Approximate an SO3FunRBF from some given orientation dependent function.
1) Spatial Method: Choose a grid of orientations \(R_i\) and evaluate \(f\). Do SO3FunRBF.interpolation afterwards.
2) Harmonic Method: Compute the weights of the SO3FunRBF such that the Fourier coefficients are similar to the Fourier coefficients of the given odf.
The spatial method is well suited for sharp functions (i.e. high bandwidth), whereas the harmonic method is better suited for low bandwidth, since the system matrix becomes very large for high bandwidths.
For the spatial method, instead of modified least squares (mlsq) also the maximum-likelihood estimate (mlrl) can be computed. Note that both of this methods have the condition that we approximate a odf (the mean of the SO3Fun is 1). We can also use some standard least squares methods (for example lsqr).
Syntax
SO3F = SO3FunRBF.approximate(f)
SO3F = SO3FunRBF.approximate(f,'kernel',psi)
SO3F = SO3FunRBF.approximate(f,'SO3Grid',S3G)
SO3F = SO3FunRBF.approximate(f,'kernel',psi,'SO3Grid',S3G)
SO3F = SO3FunRBF.approximate(f,'kernel',psi,'resolution',5*degree)
SO3F = SO3FunRBF.approximate(f,'kernel',psi,'resolution',5*degree,'approxresolution',2*degree)
SO3F = SO3FunRBF.approximate(f,'mlsq','tol',1e-3,'maxit',100,'density')SO3F = SO3FunRBF.approximate(f,'harmonic')
SO3F = SO3FunRBF.approximate(f,'harmonic','tol',1e-3,'maxit',100)
SO3F = SO3FunRBF.approximate(f,'harmonic','kernel',psi,'SO3Grid',S3G,'density')[SO3F,lsqrIter] = SO3FunRBF.approximate(___)Input
| f | SO3Fun |
| psi | SO3Kernel of the approximated SO3FunRBF (default: SO3DeLaValleePoussinKernel('halfwidth',5*degree)) |
| S3G | rotation |
Output
| SO3F | SO3FunRBF |
| lsqrIter | number of iterations of the LSQR-Solver |
Options
| halfwidth | halfwidth of the SO3Kernel of the result SO3FunRBF |
| kernel | SO3Kernel of the result SO3FunRBF |
| SO3Grid | center of the result SO3FunRBF |
| resolution | resolution of the SO3Grid which is the center of the result SO3FunRBF |
| approxresolution | resolution of the approximation grid, which is used to evaluate the input odf, if we use the spatial method (not the harmonic method) |
| tol | tolerance as termination condition for lsqr/mlsq/... |
| maxit | maximum number of iterations as termination condition for lsqr/mlsq/... |
Flags
| 'density' | ensure that result SO3FunRBF is a density function (i.e. positiv and mean is 1) |
| LSQRsolver | ('lsqr'|'mlsq'|'mlrl') specify least square solver for spatial method --> default: lsqr |
| 'harmonic' | if an SO3Fun is given, then use harmonic method for approximation, see above (spatial method is default) |
| LSQR | Solvers |
| lsqr | least squares (MATLAB) |
| mlsq | modified least squares (with positivity condition and normalization to mean 1) |
| mlrl | maximum likelihood estimate (with positivity condition and normalization to mean 1) |