Index: /trunk/doc/pslib/psLibADD.tex =================================================================== --- /trunk/doc/pslib/psLibADD.tex (revision 317) +++ /trunk/doc/pslib/psLibADD.tex (revision 318) @@ -337,5 +337,5 @@ % \[ T_{ij} = M_{ji} \] -where $M_{ji}$ is the matrix to be transposed. +where $M_{ij}$ is the matrix to be transposed. \subsubsection{Convert a matrix to a vector} @@ -482,7 +482,7 @@ \subsection{(Fast) Fourier Transforms} -(Fast) Fourier Transforms (FFTs) shall be implemented using the {\em -Fastest Fourier Transform in the West} (FFTW) library -(\href{www.fftw.org}). +(Fast) Fourier Transforms (FFTs) shall be implemented using the +\href{www.fftw.org}{{\em Fastest Fourier Transform in the West} (FFTW) +library}. \subsubsection{FFTW Plans} @@ -508,8 +508,8 @@ \begin{tabular}{ll} - PSLib function & Major FFTW call \\ \hline -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - psFFTForward() & fftw_plan_dft_r2c_2d() - psFFTReverse() & fftw_plan_dft_c2r_2d() + PSLib function & Major FFTW call \\ \hline +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \code{psFFTForward()} & \code{fftw_plan_dft_r2c_2d()} \\ + \code{psFFTReverse()} & \code{fftw_plan_dft_c2r_2d()} \\ \end{tabular} @@ -517,32 +517,34 @@ which will allow FFTW to default to a short planning time if the wisdom has not been loaded. Transforms should be performed out of -place to avoid the need to pad the input array. +place to avoid the need to pad the input array to hold the output. \subsubsection{More Complicated Functions} The \code{psFFTFilter} and \code{psFFTFilterComplex} functions provide -a means to apply a filter (purely real and imaginary multipliers, +the means to apply a filter (purely real and imaginary multipliers, respectively) to the data in the Fourier plane. If the filter function specified for \code{psFFTFilter} returns a real value, $r$, then the corresponding value in the Fourier plane should be multiplied by $r$. If the real and imaginary filter functions specified for -\code{psFFTFilterComplex} returns the values $r$ and $s$, -respectively, then the corresponding value in the Fourier plane should be -multiplied by the complex number $(r + si)$. +\code{psFFTFilterComplex} return the values $r$ and $s$, respectively, +then the corresponding value in the Fourier plane should be multiplied +by the complex number $r + si$. \code{psFFTCrossCorrelate()} and \code{psFFTConvolve()} both involve multiplication of two Fourier transforms. In the former, the first Fourier transform is multiplied by the complex conjugate of the second -Fourier transform to yield the Fourier transform of the cross-correlation. -In the latter, the two Fourier transforms are multiplied directly to yield -the Fourier transform of the convolution. +Fourier transform to yield the Fourier transform of the +cross-correlation (NR 13.2). In the latter, the two Fourier +transforms are multiplied directly to yield the Fourier transform of +the convolution (NR 13.1). If the elements of the discrete Fourier transform are $C_k$, then the -the elements of the power spectrum are defined (NR 13.4): -\begin{eqnarray} -P_0 & = & \| C_0 \|^2 / N^2 \\ -P_j & = & \left( \| C_j \|^2 + \| C_{N-j} \|^2 \right)/ N^2 & j = 1, 2, \ldots, (N/2 - 1) \\ -P_{N/2} & = & \| C_{N/2} \|^2 / N^2 \\ -\end{eqnarray} +the elements of the power spectrum are (NR 13.4): +\begin{eqnarray} +P_0 & = & \left| C_0 \right|^2 / N^2 \\ +P_j & = & \left( \left| C_j \right|^2 + \left| C_{N-j} \right|^2 \right)/ N^2 \\ +P_{N/2} & = & \left| C_{N/2} \right|^2 / N^2 \\ +\end{eqnarray} +where $j = 1, 2, \ldots, (N/2 - 1)$. Note that we leave the issue of ``windowing'' the data up to the @@ -550,4 +552,107 @@ \subsection{Astronomy Utilities} + +Most of the astronomy utilities will be implemented through wrapping +the +\href{http://star-www.rl.ac.uk/star/docs/sun67.htx/sun67.html}{SLALIB +Positional Astronomy Library}. + +\subsubsection{Celestial Coordinate Conversions} + +These will be implemented using the corresponding SLALIB functions: + +\begin{tabular}{ll} + PSLib function & SLALIB function \\ \hline +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \code{psCoordinatesItoE()} & \code{sla_EQECL} \\ + \code{psCoordinatesEtoI()} & \code{sla_ECLEQ} \\ + \code{psCoordinatesItoG()} & \code{sla_EQGAL} \\ + \code{psCoordinatesGtoI()} & \code{sla_GALEQ} \\ +\end{tabular} + +\subsubsection{Projections} + +The following information is from +\href{http://www.cv.nrao.edu/fits/documents/wcs/wcs.all.ps}{Greisen \& +Calabretta (1995, ADASS, 4, 233)}. + +Let the latitude be $\phi$ and the longitude $\theta$. The domains of +these are $-\pi < \phi \le \pi$ and $-\pi/2 \le \theta \le \pi/2$. + +For zenithal projections (e.g.\ Gnomonic and Orthographic) the +following hold: + +\begin{eqnarray} +x & = & R \sin \phi \\ +y & = & -R \cos \phi +\end{eqnarray} + +and + +\begin{eqnarray} +R & = & \sqrt{x^2 + y^2} \\ +\phi & = & {\rm arg} (-y,x) +\end{eqnarray} + +\subsubsubsection{Gnomonic} + +The Gnomonic projection (``TAN'') is a zenithal projection. + +\begin{eqnarray} +R & = & \cot \theta 180^\circ/\pi \\ +\theta & = & \arctan (180^\circ/(\pi R)) +\end{eqnarray} + +\subsubsubsection{Orthographic} + +The Orthographic projection (``SIN'') is a zenithal projection. + +\begin{eqnarray} +R & = & \cos \theta 180^\circ/\pi \\ +\theta & = & \arccos (\pi R / 180^\circ) +\end{eqnarray} + +\subsubsubsection{Cartesian} + +The Cartesian projection (``CAR'') is a very simple cylindrical projection. + +\begin{eqnarray} +x & = & \phi \\ +y & = & \theta +\end{eqnarray} + +\subsubsubsection{Mercator} + +The Mercator projection (``MER'') is a cylindrical projection. + +\begin{eqnarray} +x & = & \phi \\ +y & = & \ln \left( \tan (45^\circ + \theta/2) \right) 180^\circ/\pi \\ +{\rm and}\hspace{1cm} \theta & = & 2 \arctan \left( e^{y\pi/180^\circ} \right) - 90^\circ +\end{eqnarray} + +\subsubsubsection{Hammer-Aitoff} + +The Hammer-Aitoff projection is a general projection, and is defined: + +\begin{eqnarray} +x & = & 2 \alpha \cos \theta \sin (\phi/2) \\ +y & = & \alpha \sin \theta \\ +{\rm where}\hspace{1cm} \alpha^{-1} & \equiv & (180^\circ/\pi) \sqrt{\left(1 + \cos \theta \cos (\phi/2) \right) / 2} +\end{eqnarray} + +And in reverse: + +\begin{eqnarray} +\phi & = & 2 {\rm arg} (2z^2 - 1, xz \pi/360^\circ) \\ +\theta & = & \arcsin (yz\pi/180^\circ) \\ +{\rm where}\hspace{1cm} z & \equiv & \sqrt{1 - (x\pi/720^\circ)^2 - (y\pi/360^\circ)^2} +\end{eqnarray} + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \section{Modules}