@@ -58,21 +58,21 @@ The analysis process starts by gathering the 3D pixels real values \f$f_{ijk} \e
are the 2D angles defined by a pixelization scheme.
For instance, one can use the ECP Fejer's sum rule with \f$N_\theta = 2L-1\f$ colatitude rings of \f$N_\phi \geq 2L-1\f$ iso-latitude pixels.
Then, one proceeds as followed
+ for the \f$k\f$th-radial shell, one performs a Spherical Harmonic decomposition using for instance the `map2alm` routine of `libsharp` to get the set of complex coefficients \f$f_{lmk}\f$ with \f$l\in\{0,\dots,L-1\}\f$ and \f$m \in\{0,\dots,l\}\f$;
+ for each \f$(l,m)\f$, one uses the set of values \f$f_{lmk}\f$ with \f$k\in\{0,\dots,N-1\}\f$ to determine the \f$f_{lmn}\f$ complex coefficients thanks to the following simplified relation
+ for the \f$k\f$th-radial shell, one performs a Spherical Harmonic decomposition using for instance the `map2alm` routine of `libsharp` to get the set of complex coefficients \f$a_{lmk}=a_{lm}(r_k)\f$ with \f$l\in\{0,\dots,L-1\}\f$ and \f$m \in\{0,\dots,l\}\f$ (the notation remind the underlying 2D Spherical Harmonic transform and the usual way to note the resulting coefficients \f$a_{lm}\f$);
+ for each \f$(l,m)\f$, one uses the set of values \f$a_{lmk}\f$ with \f$k\in\{0,\dots,N-1\}\f$ to determine the \f$f_{lmn}\f$ complex coefficients thanks to the following simplified relation
+ then, thanks to the Inverse Spherical Harmonic Transform, `alm2map` routine of `libsharp`, ones determines the real \f$f_{ijk}\f$ pixel values.
We take advantage from the Matrix Multiplication writing of the \f$f_{lmn} \leftrightarrow a_{lmk}\f$ passage to use efficient algorithm or even more efficient the BLAS-like libraries (OpenBLAS for Linux and the native Accelerate framework on Mac OS X).