void backwardSHT( fftw_complex *alm, coordStruct coords, int lmax, void *map, int outputIsComplex );
#include ccSHTmpi.h
void backwardSHTmpi( fftw_complex *alm, coordStruct coords, int lmax, void *map, int outputIsComplex, int outputRoot, MPI_Comm theComm );
These functions are the members of the ccSHT library which perform the backward discrete spherical harmonic transform (SHT). backwardSHT() performs the computation in serial and backwardSHTmpi() performs the computation in parallel. For some background, and more information about the library see the man page for ccSHT.
backwardSHT() and backwardSHTmpi() perform a band limited discrete SHT, where the band limit is set by the input parameter called lmax. That is to say that the transform assumes a_{l,m} is zero for all l greater than lmax. For this reason the user supplies the spherical harmonic coefficients only for values of l less than or equal to lmax (l in the set {0, 1, 2, ..., lmax} and m in the set {-l, -l+1, -l+2, ..., l}).
The backward discrete spherical harmonic transform creates a pixel domain vector (g) from a vector of spherical harmonic coefficients (a). This is done by the following computation:
lmax l
----- -----
\ \
g = ) ) a*Y
/ /
----- -----
l = 0 m = -l
where
g = g(theta,phi)
a = a_{l,m}
Y = Y_{l,m}(theta,phi)
In the above formulation, Y are the spherical harmonics. More explicitly, the spherical harmonics are defined
________________
/ (2*l+1)*(l-m)!
Y = / ----------------*P(cos(theta))*exp(i*m*phi)
\/ 4*pi*(l+m)!
where P(cos(theta)) = P_{l,m}(cos(theta)) are the associated
Legendre polynomials, and i is the imaginary number. The
above formulation is only valid for non-negative m
since P is only defined for non-negative m. Y_{l,m}
is related to Y_{l,-m} as follows:
_
Y_{l,-m} = (-1)^m*Y_{l,m}
so we can infer the values of Y for negative m.
The input to the transform is a vector of spherical harmonic coefficients (alm). The indexing of the spherical harmonic coefficient vector is described in the man page for ccSHT(1). The output of the transform is the pixel domain vector g (map). The locations of the positions on the sphere where the function g is evaluated are stored in the structure called coords (see man page ccSHT for more details about this structure).
For more information about the calculation of the associated Legendre polynomials see the man page for calculateQlm(). The transform doesn't actually call calculateQlm() for optimization purposes. Instead most of the coding required to do the calculation of the associated Legendre polynomials is done in the functions calculateSmallQlm() and lmCalculations() which the transform does call.
Below is a table describing the parameters passed to backwardSHT() and backwardSHTmpi().
Copyright (C) 2003 Christopher M. Cantalupo
backwardSHT and backwardSHTmpi are free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version.
This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.
Christopher Cantalupo <cmc@nersc.gov>
Send bug reports or comments to the above address.