X-Git-Url: https://git.ucc.asn.au/?p=ipdf%2Fsam.git;a=blobdiff_plain;f=chapters%2FBackground.tex;h=8b8ad5038f4882533be7437686828d2ad112de87;hp=53983512eaaf6813261178c3c2a5bd1bdca20c07;hb=1e1740165abac91f4f620ef8223a30e37e7124ab;hpb=df1e38d148d992b7caf24932ae89cf5cd610d5b8

diff --git a/chapters/Background.tex b/chapters/Background.tex
index 5398351..8b8ad50 100644
--- a/chapters/Background.tex
+++ b/chapters/Background.tex
@@ -31,7 +31,7 @@ It is of some historical significance that vector display devices were popular d
 
 %\cite{brassel1979analgorithm}; %\cite{lane1983analgorithm}.
 
-\subsection{Compositing}\label{Compositing and the Painter's Model}
+\subsection{Compositing}\label{Compositing}
 
 %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.
 
@@ -50,9 +50,9 @@ The PostScript and PDF standards, as well as the OpenGL API also use a painter's
 
 \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
+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\cite{openglspec} 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.
 
-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
+It is not entirely clear how well supported the IEEE-754 standard for floating point computation is amongst GPUs\footnote{Informal technical articles are abundant 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}
@@ -107,7 +107,7 @@ The Number type does differ slightly from IEEE-754 in that there is only a singl
 
 \section{Number Representations}\label{Number Representations}
 
-Consider a value of $7.25 = 2^2 + 2^1 + 2^0 + 2^{-2}$. In binary (base 2), this could be written as $111.01_2$ Such a value would require 5 binary digits (bits) of memory to represent exactly in computer hardware. Some values, for example $7.3$ can not be represented exactly in one base (decimal) but not another; in binary the sequence $111.010\text{...}_2$ will never terminate. A rational value such as $\frac{7}{3}$ could not be represented exactly in any base, but could be represented by the combination of a numerator $7 = 111_2$ and denominator $3 = 11_2$. Lastly, some values such as $e \approx 2.81\text{...}$ can only be expressed exactly using a symbolical system --- in this case as the result of an infinite summation --- $e = \displaystyle\sum_n=0^{\infty} (-1)^{n}\frac{1}{n!}$
+Consider a value of $7.25 = 2^2 + 2^1 + 2^0 + 2^{-2}$. In binary (base 2), this could be written as $111.01_2$ Such a value would require 5 binary digits (bits) of memory to represent exactly in computer hardware. Some values, for example $7.3$ can not be represented exactly in one base (decimal) but not another; in binary the sequence $111.010\text{...}_2$ will never terminate. A rational value such as $\frac{7}{3}$ could not be represented exactly in any base, but could be represented by the combination of a numerator $7 = 111_2$ and denominator $3 = 11_2$. Lastly, some values such as $e \approx 2.81\text{...}$ can only be expressed exactly using a symbolical system --- in this case as the result of an infinite summation --- $e = \displaystyle\sum_n=0^{\infty}\frac{1}{n!}$
 
 Modern computer hardware typically supports integer and floating-point number representations and operations. Due to physical limitations, the size of these representations is limited; this is the fundamental source of both limits on range and precision in computer based calculations.