
+\end{itemize}
+
+In consequence, approximative solvers are typically preferred over analytical ones.