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\lstset{
  language=C,
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\newcommand{\matM}{\mathbf{M}}
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\newcommand{\VEC}[1]{\left\langle{#1}\right\rangle}
\newcommand{\FLOOR}[1]{\left\lfloor{#1}\right\rfloor}

\title{Compiling probe operations for Diderot}
\author{
 John Reppy \\
  University of Chicago \\
  {\small\tt{}jhr@cs.uchicago.edu} \\
}
\date{\today}

\begin{document}

\maketitle
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\section{Introduction}

This note describes the code needed to implement a probe of a field operation.

In the discussion below, we use $\matM^{-1}$ for the homogeneous matrix that maps from world
coordinates to image coordinates and use $\vecx$ for the position vector.

\section{Probing a 1D scalar field}
The simplest case is probing a 1D scalar field $F = V\circledast{}h$, where $s$ is the support
of $h$.
The probe $F\mkw{@}x$ is computed as follows:
\begin{eqnarray*}
  \left[\begin{array}{c} x' \\ 1 \end{array}\right] & = & \matM^{-1} \left[\begin{array}{c} x \\ 1 \end{array}\right] \qquad \text{\textit{transform to image space}} \\
  n  & = & \FLOOR{x'} \qquad \text{\textit{integer part of position}} \\
  f  & = & x' - f \qquad \text{\textit{fractional part of position}} \\
  F\mkw{@}x & = & \sum_{i=1-s}^s {V(n+i) h(f - i)}
\end{eqnarray*}%
The convolution $h$ is represented as a symmetric piecewise polynomial formed
from $s$ polynomials $h_1,\,\ldots,\,h_s$.
\begin{displaymath}
  h(x) = \left\{\begin{array}{ll}
    0 & \text{$x < -s$} \\
    h_i(-x) & \text{$-s \leq -i \leq x < 1-i \leq 0$} \\
    h_i(x) & \text{$0 < i-1 \leq x < i \leq s$} \\
    0 & \text{$x > s$} \\
  \end{array}\right.
\end{displaymath}%
Thus, we can rewrite the probe computation as
\begin{displaymath}
  F\mkw{@}x = \sum_{i=1-s}^{0} {V(n+i) h_{1-i}(i - f)} + \sum_{i=1}^s {V(n+i) h_i(f - i)}
\end{displaymath}%
\figref{fig:1d-probe} illustrates the situation for a kernel $h$ with support $s = 2$,
and \figref{fig:1d-probe-code} gives the C code for the probe operation, assuming that
$h$ is represented by third-degree polynomials.
\begin{figure}[t]
  \begin{center}
    \includegraphics[scale=0.8]{pictures/convo}
  \end{center}%
  \caption{1D scalar probe}
  \label{fig:1d-probe}
\end{figure}%

\begin{figure}[t]
\begin{quote}
\begin{lstlisting}
double img[];			// image data
double h1[4], h2[4];		// kernel

double probe (double x)
{
  double imgX = transform(x);	// image-space position
  double n, f;	

  f = modf (imgPos, &n);

  double value = 0.0, x;

  w = -1.0 - f;
  value += img[n-1] * (h2[0] + w*(h2[1] + w*(h2[2] + w*h2[3])));
  w = -f;
  value += img[n] * (h1[0] + w*(h1[1] + w*(h1[2] + w*h1[3])));
  w = f - 1.0;
  value += img[n+1] * (h1[0] + w*(h1[1] + w*(h1[2] + w*h1[3])));
  w = f - 2.0;
  value += img[n+2] * (h2[0] + w*(h2[1] + w*(h2[2] + w*h2[3])));

  return value;
}
\end{lstlisting}%
\end{quote}%
\caption{Computing $F\mkw{@}x$ for a 1D scalar field in C}
\label{fig:1d-probe-code}
\end{figure}

\begin{figure}[t]
\begin{quote}
\begin{lstlisting}
double4 d = (double4)(h2[0], h1[0], h1[0], h2[0]); // x^0 coeffs
double4 c = (double4)(h2[1], h1[1], h1[1], h2[1]); // x^1 coeffs
double4 b = (double4)(h2[2], h1[2], h1[2], h2[2]); // x^2 coeffs
double4 a = (double4)(h2[3], h1[3], h1[3], h2[3]); // x^3 coeffs

double probe (double x)
{
  double imgX = transform(x);	// image-space position
  double n, f;	

  f = modf (imgPos, &n);

  double4 v = (double4)(img[n-1], img[n], img[n+1], img[n+2]);
  double4 w = (double4)(-1.0 - f, -f, f - 1.0, f - 2.0);
  return dot(v, d + w*(c + w*(b + w*a)));
}
\end{lstlisting}%
\end{quote}%
\caption{Computing $F\mkw{@}x$ for a 1D scalar field in OpenCL}
\label{fig:1d-probe-code-opencl}
\end{figure}

\section{Probing a 3D scalar field}

The more common case is when the field is a convolution of a scalar 3-dimensional
field ($F = V\circledast{}h$).
Let  $s$ be the support of $h$.
Then the probe $F\mkw{@}\vecp$ is computed as follows:
\begin{eqnarray*}
  \vecx & = & \matM^{-1} \vecp \qquad \text{\textit{transform to image space}} \\
  \vecn & = & \FLOOR{\vecx} \qquad \text{\textit{integer part of position}} \\
  \vecf & = & \vecx - \vecn \qquad \text{\textit{fractional part of position}} \\
  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)}}}
\end{eqnarray*}%

\begin{figure}[t]
\begin{quote}
\begin{lstlisting}
typedef double vec3[3];

typedef struct {
    int		degree;
    double	coeff[];
} polynomial;

typedef struct {
    int		support;
    polynomial	*segments[];
} kernel;

double probe (vec3 ***img, kernel *h, vec3 pos)
{
}
\end{lstlisting}%
\end{quote}%
\caption{Computing $F\mkw{@}\vecx$ for a 3D scalar field in C}
\end{figure}

\section{Probing a 3D derivative field}
We next consider the case of probing the derivative of a scalar field $F = V\circledast{}h$, where $s$ is the support
of $h$.
The probe $(\mkw{D}\;F)\mkw{@}\vecp$ produces a vector result as follows:
\begin{eqnarray*}
  \vecx & = & \matM^{-1} \vecp \qquad \text{\textit{transform to image space}} \\
  \vecn & = & \FLOOR{\vecx} \qquad \text{\textit{integer part of position}} \\
  \vecf & = & \vecx - \vecn \qquad \text{\textit{fractional part of position}} \\
  (\mkw{D}\;F)\mkw{@}\vecp & = & \left[\begin{array}{c}
    \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)}}} \\
    \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)}}} \\
    \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)}}} \\
  \end{array}\right]
\end{eqnarray*}%

\end{document}