| 20 |
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|
| 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} |
| 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] |
| 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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