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%%%%%%%	     Wooster MAA/SIAM Presentation, October , 1999	%%%%%%

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\Header{MPI}
\begin{slide*}
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                          framesep=2mm]{\Large\bfseries\color{yellow}%
                Message Passing Interface}}
  \vspace{2cm}
  \centerline{\psframebox[fillstyle=solid,fillcolor=Beige,
                          framearc=.3]{\bfseries%
              \begin{tabular}{c}
                Jalel \textsc{Chergui}\\
                Isabelle \textsc{Dupays}\\
                Denis \textsc{Girou}\\
                Stephane \textsc{Requena}\\[0.8mm]
                {\small\texttt{<Prenom.Nom@idris.fr>}}
              \end{tabular}}}
\end{slide*}


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%			slide number: 1
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\begin{slide*}{}
\heading{Simple Skins for Matrices}
\heading{and Other Meshes:}
\heading{Surface Fitting via Matrix SVD}
\large

\bigskip
{\centerline{{\blue{Andy Long and Cliff Long}}}}

\bigskip
{\centerline{October 22, 1999}}

\bigskip
{\centerline{Ohio Section of the MAA}}

\bigskip
{\centerline{College of Wooster}}

\end{slide*}{}

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\SlideColours{white}{red}
\begin{slide*}{}
\heading{The Idea:}

\begin{itemize}
\item
	Decompose Matrix $A_{m \times n}$ by SVD: $A=U \Lambda V^T$ 
\item
	Associate $A$ with spatial coordinates by mapping to a grid in the
	plane (usually simply associate indices with space coordinates):
	$a_{ij}=a(i,j)$.
\item
	Column vectors of $U$ and $V$ form natural bases of the column and row
	spaces of $A$, respectively.
\item
	Interpolate these vectors, and recombine to get (you'll see!)
\[
	A(x,y) = \u{U}(x)^T \Lambda \u{V}(y)
\]
\end{itemize}

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\end{slide*}{}
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\begin{slide*}{}
\heading{An Example of SVD}
\small
{\bf Let's keep it simple:}
\[
A =
	\left [
\begin{array}{cc}
{-0.4}&{2}\cr
{2.2}&{-1}
\end{array}
\right ] = 
\]

\[
\left [
\begin{array}{cc}
{-.6}&{.8}\cr
{.8}&{.6}
\end{array}
\right ]
\left [
\begin{array}{cc}
{2\sqrt{2}}&{0}\cr
{0}&{\sqrt{2}}
\end{array}
\right ]
\left [
\begin{array}{cc}
{\frac{1}{\sqrt{2}}}&{-\frac{1}{\sqrt{2}}}\cr
{\frac{1}{\sqrt{2}}}&{\frac{1}{\sqrt{2}}}
\end{array}
\right ] =
\]

\[
\left [
\begin{array}{c}
{-.6}\cr
{.8}
\end{array}
\right ]
2\sqrt{2}
\left [
\begin{array}{cc}
{\frac{1}{\sqrt{2}}}&{-\frac{1}{\sqrt{2}}}
\end{array}
\right ]
+
\left [
\begin{array}{cc}
{.8}\cr
{.6}
\end{array}
\right ]
\sqrt{2}
\left [
\begin{array}{cc}
{\frac{1}{\sqrt{2}}}&{\frac{1}{\sqrt{2}}}
\end{array}
\right ]
\]
{\bf{Insight:}} we can think of the SVD as
\[
	A = \sum_{k=1}^{m} \u{u}^k \lambda^k (\u{v}^k)^T,
\]
where $\u{u}^k$ is the $k^{th}$ column vector of $U$, etc. Matrices can be
represented using {\bf{one-dimensional objects}}.

\end{slide*}{}

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\begin{slide*}{}
\heading{How does the fitting process work?}
\small
Thus, in the matrix sense,
\[
	a_{ij} = \sum_{k=1}^{n} \u{u}^k_{i} \lambda^k \u{v}^k_{j}.
\]
Using our spatial indexing we have
\[
	a(i,j) = \sum_{k=1}^{n} \u{u}^k(i) \lambda^k \u{v}^k(j),
\]
or
\huge
\[
	A(x,y) = \sum_{k=1}^{n} u^k(x) \lambda^k v^k(y)
\]
\small
	If $u^k(x)$ and $v^k(y)$ are interpolators of their respective vectors,
then it is clear that we have interpolated the matrix by this scheme; if they
smooth (rather than interpolate), then we have a smoothed fit.

\end{slide*}{}

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%
%\figger{/afs/sph.umich.edu/user/a/aelon/dad/CMJ/nnfig1.eps}{3.5}{The
%``skeleton'' of an example data matrix $A$.}{67 212 562 583}
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%
%\figger{/afs/sph.umich.edu/user/a/aelon/dad/CMJ/nnfig2.eps}{3.5}{Top row: The
%singular vectors corresponding to the largest singular value and both their
%linear-spline and Hermite interpolants. Bottom row: at left the outer product
%skeleton of the singular vectors, and at right the product of the Hermite
%interpolating functions - a skin of the first rank-one matrix.}{66 211 549 608}
%
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%
%\figger{/afs/sph.umich.edu/user/a/aelon/dad/CMJ/nnfig3.eps}{3.5}{Top row: at
%left the outer product skeleton of the singular vectors corresponding to the
%second singular value, and at right its Hermite skin. At bottom we show the
%data matrix $A$ and the result of adding together the
%rank-one interpolants to form a $C^1$-continuous interpolant to $A$.}{67 211
%545 583}
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%\figger{/afs/sph.umich.edu/user/a/aelon/dad/CMJ/fig-abe.ps}{4}{At left is the
%scanned image of Abe; at right the improved Abe, using roughly 25 times as many
%points obtained via Hermite interpolation.}{-93 196 707 596}
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%
%\figger{/afs/sph.umich.edu/user/a/aelon/dad/CMJ/abescanner.ps}{4}{This crude
%scanner was used to read points off a bust of Abraham Lincoln, and a ``mere''
%1728 points required a good deal of labor!}{0 0 435 262}
%
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%
%% \figgertoo{/afs/sph.umich.edu/user/a/aelon/dad/CMJ/out/faroukih.rle.ps}
%% {/afs/sph.umich.edu/user/a/aelon/dad/CMJ/out/farouki.rle.ps}
%% {/afs/sph.umich.edu/user/a/aelon/dad/CMJ/out/faroukis.rle.ps}
%% {Top: Farouki's $6 \times 3$ matrix of points.  The original triplets are
%% points in three-space which fall roughly on a pinched grid. At top and bottom
%% are two different skins generated for Farouki's mesh: at top, a Hermite
%% interpolant, and, at bottom, a cubic spline interpolant.}{106 256 506 536}
%
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%
%\figger{SVDfarouki.eps}{3.5}{Farouki mesh: interpolated coordinate by
%coordinate.}{74 420 543 583}
%
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%\figger{/afs/sph.umich.edu/user/a/aelon/dad/cagd/fig3nn.eps}{3.5}{Using this
%method we derived a method for matching arbitrary curve meshes.}{81 414 552
%583}
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%
%\figger{recycle2.eps}{3.5}{Twisted
%recycling symbol, constructed from data.}{67 208 567 604}
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\begin{slide*}{}
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%			slide number: 
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\heading{Conclusion}
\begin{itemize}
\item
\small
	In a nice, concise form,
\[
	A(x,y) = \u{U}(x)^T \Lambda \u{V}(y)
\]
where $u^k(x) = \u{U}(x)_k$, etc.

\item
	Our method provides a simple, intuitive, and easily implemented way to
create skins for mesh objects (particularly matrices). An example of one type
of odd mesh is Abe, in cylindrical coordinates; odder examples include the
Farouki data and recycling symbol.

\item
	Although we haven't gone into the details, this method allows for the
matching of boundary curves.

\item
	Differentiability of the resulting surface is a direct consequence of
the differentiability of the interpolating functions.

\end{itemize}

\end{slide*}{}
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