quadrature edit page

Compute the SO(3)-Fourier/Wigner coefficients of an given SO3Fun or given evaluations on a specific quadrature grid.

Therefore we obtain the Fourier coefficients with numerical integration (quadrature), i.e. we choose a quadrature scheme of meaningful quadrature nodes \(R_m\) and quadrature weights \(\omega_m\) and compute

\[ \hat f_n^{k,l} = \int_{SO(3)} f®\, \overline{D_n^{k,l}(R)} \mathrm{d}\my® \approx \sum_{m=1}^M \omega_m \, f(R_m) \, \overline{D_n^{k,l}(R_m)}, \]

for all \(n=0,\dots,N\) and \(k,l=-n,\dots,n\).

Therefore this method evaluates the given SO3Fun on a with respect to symmetries fundamental Region. Afterwards it uses a inverse trivariate nfft/fft and an adjoint coefficient (Wigner) transform which is based on a representation property of Wigner-D functions.

Hence it do not use the NFSOFT (which includes a fast polynom transform) as in the older method SO3FunHarmonic.quadratureNFSOFT.

Syntax

SO3F = SO3FunHarmonic.quadrature(f)
SO3F = SO3FunHarmonic.quadrature(f,'bandwidth',bandwidth)
SO3F = SO3FunHarmonic.quadrature(f,'bandwidth',bandwidth,quadratureScheme)
SO3F = SO3FunHarmonic.quadrature(f,'bandwidth',bandwidth,'SO3Grid',S3G,'weights',w)
SO3F = SO3FunHarmonic.quadrature(nodes,values)
SO3F = SO3FunHarmonic.quadrature(nodes,values,'bandwidth',48,'weights',w)

Input

f SO3Fun, function_handle in orientation (first dimension has to be the evaluations)
nodes quadratureSO3Grid, rotation, orientation
values double (first dimension has to be the evaluations)

Output

SO3F SO3FunHarmonic

Options

bandwidth maximal harmonic degree (default: 64)
weights quadrature weights
SO3Grid quadrature nodes

Flags

quadratureScheme ('ClenshawCurtis'|'GaussLegendre') --> default: CC

See also

SO3FunHarmonic.adjoint SO3FunHarmonic.approximate SO3FunHarmonic