Approximate an SO3FunRBF by given function values at given nodes w.r.t. some noise as described by [1].
For \(M\) given orientations \(R_i\) and corresponding function values \(y_i\) we compute the SO3FunRBF \(f\) which minimizes the least squares problem
\[\sum_{i=1}^M|f(R_i)-y_i|^2.\]
where \(f\) is
\[ f®=\sum_{j=1}^{N}w_j\psi(\omega(R_j,R)) \]
with specific kernel \(\psi\) centered at \(N\) nodes weighted by \(w_j,\sum_{j}^{N}w_{j}=1\) as described by [1].
Therefore we set up a (sparse) system matrix \(\Psi\in\mathbb{R}^{M\times N}\) with entries
\[ \Psi_{i,j}=\psi(\omega(R_i,R_j)) \]
of the kernel values of the angle between the evaluation nodes \(R_i,i=1,...,M\) and grid nodes \(R_j,j=1,...,N\). This system is solved by least squares gradient descent.
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).
Reference: [1] Schaeben, H., Bachmann, F. & Fundenberger, JJ. Construction of weighted crystallographic orientations capturing a given orientation density function. J Mater Sci 52, 2077–2090 (2017). https://doi.org/10.1007/s10853-016-0496-1
Syntax
SO3F = SO3FunRBF.interpolate(nodes,y)
SO3F = SO3FunRBF.interpolate(nodes,y,'halfwidth',1*degree)
SO3F = SO3FunRBF.interpolate(nodes,y,'kernel',psi)
SO3F = SO3FunRBF.interpolate(nodes,y,'kernel',psi,'exact')
SO3F = SO3FunRBF.interpolate(nodes,y,'kernel',psi,'resolution',5*degree)
SO3F = SO3FunRBF.interpolate(nodes,y,'kernel',psi,'SO3Grid',S3G)
SO3F = SO3FunRBF.interpolate(nodes,y,'mlsq','tol',1e-3,'maxit',100,'density')[SO3F,lsqrIter] = SO3FunRBF.interpolate(___)Input
| nodes | rotational grid SO3Grid, orientation, rotation or harmonic coefficents |
| y | function values on the grid (maybe multidimensional) or empty |
| 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
| 'exact' | if rotations are given, then use nodes as center of result SO3FunRBF and try to do exact computations |
| 'density' | ensure that result SO3FunRBF is a density function (i.e. positiv and mean is 1) |
| LSQRsolver | ('lsqr'|'lsqnonneg'|'lsqlin'|'nnls'|'mlsq'|'mlrl') specify least square solver for spatial method --> default: lsqr |
| LSQR | Solvers |
| lsqr | least squares (MATLAB) |
| lsqnonneg | non negative least squares (MATLAB, fast) |
| lsqlin | interior point non negative least squares (optimization toolbox, slow) |
| nnls | non negative least squares (W.Whiten) |
| mlsq | modified least squares (with positivity condition and normalization to mean 1) |
| mlrl | maximum likelihood estimate (with positivity condition and normalization to mean 1) |