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\title{Multi-Resolution Analysis for the Haar Wavelet: \\A Minimalist Approach}
\institute{{\large Helmut Knaust}\medskip\\Department of Mathematical Sciences\\The University of Texas at El Paso\\El Paso TX 79968-0514\medskip\\\tt{hknaust@utep.edu}}
\date{{\small Joint Mathematics Meetings\\San Antonio TX\\\smallskip January 12, 2015}}
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\begin{document}
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\titlepage
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\section{Math 5311: Applied Mathematics}
\subsection{Course Objectives}
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I am regularly teaching a discrete wavelets course as part of my department's \textit{Master of Arts in Teaching Mathematics} program. 
\begin{itemize}
\item Diverse student population:
\begin{itemize}
\item  Current high school (and middle school) teachers
\item Recent graduates of our B.S. program, \item Career switchers
\end{itemize}
\item This is the only required applied mathematics course in the program.
\end{itemize}
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\begin{frame}
Course Objectives:
\begin{enumerate}
\item Get a flavor of the ideas and issues involved in applying mathematics to a relevant engineering problem
\item<2-> Learn how to use a computer algebra system for mathematical investigations, as a computational and visualization aid, and for the implementation of mathematical algorithms
\item<3-> Develop an understanding of the theoretical underpinnings of wavelet transforms and their applications
\end{enumerate}
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Image compression is used as a model of how applied math works:
\begin{itemize}
\item Start with a ``vague'' engineering problem
\item<2-> Introduce the necessary concepts (PSNR, entropy) to turn this into a mathematical problem 
\item<3->Open the mathematical toolbox to find solutions (Fourier analysis)
\item<4-> Find better tools (wavelets) or improve the existing ones.
\item<5-> Understand what else the tools can do in the given context (e.g., edge detection).
\end{itemize}
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\subsection{Prerequisites}
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Student Prerequisites:
\begin{itemize}
\item A thorough understanding of Calculus
\item Some familiarity with matrices
\item Mathematical maturity
\item Willingness to learn \emph{Mathematica}
\end{itemize}
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\subsection{Textbook}
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Textbook:
\begin{columns}[c]
\column{1.5in}
Patrick Van Fleet,\\
\textit{Discrete Wavelet Transformations: An Elementary Approach with Applications}.\\Wiley-Interscience.
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\section{MRA for the Haar Wavelet}
\subsection{Motivation}
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Multi-Resolution Analysis for the Haar wavelet
\begin{itemize}
\item<1-> A concrete introduction to an abstract topic
\item<2-> Easier than the general case (unit interval, only one direction)
\item<3-> Reinforces the notion of orthogonality of functions
\item<4-> Seamlessly leads to discrete transformations via matrices
\end{itemize}
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\subsection{The steps}
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\begin{equation*}
	\phi(x)=\left\{\begin{array}{rl}1,&\mbox{ if } x\in [0,1)\\0,&\mbox{ if } x< 0 \mbox{ or } x\geq 1\end{array}\right.
\end{equation*}
\begin{center}
\resizebox{2.5in}{!} {\includegraphics{nb/002.pdf}}
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$\phi:\real\to\real$ is called the \textsc{Haar scaling function}. Throughout we will identify $\phi(x)$ with its restriction to $[0,1)$.
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Notice that we obtain the dilation equation
\[\phi(x)=\phi(2x)+\phi(2x-1).\]

Use this to define a sequence of vector spaces $V_0\subseteq V_1\subseteq V_2, \ldots$, generated by the dilations of $\phi$, i.e.,

$V_j$ is spanned by the orthogonal functions 
\[2^{j/2}\phi(2^j x), 2^{j/2}\phi(2^j x-1),\ldots,2^{j/2}\phi(2^j x-(2^j-1)).\]

The vector space $V_j$ consists of all functions on $[0,1)$ that are constant on the standard  intervals of length $2^{-j}$. 
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Gram-Schmidt orthogonalization to replace the basis in $V_1$ by the basis vector in $V_0$ plus an orthonormal vector $\psi(x)$ in $V_1$ yields the Haar wavelet function 
\begin{eqnarray*}
\psi(x)	&=& \phi(2x)-\phi(2x-1)\\&=&\left\{\begin{array}{rl}1,&\mbox{ if } x\in [0,\frac{1}{2})\\-1,&\mbox{ if } x\in[\frac{1}{2},1)\\0,&\mbox{ otherwise}\end{array}\right.
\end{eqnarray*}
Then use dilations of $\psi(x)$ to create the multi-resolution analysis
\[V_j=V_0\oplus W_0 \oplus W_1\oplus \cdots \oplus W_{j-1}\]
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\subsection{The DWT matrix}
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Consider the example $V_4=V_3\oplus W_3$.

($V_4$ has dimension 16, while both $V_3$ and $W_3$ have dimension 8.)

Given a function in $V_4$,  an elementary calculation yields that the coefficient $c_k$ of the corresponding function in $W_3$ is given by
\begin{equation*}
	c_k=\frac{a_{2k}-a_{2k+1}}{\sqrt{2}},
\end{equation*}
where  $a_k$ denotes the $k$th coefficient of the function in $V_4$. 

In other words, we obtain the vector representing the function in $W_3$ by multiplying the vector $(a_0,a_2,\ldots,a_{15})$ of the function in $V_4$ by the following matrix.
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 \begin{equation*}	\left(
\begin{array}{llllllll}
 \frac{1}{\sqrt{2}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
 -\frac{1}{\sqrt{2}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
 0 & \frac{1}{\sqrt{2}} & 0 & 0 & 0 & 0 & 0 & 0 \\
 0 & -\frac{1}{\sqrt{2}} & 0 & 0 & 0 & 0 & 0 & 0 \\
 &&&&\vdots&&&\\
 0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{\sqrt{2}} & 0 \\
 0 & 0 & 0 & 0 & 0 & 0 & -\frac{1}{\sqrt{2}} & 0 \\
 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{\sqrt{2}} \\
 0 & 0 & 0 & 0 & 0 & 0 & 0 & -\frac{1}{\sqrt{2}}
\end{array}
\right)
\end{equation*}
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Similarly the coefficients in $V_3$ are generated by multiplying the coefficients in $V_4$ by the matrix
\begin{equation*}
	\left(
\begin{array}{llllllll}
 \frac{1}{\sqrt{2}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
 \frac{1}{\sqrt{2}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
 0 & \frac{1}{\sqrt{2}} & 0 & 0 & 0 & 0 & 0 & 0 \\
 0 & \frac{1}{\sqrt{2}} & 0 & 0 & 0 & 0 & 0 & 0 \\
 &&&&\vdots&&&\\
 0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{\sqrt{2}} & 0 \\
 0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{\sqrt{2}} & 0 \\
 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{\sqrt{2}} \\
 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{\sqrt{2}}
\end{array}
\right)
\end{equation*}
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Contact:\medskip\\
{\tt{hknaust@utep.edu}}
\medskip\\
Handout:\medskip\\
\href{http://helmut.knaust.info/paper/MRA.pdf}{\tt helmut.knaust.info/paper/MRA.pdf}
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