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[diderot] Diff of /trunk/doc/probe/paper.tex
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Diff of /trunk/doc/probe/paper.tex

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revision 158, Mon Jul 12 21:56:50 2010 UTC revision 159, Tue Jul 13 02:46:31 2010 UTC
# Line 20  Line 20 
20    
21  \newcommand{\matM}{\mathbf{M}}  \newcommand{\matM}{\mathbf{M}}
22  \newcommand{\vecx}{\mathbf{x}}  \newcommand{\vecx}{\mathbf{x}}
23    \newcommand{\vecp}{\mathbf{p}}
24  \newcommand{\vecn}{\mathbf{n}}  \newcommand{\vecn}{\mathbf{n}}
25  \newcommand{\vecf}{\mathbf{f}}  \newcommand{\vecf}{\mathbf{f}}
26  \newcommand{\VEC}[1]{\left\langle{#1}\right\rangle}  \newcommand{\VEC}[1]{\left\langle{#1}\right\rangle}
# Line 145  Line 146 
146  The more common case is when the field is a convolution of a scalar 3-dimensional  The more common case is when the field is a convolution of a scalar 3-dimensional
147  field ($F = V\circledast{}h$).  field ($F = V\circledast{}h$).
148  Let  $s$ be the support of $h$.  Let  $s$ be the support of $h$.
149  Then the probe $F\mkw{@}\vecx$ is computed as follows:  Then the probe $F\mkw{@}\vecp$ is computed as follows:
150  \begin{eqnarray*}  \begin{eqnarray*}
151    \vecx' & = & \matM^{-1} \vecx \qquad \text{\textit{transform to image space}} \\    \vecx & = & \matM^{-1} \vecp \qquad \text{\textit{transform to image space}} \\
152    \vecn  & = & \FLOOR{\vecx'} \qquad \text{\textit{integer part of position}} \\    \vecn & = & \FLOOR{\vecx} \qquad \text{\textit{integer part of position}} \\
153    \vecf  & = & \vecx' - \vecn \qquad \text{\textit{fractional part of position}} \\    \vecf & = & \vecx - \vecn \qquad \text{\textit{fractional part of position}} \\
154    F\mkw{@}\vecx & = & \sum_{i=1-s}^s {\sum_{j=1-s}^s {\sum_{k=1-s}^s {V(\vecn+\VEC{i,j,k}) h(\vecf_x - i) h(\vecf_y - j) h(\vecf_z - k)}}}    F\mkw{@}\vecp & = & \sum_{i=1-s}^s {\sum_{j=1-s}^s {\sum_{k=1-s}^s {V(\vecn+\VEC{i,j,k}) h(\vecf_x - i) h(\vecf_y - j) h(\vecf_z - k)}}}
155  \end{eqnarray*}%  \end{eqnarray*}%
156    
157  \begin{figure}[t]  \begin{figure}[t]
# Line 176  Line 177 
177  \caption{Computing $F\mkw{@}\vecx$ for a 3D scalar field in C}  \caption{Computing $F\mkw{@}\vecx$ for a 3D scalar field in C}
178  \end{figure}  \end{figure}
179    
180    \section{Probing a 3D derivative field}
181    We next consider the case of probing the derivative of a scalar field $F = V\circledast{}h$, where $s$ is the support
182    of $h$.
183    The probe $(\mkw{D}\;F)\mkw{@}\vecp$ produces a vector result as follows:
184    \begin{eqnarray*}
185      \vecx & = & \matM^{-1} \vecp \qquad \text{\textit{transform to image space}} \\
186      \vecn & = & \FLOOR{\vecx} \qquad \text{\textit{integer part of position}} \\
187      \vecf & = & \vecx - \vecn \qquad \text{\textit{fractional part of position}} \\
188      (\mkw{D}\;F)\mkw{@}\vecp & = & \left[\begin{array}{c}
189        \sum_{i=1-s}^s {\sum_{j=1-s}^s {\sum_{k=1-s}^s {V(\vecn+\VEC{i,j,k}) h'(\vecf_x - i) h(\vecf_y - j) h(\vecf_z - k)}}} \\
190        \sum_{i=1-s}^s {\sum_{j=1-s}^s {\sum_{k=1-s}^s {V(\vecn+\VEC{i,j,k}) h(\vecf_x - i) h'(\vecf_y - j) h(\vecf_z - k)}}} \\
191        \sum_{i=1-s}^s {\sum_{j=1-s}^s {\sum_{k=1-s}^s {V(\vecn+\VEC{i,j,k}) h(\vecf_x - i) h(\vecf_y - j) h'(\vecf_z - k)}}} \\
192      \end{array}\right]
193    \end{eqnarray*}%
194    
195  \end{document}  \end{document}

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