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Use \ccSum.

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This commit is contained in:
Sylvain Pion
2009-02-02 17:07:31 +00:00
parent 40200e96cf
commit 2c5ebdb4f4
3 changed files with 24 additions and 24 deletions
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Interpolation/doc_tex/Interpolation/interpolation.tex
+15 -15
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@@ -7,11 +7,11 @@ re-produces linear functions exactly. The interpolation of
$\Phi(\mathbf{x})$ is given as the linear combination of the neighbors' function
values weighted by the coordinates:
\begin{displaymath}
Z^0(\mathbf{x}) = \sum_i \lambda_i(\mathbf{x}) z_i.
Z^0(\mathbf{x}) = \ccSum{i}{}{ \lambda_i(\mathbf{x}) z_i}.
\end{displaymath}
Indeed, if $z_i=a + \mathbf{b}^t \mathbf{p_i}$ for all natural
neighbors of $\mathbf{x}$, we have
\[ Z^0(\mathbf{x}) = \sum_i \lambda_i(\mathbf{x}) (a + \mathbf{b}^t\mathbf{p_i}) = a+\mathbf{b}^t \mathbf{x}\]
\[ Z^0(\mathbf{x}) = \ccSum{i}{}{ \lambda_i(\mathbf{x}) (a + \mathbf{b}^t\mathbf{p_i})} = a+\mathbf{b}^t \mathbf{x}\]
by the barycentric coordinate property. The first example in
Subsection~\ref{subsec:interpol_examples} shows how the function is
called.
@@ -31,20 +31,20 @@ Sibson's $Z^1$ interpolant is a combination of the linear interpolant
$Z^0$ and an interpolant $\xi$ which is the weighted sum of the first
degree functions
$$\xi_i(\mathbf{x}) = z_i
+\mathbf{g_i}^t(\mathbf{x}-\mathbf{p_i}),\qquad \xi(\mathbf{x})= \frac{\sum_i \frac{\lambda_i(\mathbf{x})}
{\|\mathbf{x}-\mathbf{p_i}\|}\xi_i(\mathbf{x}) }{\sum_i
\frac{\lambda_i(\mathbf{x})}{\|\mathbf{x}-\mathbf{p_i}\|}}.$$
+\mathbf{g_i}^t(\mathbf{x}-\mathbf{p_i}),\qquad \xi(\mathbf{x})= \frac{\ccSum{i}{}{ \frac{\lambda_i(\mathbf{x})}
{\|\mathbf{x}-\mathbf{p_i}\|}\xi_i(\mathbf{x}) } }{\ccSum{i}{}{
\frac{\lambda_i(\mathbf{x})}{\|\mathbf{x}-\mathbf{p_i}\|}}}.$$
Sibson observed that the combination of $Z^0$ and $\xi$ reconstructs exactly
a spherical quadric if they are mixed as follows:
$$
Z^1(\mathbf{x}) = \frac{\alpha(\mathbf{x}) Z^0(\mathbf{x}) +
\beta(\mathbf{x}) \xi(\mathbf{x})}{\alpha(\mathbf{x}) +
\beta(\mathbf{x})} \textrm{ where } \alpha(\mathbf{x}) =
\frac{\sum_i \lambda_i(\mathbf{x}) \frac{\|\mathbf{x} -
\mathbf{p_i}\|^2}{f(\|\mathbf{x} - \mathbf{p_i}\|)}}{\sum_i
\frac{\lambda_i(\mathbf{x})} {f(\|\mathbf{x} - \mathbf{p_i}\|)}}
\textrm{ and } \beta(\mathbf{x})= \sum_i \lambda_i(\mathbf{x})
\|\mathbf{x} - \mathbf{p_i}\|^2,$$
\beta(\mathbf{x})} \mbox{ where } \alpha(\mathbf{x}) =
\frac{\ccSum{i}{}{ \lambda_i(\mathbf{x}) \frac{\|\mathbf{x} -
\mathbf{p_i}\|^2}{f(\|\mathbf{x} - \mathbf{p_i}\|)}}}{\ccSum{i}{}{
\frac{\lambda_i(\mathbf{x})} {f(\|\mathbf{x} - \mathbf{p_i}\|)}}}
\mbox{ and } \beta(\mathbf{x})= \ccSum{i}{}{ \lambda_i(\mathbf{x})
\|\mathbf{x} - \mathbf{p_i}\|^2},$$
where in Sibson's original work,
$f(\|\mathbf{x} - \mathbf{p_i}\|) = \|\mathbf{x} - \mathbf{p_i}\|$.
@@ -77,8 +77,8 @@ Knowing the gradient $\mathbf{g_i}$ for all $\mathbf{p_i} \in
exactly quadratic functions. This interpolant is not $C^1$ continuous
in general. It is defined as follows:
\begin{displaymath}
I^1(\mathbf{x}) = \sum_i \lambda_i(\mathbf{x})
(z_i + \frac{1}{2} \mathbf{g_i}^t (\mathbf{x} - \mathbf{p_i}))
I^1(\mathbf{x}) = \ccSum{i}{}{ \lambda_i(\mathbf{x})
(z_i + \frac{1}{2} \mathbf{g_i}^t (\mathbf{x} - \mathbf{p_i}))}
\end{displaymath}
@@ -89,9 +89,9 @@ $f$ from the function values on the data sites. For the data point
$\mathbf{p_i}$, we determine
$$\mathbf{g_i}
= \min_{\mathbf{g}}
\sum_j
\ccSum{j}{}{
\frac{\lambda_j(\mathbf{p_i})}{\|\mathbf{p_i} - \mathbf{p_j}\|^2}
\left( z_j - (z_i + \mathbf{g}^t (\mathbf{p_j} -\mathbf{p_i})) \right),
\left( z_j - (z_i + \mathbf{g}^t (\mathbf{p_j} -\mathbf{p_i})) \right)},
$$
where $\lambda_j(\mathbf{p_i})$ is the natural neighbor coordinate
of $\mathbf{p_i}$ with respect to $\mathbf{p_i}$ associated to