Changeset 1028 for trunk/doc/pslib/psLibADD.tex

Timestamp:

Jun 14, 2004, 11:19:36 AM (22 years ago)

Author:

Paul Price

Message:

Removed incorrect algorithm for 2D Chebyshev fitting.

File:

• trunk/doc/pslib/psLibADD.tex (modified) (2 diffs)

trunk/doc/pslib/psLibADD.tex

@@ -1 @@
-%%% $Id: psLibADD.tex,v 1.17 2004-05-24 20:14:06 eugene Exp $
+%%% $Id: psLibADD.tex,v 1.18 2004-06-14 21:19:36 price Exp $
 \documentclass[panstarrs]{panstarrs}
 
@@ -520 +520 @@
 \end{center}
 
-\paragraph{Fitting a 2D Chebyshev Polynomial}
-
-Suppose we have an image, $z = z(x,y)$ with corresponding error
-estimates, $\sigma_z(x,y)$, and want to fit this with a 2D Chebyshev
-polynomial,
-\begin{equation}
-T(x,y) = \sum_{i,j} P_{i,j} T_i(x) T_j(y)
-\end{equation}
-where $T_i(x)$ is a Chebyshev polynomial in $x$ of order $i$
-(\S\ref{sec:polynomials}).  Then we can calculate the coefficients,
-$P_{i,j}$ by minimising $\chi^2$ in the standard manner:
-
-\begin{eqnarray}
-\chi^2 & = & \sum_{x,y} \left[ \frac{z(x,y) - P_{i,j} T_i(x) T_j(y)}{\sigma_z(x,y)} \right] ^2 \\
-\frac{\partial \chi^2}{\partial P_{k,l}} & = & -2 \sum_{x,y} \left[ \frac{z(x,y) - \sum_{i,j} P_{i,j} T_i(x) T_j(y)}{\sigma_z(x,y)} \right] \frac{T_k(x) T_l(y)}{\sigma_z(x,y)}
-\end{eqnarray}
-
-Setting the partial derivative to zero and assuming that the errors
-are approximately equal over the image (as is the case for background
-fitting), $\sigma_z(x,y) = \sigma = {\rm const}$, then:
-\begin{equation}
-\sum_{x,y} z(x,y) T_k(x) T_l(y) = \sum_{i,j} \sum_{x,y} P_{i,j} T_i(x) T_j(y) T_k(x) T_l(y)
-\end{equation}
-Then we use the orthogonality property of Chebyshev polynomials, namely,
-\begin{equation}
-\sum_{x=0}^N T_i(x) T_j(x) = \begin{cases} 0 & i \ne j \\
-N/2 & i=j \ne 0 \\
-N & i=j=0 \\
-\end{cases}
-\end{equation}
-
-\begin{eqnarray}
-\sum_{x,y} z(x,y) T_k(x) T_l(y) & = & \sum_{i,j} P_{i,j} (\delta_{ik} N_x/2 + \delta_{i0} N_x/2) (\delta_{jl} N_y/2 + \delta_{j0} N_y/2) \\
-& = & ( P_{k,l} + P_{k,0} + P_{0,l} + P_{0,0} ) N_x N_y / 4
-\end{eqnarray}
-where $N_x$ and $N_y$ are the order of the polynomials in $x$ and $y$.
-
-Note that this is not a matrix equation, but simply requires a single pass
-through the data to calculate each coefficient.
-
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