+\subsection{Compositing}\label{Compositing and the Painter's Model}
+
+%So far we have discussed techniques for rendering vector graphics primitives in isolation, with no regard to the overall structure of a document which may contain many thousands of primitives. A straight forward approach would be to render all elements sequentially to the display, with the most recently drawn pixels overwriting lower elements. Such an approach is particularly inconvenient for anti-aliased images where colours must appear to smoothly blur between the edge of a primitive and any drawn underneath it.
+
+Colour raster displays are based on an additive red-green-blue $(r,g,b)$ colour representation which matches the human eye's response to light\cite{computergraphics2}. In 1984, Porter and Duff introduced a fourth colour channel for rasterised images called the ``alpha'' channel, analogous to the transparency of a pixel\cite{porter1984compositing}. In compositing models, elements can be rendered seperately, with the four colour channels of successively drawn elements being combined according to one of several possible operations.
+
+In the ``painter's model'' as described by the SVG standard the ``over'' operation is used when rendering one primitive over another\cite{svg2011-1.1}.
+Given an existing pixel $P_1$ with colour values $(r_1, g_1, b_1, a_1)$ and a pixel $P_2$ with colours $(r_2, g_2, b_2, a_2)$ to be painted over $P_1$, the resultant pixel $P_T$ has colours given by:
+\begin{align}
+ a_T &= 1 - (1-a_1)(1-a_2) \\
+ r_T &= (1 - a_2)r_1 + r_2 \quad \text{(similar for $g_T$ and $b_T$)}
+\end{align}
+It should be apparent that alpha values of $1$ correspond to an opaque pixel; that is, when $a_2 = 1$ the resultant pixel $P_T$ is the same as $P_2$.
+When the final pixel is actually drawn on an rgb display, the $(r, g, b)$ components are $(r_T/a_T, g_T/a_T, b_T/a_T)$.
+
+The PostScript and PDF standards, as well as the OpenGL API also use a painter's model for compositing. However, PostScript does not include an alpha channel, so $P_T = P_2$ always\cite{plrm}. Figure \ref{SVG} illustrates the painter's model for partially transparent shapes as they would appear in both the SVG and PDF models.
+
+\subsection{Rasterisation on the CPU and GPU}
+
+Traditionally, vector images have been rasterized by the CPU before being sent to a specialised Graphics Processing Unit (GPU) for drawing\cite{computergraphics2}. Rasterisation of simple primitives such as lines and triangles have been supported directly by GPUs for some time through the OpenGL standard\cite{openglspec}. However complex shapes (including those based on B{\'e}zier curves such as font glyphs) must either be rasterised entirely by the CPU or decomposed into simpler primitives that the GPU itself can directly rasterise. There is a significant body of research devoted to improving the performance of rendering such primitives using the latter approach, mostly based around the OpenGL API\cite{robart2009openvg, leymarie1992fast, frisken2000adaptively, green2007improved, loop2005resolution, loop2007rendering}. Recently Mark Kilgard of the NVIDIA Corporation described an extension to OpenGL for NVIDIA GPUs capable of drawing and shading vector paths\cite{kilgard2012gpu,kilgard300programming}. From this development it seems that rasterization of vector graphics may eventually become possible upon the GPU.openglspec
+
+It is not entirely clear how well supported the IEEE-754 standard for floating point computation (which we will discuss in Section \ref{}) is amongst GPUs\footnote{Informal technical articles are prevelant on the internet --- Eg: Regarding the Dolphin Wii GPU Emulator: \url{https://dolphin-emu.org/blog} (accessed 2014-05-22)}. Although the OpenGL API does use IEEE-754 number representations, research by Hillesland and Lastra in 2004 suggested that many GPUs were not internally compliant with the standard\cite{hillesland2004paranoia}. %Arbitrary precision arithmetic, is provided by many software libraries for CPU based calculations
+
+ \pagebreak
+\section{Document Representations}\label{Document Representations}
+
+The representation of information, particularly for scientific purposes, has changed dramatically over the last few decades. For example, Brassel's 1979 paper referenced earlier\cite{brassel1979analgorithm} has been produced on a mechanical type writer. Although the paper discusses an algorithm for shading on computer displays, the figures illustrating this algorithm have not been generated by a computer, but drawn by Brassel's assistant. In contrast, modern papers such as Barnes et. al's 2013 paper on embedding 3d images in PDF documents\cite{barnes2013embedding} can themselves be an interactive proof of concept.
+
+Haye's 2012 article ``Pixels or Perish'' discusses the recent history and current state of the art in documents for scientific publications\cite{hayes2012pixels}. Hayes argued that there are currently two different approaches to representing a document: As a sequence of static sheets of paper (Programmed Documents) or as a dynamic and interactive way to convey information, using the Document Object Model. We will now explore these two approaches and the extent to which they overlap.
+
+
+\subsection{Programmed Documents}
+\input{chapters/Background_Interpreted}
+
+\pagebreak
+\subsection{Document Object Model}\label{Document Object Model}
+\input{chapters/Background_DOM}
+
+\subsection{The Portable Document Format}
+
+Adobe's Portable Document Format (PDF) is currently used almost universally for sharing documents; the ability to export or print to PDF can be found in most graphical document editors and even some plain text editors\cite{cheng2002finally}.
+
+Hayes describes PDF as ``... essentially 'flattened' PostScript; it’s what’s left when you remove all the procedures and loops in a program, replacing them with sequences of simple drawing commands.''\cite{hayes2012pixels}. Consultation of the PDF 1.7 standard shows that this statement does not a give a complete picture --- despite being based on the Adobe PostScript model of a document as a series of ``pages'' to be printed by executing sequential instructions, from version 1.5 the PDF standard began to borrow some ideas from the Document Object Model. For example, interactive elements such as forms may be included as XHTML objects and styled using CSS. ``Actions'' are objects used to modify the data structure dynamically. In particular, it is possible to include Javascript Actions. Adobe defines the API for Javascript actions seperately to the PDF standard\cite{js_3d_pdf}. There is some evidence in the literature of attempts to exploit these features, with mixed success\cite{barnes2013embedding, hayes2012pixels}.
+
+%\subsection{Scientific Computation Packages}