Changeset 866 in Sophya


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Timestamp:
Apr 10, 2000, 4:22:15 PM (25 years ago)
Author:
ansari
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mise a jour 04/2000

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1 edited

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  • trunk/SophyaLib/Samba/sphericaltransformserver.h

    r858 r866  
    1313//
    1414/*! Class for performing analysis and synthesis of sky maps using spin-0 or spin-2 spherical harmonics.
     15
     16Maps must be SOPHYA SphericalMaps (SphereGorski or SphereThetaPhi).
     17
     18Temperature and polarization (Stokes parameters) can be developped on spherical harmonics :
     19\f[
     20\frac{\Delta T}{T}(\hat{n})=\sum_{lm}a_{lm}^TY_l^m(\hat{n})
     21\f]
     22\f[
     23Q(\hat{n})=\frac{1}{\sqrt{2}}\sum_{lm}N_l\left(a_{lm}^EW_{lm}(\hat{n})+a_{lm}^BX_{lm}(\hat{n})\right)
     24\f]
     25\f[
     26U(\hat{n})=-\frac{1}{\sqrt{2}}\sum_{lm}N_l\left(a_{lm}^EX_{lm}(\hat{n})-a_{lm}^BW_{lm}(\hat{n})\right)
     27\f]
     28\f[
     29\left(Q \pm iU\right)(\hat{n})=\sum_{lm}a_{\pm 2lm}\, _{\pm 2}Y_l^m(\hat{n})
     30\f]
     31
     32\f[
     33Y_l^m(\hat{n})=\lambda_l^m(\theta)e^{im\phi}
     34\f]
     35\f[
     36_{\pm}Y_l^m(\hat{n})=_{\pm}\lambda_l^m(\theta)e^{im\phi}
     37\f]
     38\f[
     39W_{lm}(\hat{n})=\frac{1}{N_l}\,_{w}\lambda_l^m(\theta)e^{im\phi}
     40\f]
     41\f[
     42X_{lm}(\hat{n})=\frac{-i}{N_l}\,_{x}\lambda_l^m(\theta)e^{im\phi}
     43\f]
     44
     45(see LambdaLMBuilder, LambdaPMBuilder, LambdaWXBuilder classes)
     46
     47power spectra :
     48
     49\f[
     50C_l^T=\frac{1}{2l+1}\sum_{m=0}^{+ \infty }\left|a_{lm}^T\right|^2=\langle\left|a_{lm}^T\right|^2\rangle
     51\f]
     52\f[
     53C_l^E=\frac{1}{2l+1}\sum_{m=0}^{+\infty}\left|a_{lm}^E\right|^2=\langle\left|a_{lm}^E\right|^2\rangle
     54\f]
     55\f[
     56C_l^B=\frac{1}{2l+1}\sum_{m=0}^{+\infty}\left|a_{lm}^B\right|^2=\langle\left|a_{lm}^B\right|^2\rangle
     57\f]
     58
     59\arg
     60\b Synthesis : Get temperature and polarization maps  from \f$a_{lm}\f$ coefficients or from power spectra, (methods GenerateFrom...).
     61
     62\b Temperature:
     63\f[
     64\frac{\Delta T}{T}(\hat{n})=\sum_{lm}a_{lm}^TY_l^m(\hat{n}) = \sum_{-\infty}^{+\infty}b_m(\theta)e^{im\phi}
     65\f]
     66
     67with
     68\f[
     69b_m(\theta)=\sum_{l=\left|m\right|}^{+\infty}a_{lm}^T\lambda_l^m(\theta)
     70\f]
     71
     72\b Polarisation
     73\f[
     74Q \pm iU = \sum_{-\infty}^{+\infty}b_m^{\pm}(\theta)e^{im\phi}
     75\f]
     76
     77where :
     78\f[
     79b_m^{\pm}(\theta) = \sum_{l=\left|m\right|}^{+\infty}a_{\pm 2lm}\,_{\pm}\lambda_l^m(\theta)
     80\f]
     81
     82or :
     83\f[
     84Q  = \sum_{-\infty}^{+\infty}b_m^{Q}(\theta)e^{im\phi}
     85\f]
     86\f[
     87U  = \sum_{-\infty}^{+\infty}b_m^{U}(\theta)e^{im\phi}
     88\f]
     89
     90where:
     91\f[
     92b_m^{Q}(\theta) = \frac{1}{\sqrt{2}}\sum_{l=\left|m\right|}^{+\infty}\left(a_{lm}^E\,_{w}\lambda_l^m(\theta)-ia_{lm}^B\,_{x}\lambda_l^m(\theta)\right)
     93\f]
     94\f[
     95b_m^{U}(\theta) = \frac{1}{\sqrt{2}}\sum_{l=\left|m\right|}^{+\infty}\left(ia_{lm}^E\,_{x}\lambda_l^m(\theta)+a_{lm}^B\,_{w}\lambda_l^m(\theta)\right)
     96\f]
     97
     98Since the pixelization provides "slices" with constant \f$\theta\f$ and \f$\phi\f$ equally distributed  on \f$2\pi\f$  \f$\frac{\Delta T}{T}\f$, \f$Q\f$,\f$U\f$  can be computed by FFT.
     99
     100
     101\arg
     102\b Analysis :  Get \f$a_{lm}\f$ coefficients or  power spectra from temperature and polarization maps   (methods DecomposeTo...).
     103
     104\b Temperature:
     105\f[
     106a_{lm}^T=\int\frac{\Delta T}{T}(\hat{n})Y_l^{m*}(\hat{n})d\hat{n}
     107\f]
     108
     109approximated as :
     110\f[
     111a_{lm}^T=\sum_{\theta_k}\omega_kC_m(\theta_k)\lambda_l^m(\theta_k)
     112\f]
     113where :
     114\f[
     115C_m (\theta _k)=\sum_{\phi _{k\prime}}\frac{\Delta T}{T}(\theta _k,\phi_{k\prime})e^{-im\phi _{k\prime}}
     116\f]
     117Since the pixelization provides "slices" with constant \f$\theta\f$ and \f$\phi\f$ equally distributed  on \f$2\pi\f$ (\f$\omega_k\f$ is the solid angle of each pixel of the slice \f$\theta_k\f$) \f$C_m\f$ can be computed by FFT.
     118
     119\b polarisation:
     120
     121\f[
     122a_{\pm 2lm}=\sum_{\theta_k}\omega_kC_m^{\pm}(\theta_k)\,_{\pm}\lambda_l^m(\theta_k)
     123\f]
     124where :
     125\f[
     126C_m^{\pm} (\theta _k)=\sum_{\phi _{k\prime}}\left(Q \pm iU\right)(\theta _k,\phi_{k\prime})e^{-im\phi _{k\prime}}
     127\f]
     128or :
     129
     130\f[
     131a_{lm}^E=\frac{1}{\sqrt{2}}\sum_{\theta_k}\omega_k\left(C_m^{Q}(\theta_k)\,_{w}\lambda_l^m(\theta_k)-iC_m^{U}(\theta_k)\,_{x}\lambda_l^m(\theta_k)\right)
     132\f]
     133\f[
     134a_{lm}^B=\frac{1}{\sqrt{2}}\sum_{\theta_k}\omega_k\left(iC_m^{Q}(\theta_k)\,_{x}\lambda_l^m(\theta_k)+C_m^{U}(\theta_k)\,_{w}\lambda_l^m(\theta_k)\right)
     135\f]
     136
     137where :
     138\f[
     139C_m^{Q} (\theta _k)=\sum_{\phi _{k\prime}}Q(\theta _k,\phi_{k\prime})e^{-im\phi _{k\prime}}
     140\f]
     141\f[
     142C_m^{U} (\theta _k)=\sum_{\phi _{k\prime}}U(\theta _k,\phi_{k\prime})e^{-im\phi _{k\prime}}
     143\f]
     144
    15145 */
    16146template <class T>
    … …  
    53183 /*!return the Alm coefficients from analysis of a temperature map.
    54184
     185    \param<nlmax> : maximum value of the l index
     186
     187     \param<cos_theta_cut> : cosinus of the symmetric cut EULER angle theta : cos_theta_cut=0 means no cut ; cos_theta_cut=1 all the sphere is cut.
     188  */
     189
     190
     191Alm<T> DecomposeToAlm(const SphericalMap<T>& map, int_4 nlmax, r_8 cos_theta_cut) const;
     192 /*!analysis of a polarization map into Alm coefficients.
     193
     194 The spheres \c mapq and \c mapu contain respectively the Stokes parameters.
     195
     196 \c a2lme and \c a2lmb will receive respectively electric and magnetic Alm's
    55197    nlmax : maximum value of the l index
    56198
    57      cos_theta_cut : cosinus of the symmetric cut EULER angle theta : cos_theta_cut=0 means no cut ; cos_theta_cut=1 all the sphere is cut.
    58   */
    59 
    60 
    61 Alm<T> DecomposeToAlm(const SphericalMap<T>& map, int_4 nlmax, r_8 cos_theta_cut) const;
    62  /*analysis of a polarization map into Alm coefficients.
    63 
    64  The spheres mapq and mapu contain respectively the Stokes parameters.
    65 
    66  a2lme and a2lmb will receive respectively electric and magnetic Alm's
    67     nlmax : maximum value of the l index
    68 
    69     cos_theta_cut : cosinus of the symmetric cut EULER angle theta : cos_theta_cut=0 means no cut ; cos_theta_cut=1 all the sphere is cut.
     199 \c cos_theta_cut : cosinus of the symmetric cut EULER angle theta : cos_theta_cut=0 means no cut ; cos_theta_cut=1 all the sphere is cut.
    70200 */
    71201
    … …  
    74204                     int_4 nlmax, r_8 cos_theta_cut) const;
    75205
    76  // /*return power spectrum from analysis of a temperature map.
    77 
    78  //    nlmax : maximum value of the l index
    79 
    80  //    cos_theta_cut : cosinus of the symmetric cut EULER angle theta : cos_theta_cut=0 means no cut ; cos_theta_cut=1 all the sphere is cut.
    81  // */
     206/*!return power spectrum from analysis of a temperature map.
     207
     208     \param<nlmax> : maximum value of the l index
     209
     210     \param<cos_theta_cut> : cosinus of the symmetric cut EULER angle theta : cos_theta_cut=0 means no cut ; cos_theta_cut=1 all the sphere is cut.
     211  */
    82212 TVector<T>  DecomposeToCl(const SphericalMap<T>& sph, 
    83213                           int_4 nlmax, r_8 cos_theta_cut) const;
    … …  
    109239Compute polarized Alm's as :
    110240\f[
    111 a_{lm}^E=\frac{1}{\sqrt{2}}\sum_{slices}{\omega_{pix}\left(_{w}\lambda_l^m\tilde{Q}-i_{x}\lambda_l^m\tilde{U}\right)}
    112 \f]
    113 \f[
    114 a_{lm}^B=\frac{1}{\sqrt{2}}\sum_{slices}{\omega_{pix}\left(i_{x}\lambda_l^m\tilde{Q}+_{w}\lambda_l^m\tilde{U}\right)}
     241a_{lm}^E=\frac{1}{\sqrt{2}}\sum_{slices}{\omega_{pix}\left(\,_{w}\lambda_l^m\tilde{Q}-i\,_{x}\lambda_l^m\tilde{U}\right)}
     242\f]
     243\f[
     244a_{lm}^B=\frac{1}{\sqrt{2}}\sum_{slices}{\omega_{pix}\left(i\,_{x}\lambda_l^m\tilde{Q}+\,_{w}\lambda_l^m\tilde{U}\right)}
    115245\f]
    116246
    … …  
    129259Compute polarized Alm's as :
    130260\f[
    131 a_{lm}^E=-\frac{1}{2}\sum_{slices}{\omega_{pix}\left(_{+}\lambda_l^m\tilde{P^+}+_{-}\lambda_l^m\tilde{P^-}\right)}
    132 \f]
    133 \f[
    134 a_{lm}^B=\frac{i}{2}\sum_{slices}{\omega_{pix}\left(_{+}\lambda_l^m\tilde{P^+}-_{-}\lambda_l^m\tilde{P^-}\right)}
     261a_{lm}^E=-\frac{1}{2}\sum_{slices}{\omega_{pix}\left(\,_{+}\lambda_l^m\tilde{P^+}+\,_{-}\lambda_l^m\tilde{P^-}\right)}
     262\f]
     263\f[
     264a_{lm}^B=\frac{i}{2}\sum_{slices}{\omega_{pix}\left(\,_{+}\lambda_l^m\tilde{P^+}-\,_{-}\lambda_l^m\tilde{P^-}\right)}
    135265\f]
    136266
    … …  
    161291
    162292\f[
    163 b_m^q=-\frac{1}{\sqrt{2}}\sum_{l=|m|}^{lmax}{\left(_{w}\lambda_l^ma_{lm}^E-i_{x}\lambda_l^ma_{lm}^B\right) }
    164 \f]
    165 \f[
    166 b_m^u=\frac{1}{\sqrt{2}}\sum_{l=|m|}^{lmax}{\left(i_{x}\lambda_l^ma_{lm}^E+_{w}\lambda_l^ma_{lm}^B\right) }
     293b_m^q=-\frac{1}{\sqrt{2}}\sum_{l=|m|}^{lmax}{\left(\,_{w}\lambda_l^ma_{lm}^E-i\,_{x}\lambda_l^ma_{lm}^B\right) }
     294\f]
     295\f[
     296b_m^u=\frac{1}{\sqrt{2}}\sum_{l=|m|}^{lmax}{\left(i\,_{x}\lambda_l^ma_{lm}^E+\,_{w}\lambda_l^ma_{lm}^B\right) }
    167297\f]
    168298 */
    … …  
    185315
    186316\f[
    187 b_m^+=-\sum_{l=|m|}^{lmax}{_{+}\lambda_l^m \left( a_{lm}^E+ia_{lm}^B \right) }
    188 \f]
    189 \f[
    190 b_m^-=-\sum_{l=|m|}^{lmax}{_{+}\lambda_l^m \left( a_{lm}^E-ia_{lm}^B \right) }
     317b_m^+=-\sum_{l=|m|}^{lmax}{\,_{+}\lambda_l^m \left( a_{lm}^E+ia_{lm}^B \right) }
     318\f]
     319\f[
     320b_m^-=-\sum_{l=|m|}^{lmax}{\,_{+}\lambda_l^m \left( a_{lm}^E-ia_{lm}^B \right) }
    191321\f]
    192322 */
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