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Commit 62d1b573 authored by Jean-Eric Campagne's avatar Jean-Eric Campagne
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(JEC) 19/1/16 update doc for spin>0

parent f6a0af32
......@@ -11,8 +11,10 @@ The original work by **Boris Leistedt** and **Jason D. McEwen** ([see the ArXiv
* we have also extended the numerical viability of the Laguerre Transform using simple 64-bits floating-point representation thanks to an on fly re-scaling technics to compute the Laguerre polynomial values. The same technics was used in the `libsharp` library to compute the Wigner d-matrix coefficients. This is particularly important to compute the weights and nodes of the Laguerre Quadrature that we can manage easily with N ~ 20,000 (we do not study exactly up to which N the computation fails).
Mathematics in summary
----------------
Any function $f(r,\Omega)$ square-integrable on $\mathrm{B}^3 = \mathrm{R}^+ \times [0,\pi] \times [0,2\pi)$ can be decomposed as:
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### Spin = 0
Any real function $f(r,\Omega)$ square-integrable on $\mathrm{B}^3 = \mathrm{R}^+ \times [0,\pi] \times [0,2\pi)$ can be decomposed as:
$$\begin{eqnarray}
f(r,\Omega) &=& \sum_{n=0}^\infty \sum_{l=0}^{\infty}\sum_{m=-l}^{l}\ f_{lmn}\ K_{lmn}(r,\Omega; \tau) \label{eq:FLagfulla}\\
......@@ -77,6 +79,11 @@ $$
We take advantage from the Matrix Multiplication writing of the $f_{lmn} \leftrightarrow a_{lmk}$ passage to use efficient algorithm or even more efficiently the BLAS-like libraries (OpenBLAS for Linux and the native Accelerate framework on Mac OS X).
### Spin > 0
Since **v2.0** it is introduced the spin-weighted decomposition of complex functions. In practice it is restricted to spin up to 2. The Spherical decomposition uses the *gradient* **E** and *curl* **B** coeficients in place of the above $a_{lmk}$ and the output of the Analysis are the $E_{lmn}$ and $B_{lmn}$ complex coefficients. For praticle usage of the ligsharp library C++ interface one should feed sperately the real and imaginary parts of the complex pixelized function.
### Fourier-Bessel (spin = 0)
In addition to the **Fourier-Laguerre** transform, the computation of the **Fourier-Bessel** is performed thanks to the link between the two sets of coefficients $f_{lmp}^{FB}$ and $f_{lmn}^{FL}$:
$$\begin{eqnarray}
f_{lmp}^{FB} &=& \sum_n f_{lmn}^{FL} J_{ln}(k_{lp})\\
......
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