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[diderot] Diff of /branches/vis12-cl/doc/diderot.tex

[diderot] / branches / vis12-cl / doc / diderot.tex

Diff of /branches/vis12-cl/doc/diderot.tex

-revision 21, Sat Jan 16 18:16:32 2010 UTC
+revision 22, Thu Jan 21 05:34:09 2010 UTC
 Line 68
  \section{Operations}
  \subsection{Scalar operations}
- \noindent{}Arithmetic:
+ \noindent{}\point Arithmetic:
  \begin{displaymath}
    \begin{array}{cl}
      \BinopTy{{\odot}}{\tau}{\tau}{\tau}
 Line 78
    \end{array}%
  \end{displaymath}%
- \noindent{}Comparisons:
+ \noindent{}\point Comparisons:
  \begin{displaymath}
    \BinopTy{{\odot}}{\tau}{\tau}{\TYbool}
    \qquad\text{for $\odot\in\SET{{<},{\leq},{=},{\neq}{>},{\geq}}$ and $\tau\in\SET{\TYint,\TYreal}$}
 Line 89
  \subsection{Tensor operations}
- \noindent{}Scalar multiplication:
+ \noindent{}\point Addition:
+ \begin{displaymath}
+   \BinopTy{{\odot}}{\TYtensor{o}{d}}{\TYtensor{o}{d}}{\TYtensor{o}{d}}
+   \qquad\text{for $\odot\in\SET{{+},{-}}$}
+ \end{displaymath}%
+ \noindent{}\point Negation:
+ \begin{displaymath}
+   \UnopTy{-}{\TYtensor{o}{d}}{\TYtensor{o}{d}}
+ \end{displaymath}%
+ \noindent{}\point Scalar division:
+ \begin{displaymath}
+   \BinopTy{{/}}{\TYtensor{o}{d}}{\TYreal}{\TYtensor{o}{d}}
+ \end{displaymath}%
+ \noindent{}\point Scalar multiplication (scalar times order-N):
  \begin{displaymath}
    \begin{array}{c}
-     \BinopTy{{*}}{\TYreal}{\TYtensor{o}{d}}{\TYtensor{o}{d}} \\
+     \BinopTy{{?}}{\TYreal}{\TYtensor{o}{d}}{\TYtensor{o}{d}} \\
-     \BinopTy{{*}}{\TYtensor{o}{d}}{\TYreal}{\TYtensor{o}{d}}
+     \BinopTy{{?}}{\TYtensor{o}{d}}{\TYreal}{\TYtensor{o}{d}}
    \end{array}%
  \end{displaymath}%
+ Possible direct notation syntax (TBD): ~~ {\tt *} ~~ `` '' (space)
- \noindent{}Scalar division:
+ \noindent{}\point Tensor scalar multiplication (contraction of two order-N
+ tensors down to a scalar, \eg{} dot product of vectors, double dot
+ product of 2nd-order tensors)
  \begin{displaymath}
-   \BinopTy{{/}}{\TYtensor{o}{d}}{\TYreal}{\TYtensor{o}{d}}
+   \begin{array}{c}
+     \BinopTy{{?}}{\TYtensor{o}{d}}{\TYtensor{o}{d}}{\TYreal}
+   \end{array}%
  \end{displaymath}%
+ Defined by $c = A_{\vec{i}}B_{\vec{i}}$
+ ~(\eg{} $\mathbf{a}\cdot\mathbf{b} = a_ib_i$  or $\mathbf{A:B} = A_{ij}B_{ij}$)
+ NOTE: In this and subsequent index notation expressions, a vector over the index variable
+ (\eg{} $\vec{i}$) means that the variable is in fact standing for a sequence of
+ contiguous index variables \\
+ Possible direct notation syntax: ~~ {\tt .} (period) ~~ {\tt :} (colon) ~~ {\tt dot} ~~ {\tt o}
- \noindent{}Addition:
+ \noindent{}\point Tensor product (aka outer product; order output is sum of orders)
  \begin{displaymath}
-   \BinopTy{{\odot}}{\TYtensor{o}{d}}{\TYtensor{o}{d}}{\TYtensor{o}{d}}
+   \begin{array}{c}
-   \qquad\text{for $\odot\in\SET{{+},{-}}$}
+     \BinopTy{{?}}{\TYtensor{o}{d}}{\TYtensor{p}{d}}{\TYtensor{o+p}{d}}
+   \end{array}%
  \end{displaymath}%
+ Defined by $C_{\vec{i}\vec{j}} = A_{\vec{i}}B_{\vec{j}}$ \\
+ Possible direct notation syntax: ~~ {\tt x} ~~ {\tt (x)} ~~ {\tt out}
- \noindent{}Negation:
+ \noindent{}\point Matrix Multiply
  \begin{displaymath}
-   \UnopTy{-}{\TYtensor{o}{d}}{\TYtensor{o}{d}}
+   \begin{array}{c}
+     \BinopTy{{?}}{\TYtensor{o}{d}}{\TYtensor{2}{d}}{\TYtensor{o}{d}} \\
+     \BinopTy{{?}}{\TYtensor{2}{d}}{\TYtensor{o}{d}}{\TYtensor{o}{d}}
+   \end{array}%
+ \end{displaymath}%
+ Defined by $C_{i\vec{k}} = A_{ij}B_{j\vec{k}}$ and $C_{\vec{i}k} = B_{\vec{i}j}A_{jk}$
+ ~(\eg{} $u_i = M_{ij}v_j$ or $T_{ijl} = S_{ijk}M_{kl}$) \\
+ Possible direct notation syntax: ~~ ?
+ \noindent{}\point Contracting out last or first index of tensor (order $o \geq 1$)
+ with vector
+ \begin{displaymath}
+   \begin{array}{c}
+     \BinopTy{{?}}{\TYtensor{o}{d}}{\TYvec{d}}{\TYtensor{o-1}{d}} \\
+     \BinopTy{{?}}{\TYvec{d}}{\TYtensor{o}{d}}{\TYtensor{o-1}{d}}
+   \end{array}%
+ \end{displaymath}%
+ Defined by $C_{\vec{i}} = A_{\vec{i}j}v_j$ and $C_{\vec{j}} = v_iA_{j\vec{j}}$ \\
+ Possible direct notation syntax: ~~ ?
+ \noindent{}\point Arbitrary ``tensor comprehension''. The product can also be
+ expressed in general index notation, and which may increase, preserve, or
+ decrease the tensor order.
+ \begin{displaymath}
+   \begin{array}{c}
+     \BinopTy{{?}}{\TYtensor{o}{d}}{\TYtensor{p}{d}}{\TYtensor{q}{d}}
+   \end{array}%
  \end{displaymath}%
+ Defined by the conventions of Einstein summation notation. \\
+ Possible syntax: {\tt <A.i.j.k,B.j.k.l>} ~~ {\tt <A\_i\_j\_k,B\_j\_k\_l>}
  \subsection{Field operations}
- \noindent{}Creation from an image:
+ \noindent{}\point Creation from an image:
  \begin{displaymath}
-   \BinopTy{\OPconvolve}{\TYkern{k}}{\TYimage{d}{\mu}}{\TYfield{k}{d}{\theta}}
+   \BinopTy{\OPconvolve}{\TYkern{k}}{\TYimage{d}{\TYrawten{o}{d}{\rho}}}{\TYfield{k}{d}{\TYtensor{o}{d}}}
-   \qquad\text{where $\theta$ is the real conversion of $\mu$.}
  \end{displaymath}%
- \noindent{}Scalar multiplication:
+ \noindent{}\point Scalar multiplication:
  \begin{displaymath}
    \begin{array}{c}
      \BinopTy{{*}}{\TYreal}{\TYfield{k}{d}{\theta}}{\TYfield{k}{d}{\theta}} \\
-Line 129
+Line 186
    \end{array}%
  \end{displaymath}%
- \noindent{}Scalar division:
+ \noindent{}\point Scalar division:
  \begin{displaymath}
    \BinopTy{{/}}{\TYfield{k}{d}{\theta}}{\TYreal}{\TYfield{k}{d}{\theta}}
  \end{displaymath}%
- \noindent{}Negation:
+ \noindent{}\point Negation:
  \begin{displaymath}
    \UnopTy{-}{\TYfield{k}{d}{\theta}}{\TYfield{k}{d}{\theta}}
  \end{displaymath}%
- \noindent{}Addition:
+ \noindent{}\point Addition:
  \begin{displaymath}
    \begin{array}{c}
    \BinopTy{{\odot}}{\TYfield{k}{d}{\theta}}{\theta}{\TYfield{k}{d}{\theta}} \\
-Line 149
+Line 206
    \qquad\text{for $\odot\in\SET{{+},{-}}$}
  \end{displaymath}%
- \noindent{}Differentiation:
+ \noindent{}\point Differentiation:
  \begin{displaymath}
    \UnopTy{\OPdiff}{\TYfield{k}{d}{\TYtensor{o}{d}}}{\TYfield{k-1}{d}{\TYtensor{o+1}{d}}}
    \qquad\text{for $k > 0$}
  \end{displaymath}%
- \noindent{}Probing:
+ \noindent{}\point Probing:
  \begin{displaymath}
    \BinopTy{@}{\TYfield{k}{d}{\theta}}{\TYvec{d}}{\theta}
  \end{displaymath}%

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