X-Git-Url: https://git.ucc.asn.au/?p=ipdf%2Fdocuments.git;a=blobdiff_plain;f=LiteratureNotes.tex;h=95b299723d8f4a6b8ee5cae6aabcc7cfa2a199a5;hp=fd86b5a86b5f25a4e9b3eae8ab60b5649b716122;hb=4e29c54be001b88f60fd055ddb01d9b6bb12097f;hpb=4bc26ea901eb666201d0f02e8da789ec5ebf5c08;ds=sidebyside

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@@ -188,9 +188,204 @@ This paper uses Metaphors a lot. I never met a phor that didn't over extend itse
 	\item Title pretty much summarises it; similar to \cite{hayes2012pixels} except these guys actually did something practical
 \end{itemize}
 
+\section{27 Bits are not enough for 8 digit accuracy\cite{goldberg1967twentyseven}}
 
+Proves with maths, that rounding errors mean that you need at least $q$ bits for $p$ decimal digits. $10^p < 2^{q-1}$
 
+\begin{itemize}
+	\item Eg: For 8 decimal digits, since $10^8 < 2^{27}$ would expect to be able to represent with 27 binary digits
+	\item But: Integer part requires digits bits (regardless of fixed or floating point represenetation)
+	\item Trade-off between precision and range
+	\begin{itemize}
+		\item 9000000.0 $\to$ 9999999.9 needs 24 digits for the integer part $2^{23} = 83886008$
+	\end{itemize}
+	\item Floating point zero = smallest possible machine exponent
+	\item Floating point representation:
+	\begin{align*}
+		y &= 0.y_1 y_2 \text{...} y_q \times 2^{n}
+	\end{align*}
+	\item Can eliminate a bit by considering whether $n = -e$ for $-e$ the smallest machine exponent (???)
+	\begin{itemize}
+		\item Get very small numbers with the same precision	
+		\item Get large numbers with the extra bit of precision
+	\end{itemize}
+\end{itemize}
+
+\section{What every computer scientist should know about floating-point arithmetic\cite{goldberg1991whatevery}}
+
+\begin{itemize}
+	\item Book: \emph{Floating Point Computation} by Pat Sterbenz (out of print... in 1991)
+    \item IEEE floating point standard becoming popular (introduced in 1987, this is 1991)
+    \begin{itemize}
+		\item As well as structure, defines the algorithms for addition, multiplication, division and square root
+		\item Makes things portable because results of operations are the same on all machines (following the standard)
+		\item Alternatives to floating point: Floating slasi and Signed Logarithm (TODO: Look at these, although they will probably not be useful)
+
+	\end{itemize}
+	\item Base $\beta$ and precision $p$ (number of digits to represent with) - powers of the base can be represented exactly.
+	\item Largest and smallest exponents $e_{min}$ and $e_{max}$
+	\item Need bits for exponent and fraction, plus one for sign
+	\item ``Floating point number'' is one that can be represented exactly.
+	\item Representations are not unique! $0.01 \times 10^1 = 1.00 \times 10^{-1}$ Leading digit of one $\implies$ ``normalised''
+	\item Requiring the representation to be normalised makes it unique, {\bf but means it is impossible to represent zero}.
+	\begin{itemize}
+		\item Represent zero as $1 \times \beta^{e_{min}-1}$ - requires extra bit in the exponent
+	\end{itemize}
+	\item {\bf Rounding Error}
+	\begin{itemize}
+		\item ``Units in the last place'' eg: 0.0314159 compared to 0.0314 has ulp error of 0.159
+		\item If calculation is the nearest floating point number to the result, it will still be as much as 1/2 ulp in error
+		\item Relative error corresponding to 1/2 ulp can vary by a factor of $\beta$ ``wobble''. Written in terms of $\epsilon$
+		\item Maths $\implies$ {\bf Relative error is always bounded by $\epsilon = (\beta/2)\beta^{-p}$}
+		\item Fixed relative error $\implies$ ulp can vary by a factor of $\beta$ . Vice versa
+		\item Larger $\beta \implies$ larger errors
+	\end{itemize}
+	\item {\bf Guard Digits}
+	\begin{itemize}
+		\item In subtraction: Could compute exact difference and then round; this is expensive
+		\item Keep fixed number of digits but shift operand right; discard precision. Lead to relative error up to $\beta - 1$
+		\item Guard digit: Add extra digits before truncating. Leads to relative error of less than $2\epsilon$. This also applies to addition
+	\end{itemize}
+	\item {\bf Catastrophic Cancellation} - Operands are subject to rounding errors - multiplication
+	\item {\bf Benign Cancellation} - Subtractions. Error $< 2\epsilon$
+	\item Rearrange formula to avoid catastrophic cancellation
+	\item Historical interest only - speculation on why IBM used $\beta = 16$ for the system/370 - increased range? Avoids shifting
+	\item Precision: IEEE defines extended precision (a lower bound only)
+	\item Discussion of the IEEE standard for operations (TODO: Go over in more detail)
+	\item NaN allow continuing with underflow and Infinity with overflow
+	\item ``Incidentally, some people think that the solution to such anomalies is never to compare floating-point numbers for equality but instead to consider them equal if they are within some error bound E. This is hardly a cure all, because it raises as many questions as it answers.'' - On equality of floating point numbers
+
+\end{itemize}
+
+
+%%%%
+% David's Stuff
+%%%%
+\section{Compositing Digital Images\cite{porter1984compositing}}
+
+
+
+Perter and Duff's classic paper "Compositing Digital Images" lays the
+foundation for digital compositing today. By providing an "alpha channel,"
+images of arbitrary shapes — and images with soft edges or sub-pixel coverage
+information — can be overlayed digitally, allowing separate objects to be
+rasterized separately without a loss in quality.
 
+Pixels in digital images are usually represented as 3-tuples containing
+(red component, green component, blue component). Nominally these values are in
+the [0-1] range. In the Porter-Duff paper, pixels are stored as $(R,G,B,\alpha)$
+4-tuples, where alpha is the fractional coverage of each pixel. If the image
+only covers half of a given pixel, for example, its alpha value would be 0.5.
+
+To improve compositing performance, albeit at a possible loss of precision in
+some implementations, the red, green and blue channels are premultiplied by the
+alpha channel. This also simplifies the resulting arithmetic by having the
+colour channels and alpha channels use the same compositing equations.
+
+Several binary compositing operations are defined:
+\begin{itemize}
+\item over
+\item in
+\item out
+\item atop
+\item xor
+\item plus
+\end{itemize}
+
+The paper further provides some additional operations for implementing fades and
+dissolves, as well as for changing the opacity of individual elements in a
+scene.
+
+The method outlined in this paper is still the standard system for compositing
+and is implemented almost exactly by modern graphics APIs such as \texttt{OpenGL}. It is
+all but guaranteed that this is the method we will be using for compositing
+document elements in our project.
+
+\section{Bresenham's Algorithm: Algorithm for computer control of a digital plotter\cite{bresenham1965algorithm}}
+Bresenham's line drawing algorithm is a fast, high quality line rasterization
+algorithm which is still the basis for most (aliased) line drawing today. The
+paper, while originally written to describe how to control a particular plotter,
+is uniquely suited to rasterizing lines for display on a pixel grid.
+
+Lines drawn with Bresenham's algorithm must begin and end at integer pixel
+coordinates, though one can round or truncate the fractional part. In order to
+avoid multiplication or division in the algorithm's inner loop, 
+
+The algorithm works by scanning along the long axis of the line, moving along
+the short axis when the error along that axis exceeds 0.5px. Because error
+accumulates linearly, this can be achieved by simply adding the per-pixel
+error (equal to (short axis/long axis)) until it exceeds 0.5, then incrementing
+the position along the short axis and subtracting 1 from the error accumulator.
+
+As this requires nothing but addition, it is very fast, particularly on the
+older CPUs used in Bresenham's time. Modern graphics systems will often use Wu's
+line-drawing algorithm instead, as it produces antialiased lines, taking
+sub-pixel coverage into account. Bresenham himself extended this algorithm to
+produce Bresenham's circle algorithm. The principles behind the algorithm have
+also been used to rasterize other shapes, including B\'{e}zier curves.
+
+\section{Quad Trees: A Data Structure for Retrieval on Composite Keys\cite{finkel1974quad}}
+
+This paper introduces the ``quadtree'' spatial data structure. The quadtree structure is
+a search tree in which every node has four children representing the north-east, north-west,
+south-east and south-west quadrants of its space.
+
+\section{Xr: Cross-device Rendering for Vector Graphics\cite{worth2003xr}}
+
+Xr (now known as Cairo) is an implementation of the PDF v1.4 rendering model,
+independent of the PDF or PostScript file formats, and is now widely used
+as a rendering API. In this paper, Worth and Packard describe the PDF v1.4 rendering
+model, and their PostScript-derived API for it.
+
+The PDF v1.4 rendering model is based on the original PostScript model, based around
+a set of \emph{paths} (and other objects, such as raster images) each made up of lines
+and B\'{e}zier curves, which are transformed by the ``Current Transformation Matrix.''
+Paths can be \emph{filled} in a number of ways, allowing for different handling of self-intersecting
+paths, or can have their outlines \emph{stroked}.
+Furthermore, paths can be painted with RGB colours and/or patterns derived from either
+previously rendered objects or external raster images.
+PDF v1.4 extends this to provide, amongst other features, support for layering paths and
+objects using Porter-Duff compositing\cite{porter1984compositing}, giving each painted path
+the option of having an $\alpha$ value and a choice of any of the Porter-Duff compositing
+methods.
+
+The Cairo library approximates the rendering of some objects (particularly curved objects
+such as splines) with a set of polygons. An \texttt{XrSetTolerance} function allows the user
+of the library to set an upper bound on the approximation error in fractions of device pixels,
+providing a trade-off between rendering quality and performance. The library developers found
+that setting the tolerance to greater than $0.1$ device pixels resulted in errors visible to the
+user.
+
+\section{Glitz: Hardware Accelerated Image Compositing using OpenGL\cite{nilsson2004glitz}}
+
+This paper describes the implementation of an \texttt{OpenGL} based rendering backend for
+the \texttt{Cairo} library. 
+
+The paper describes how OpenGL's Porter-Duff compositing is easily suited to the Cairo/PDF v1.4
+rendering model. Similarly, traditional OpenGL (pre-version 3.0 core) support a matrix stack
+of the same form as Cairo.
+
+The ``Glitz'' backend will emulate support for tiled, non-power-of-two patterns/textures if
+the hardware does not support it.
+
+Glitz can render both triangles and trapezoids (which are formed from pairs of triangles).
+However, it cannot guarantee that the rasterization is pixel-precise, as OpenGL does not proveide
+this consistently.
+
+Glitz also supports multi-sample anti-aliasing, convolution filters for raster image reads (implemented
+with shaders).
+
+Performance was much improved over the software rasterization and over XRender accellerated rendering
+on all except nVidia hardware. However, nVidia's XRender implementation did slow down significantly when
+some transformations were applied.
+
+\section{Boost Multiprecision Library\cite{boost_multiprecision}}
+
+\begin{itemize}
+	\item ``The Multiprecision Library provides integer, rational and floating-point types in C++ that have more range and precision than C++'s ordinary built-in types.''
+	\item Specify number of digits for precision as a template argument.
+	\item Precision is fixed... {\bf possible approach to project:} Use \verb/boost::mpf_float<N>/ and increase \verb/N/ as more precision is required?
+\end{itemize}
 
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