Changeset 3094

Timestamp:

Jan 25, 2005, 3:10:14 PM (21 years ago)

Author:

Paul Price

Message:

Added section on inverse and combined transformations.

Location:

trunk/doc/pslib

Files:

• ChangeLogADD.tex (modified) (1 diff)
• psLibADD.tex (modified) (2 diffs)

trunk/doc/pslib/ChangeLogADD.tex

@@ -19 +19 @@
 \item Added section on astronomical object models
 \end{itemize}
+
+\subsection{Changes from version 08 to version 09}
+
+\begin{itemize}
+\item Added section on inverse and combined transformations.
+\end{itemize}

trunk/doc/pslib/psLibADD.tex

@@ -1 @@
-%%% $Id: psLibADD.tex,v 1.57 2005-01-22 01:57:42 eugene Exp $
+%%% $Id: psLibADD.tex,v 1.58 2005-01-26 01:09:07 price Exp $
 \documentclass[panstarrs]{panstarrs}
 
@@ -1317 +1317 @@
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsubsection{2D transformations}
+
+In PSLib, we implement 2-dimensional transformations using
+\code{psPlaneTransform}, which contains a matrix of polynomial
+coefficients for each dimension.  Since we are using these to model
+the real world, where, for example, a particular point on the detector
+maps to a particular point on the sky, we consider only
+transformations that are ``one-to-one''.  This makes it possible to
+speak of inverse transformations, and of combining multiple
+transformations.
+
+Given a transformation, $f(x,y)$, the inverse transformation,
+$g(x,y)$, is that for which $g(f(x,y)) = (x,y)$ for $(x,y)$ over the
+range of interest (not necessarily the entire set of real numbers).
+
+Given two transformations, $f(x,y)$ and $g(x,y)$, the combined
+transformation is the transformation, $h(x,y) = g(f(x,y))$ for $(x,y)$
+over the range of interest (not necessarily the entire set of real
+numbers).
+
+Both of these operations are straightforward if the transformation is
+linear.  If the function $(u,v) = f(x,y)$ is:
+\begin{eqnarray}
+u & = & a + bx + cy \\
+v & = & d + ex + fy
+\end{eqnarray}
+then the inverse transformation $(x,y) = g(u,v)$ is:
+\begin{eqnarray}
+x & = & (-fa+cd)/\Delta + fu/\Delta - cv/\Delta \\
+y & = & (ae-bd)/\Delta - eu/\Delta + bv/\Delta
+\end{eqnarray}
+where $\Delta = bf - ce$ is the matrix determinant.  Given two
+functions $f_i(x,y)$ for $i=1,2$:
+\begin{eqnarray}
+u & = & a_i + b_i x + c_i y \\
+v & = & d_i + e_i x + f_i y
+\end{eqnarray}
+then the combined transformation, $(u,v) = f_2(f_1(x,y))$ is:
+\begin{eqnarray}
+u & = & (a_2 + b_2 a_1 + c_2 d_1) + (b_2 b_1 + c_2 e_1) x + (b_2 c_1 + c_2 f_1) y \\
+v & = & (d_2 + e_2 a_1 + f_2 d_1) + (e_2 b_1 + f_2 e_1) x + (e_2 c_1 + f_2 f_1) y
+\end{eqnarray}
+
+When the transformations are not linear, the inverse and combined
+transformations can be estimated by sampling a grid over the region of
+interest, calculating the transformation (or double transformation)
+for each sample, and using this information to derive the best fit
+transformation that produces the inverse or combined transformation.
+The inverse transformation should be of the same order as that of the
+forward transformation, while the combined transformation should be of
+the higher order of the two component transformations.
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 \subsubsection{Projections}