Vector-Valued Orientation Functions

A scalar orientation function assigns one number to each orientation. A vector-valued orientation function assigns an array of numbers instead:

\[ f\colon \mathrm{SO}(3) \to \mathbb{R}^n. \]

This is useful when several orientation-dependent quantities share the same nodes, symmetries and approximation method. MTEX stores the components together so that they can be interpolated and manipulated as one array of functions.

How evaluations are arranged

The nodes are always interpreted as a column vector. The node index is the first dimension of an evaluated array. The dimensions of the vector-valued SO3Fun begin with the second dimension.

Consider four nodes \(R_1,\ldots,R_4\) and six scalar functions arranged as a \(3\mathbin{\times}2\) array. Evaluation returns a \(4\mathbin{\times}3\mathbin{\times}2\) numeric array \(F\):

\[ F(:,:,1)=\begin{pmatrix} f_1(R_1)&f_2(R_1)&f_3(R_1)\\ f_1(R_2)&f_2(R_2)&f_3(R_2)\\ f_1(R_3)&f_2(R_3)&f_3(R_3)\\ f_1(R_4)&f_2(R_4)&f_3(R_4) \end{pmatrix}, \qquad F(:,:,2)=\begin{pmatrix} f_4(R_1)&f_5(R_1)&f_6(R_1)\\ f_4(R_2)&f_5(R_2)&f_6(R_2)\\ f_4(R_3)&f_5(R_3)&f_6(R_3)\\ f_4(R_4)&f_5(R_4)&f_6(R_4) \end{pmatrix}. \]

The same convention applies to harmonic coefficients. The first dimension of the internal coefficient array stores all Fourier coefficients of one scalar function. If \(\widehat{\mathbf f}_j\) is the coefficient column for \(f_j\), the two slices are

\[ \widehat F(:,:,1)= \begin{pmatrix}\widehat{\mathbf f}_1&\widehat{\mathbf f}_2& \widehat{\mathbf f}_3\end{pmatrix}, \qquad \widehat F(:,:,2)= \begin{pmatrix}\widehat{\mathbf f}_4&\widehat{\mathbf f}_5& \widehat{\mathbf f}_6\end{pmatrix}. \]

This leading coefficient dimension is an implementation detail. Indexing and size report only the \(3\mathbin{\times}2\) function array seen by the user.

Interpolate several components from samples

First construct a column of approximately \(10^5\) orientations. The crystal and specimen symmetries determine the fundamental region sampled by the grid.

nodes = equispacedSO3Grid(crystalSymmetry,specimenSymmetry,'points',1e5);
nodes = nodes(:);

The first value column samples the Dubna model function. The second is a different scalar quantity evaluated at the same nodes. Assigning the Dubna crystal symmetry to the nodes makes their symmetry agree with the sampled model.

plottingConvention.default('y↑→x');
y = [SO3Fun.dubna(nodes), (nodes.a .* nodes.b).^(1/4)];
nodes.CS = SO3Fun.dubna.CS;

SO3FunHarmonic.interpolate fits both columns in one call. The result is a \(2\mathbin{\times}1\) array of harmonic functions. Here the bandwidth is 48 and LSQR is limited to 10 iterations.

SO3F1 = SO3FunHarmonic.interpolate(nodes,y,'maxit',10,'bandwidth',48)

Warning: Maximum number of iterations reached, result may not have converged to
the optimum yet. 
 
SO3F1 = SO3FunHarmonic (Quartz → y↑→x)
  isReal: false
  size: 2 × 1
  bandwidth: 48
  weights: [1 0.49]

The deliberately short iteration limit can produce an LSQR warning that the optimum has not yet been reached. Increase maxit when convergence, rather than a compact teaching run, is required.

Evaluating four orientations demonstrates the layout directly: four rows are nodes and two columns are components.

size(eval(SO3F1,nodes(1:4)))

ans =
     4     2

The generic interp route can instead build an SO3FunRBF from one value column.

SO3F2 = interp(nodes,y(:,1))

Warning: Maximum number of iterations reached, result may not have converged to
the optimum yet. 
 
SO3F2 = SO3FunRBF (Quartz → y↑→x)
 
  multimodal components
  kernel: de la Vallee Poussin, halfwidth 5°
  center: 19848 orientations, resolution: 5°
  weight: 1

This interp syntax is available only for a univariate, or scalar-valued, function. Use the explicit harmonic interpolation above when several sampled components should remain together.

Construct a function from a function handle

Quadrature provides a second route when values can be evaluated at any orientation. The handle below returns four columns for every input orientation.

f = @(rot) [exp(rot.a+rot.b+rot.c) + ...
  50*(rot.b-cos(pi/3)).^3.*(rot.b-cos(pi/3)>0), ...
  rot.a, rot.b, rot.c];

SO3FunHarmonic.quadrature evaluates the handle on a quadrature grid. It creates a \(4\mathbin{\times}1\) harmonic function with bandwidth 50 and the same crystal symmetry as SO3F1.

SO3F3 = SO3FunHarmonic.quadrature(f,'bandwidth',50,SO3F1.CS)

SO3F3 = SO3FunHarmonic (Quartz → y↑→x)
  isReal: false
  size: 4 × 1
  bandwidth: 50

Construct a function from Fourier coefficients

Known Fourier coefficients can be passed directly to the SO3FunHarmonic constructor. Each column of the coefficient matrix becomes one component.

SO3F4 = SO3FunHarmonic(eye(10))

SO3F4 = SO3FunHarmonic (y↑→x → y↑→x)
  isReal: false
  size: 10 × 1
  bandwidth: 1

This stores the first ten Wigner-D functions as a \(10\mathbin{\times}1\) function array. The coefficient dimension is hidden from the reported array size.

Index and reshape components

Function arrays follow ordinary MATLAB array conventions. They can be concatenated, indexed and reshaped. Concatenating the two- and four-component arrays below produces six components, and indexing selects components two through four.

SO3F5 = [SO3F1; SO3F3];
SO3F5(2:4)

ans = SO3FunHarmonic (Quartz → y↑→x)
  isReal: false
  size: 3 × 1
  bandwidth: 50
  weights: [0.5 1.1 -0.2]

Transpose and conjugation act on the function array and its Fourier coefficients. The nonconjugating transpose .' changes only the array shape, whereas ' also conjugates complex coefficients.

conj(SO3F1);
SO3F1.';
SO3F1';

Standard array queries report the component array dimensions.

length(SO3F1)
size(SO3F3)
SO3F4 = reshape(SO3F4,2,[])

ans =
     2
ans =
     4     1
 
SO3F4 = SO3FunHarmonic (y↑→x → y↑→x)
  isReal: false
  size: 2 × 5
  bandwidth: 1

Integrals and component-wise reductions

With no dimension argument, sum integrates every component over \(\mathrm{SO}(3)\) and mean returns its orientation-space mean. The result is a numeric array with the same component shape.

sum(SO3F1)
mean(SO3F4)

ans =
   78.8886
   38.6429
ans =
     1     0     0     0     0
     0     0     0     0     0

Passing a dimension changes the meaning to an ordinary pointwise reduction over the function array. The results below remain SO3FunHarmonic objects.

sum(SO3F1,1)
mean(SO3F4,2)

ans = SO3FunHarmonic (Quartz → y↑→x)
  isReal: false
  bandwidth: 48
  weight: 1.49+0.2i
 
 
ans = SO3FunHarmonic (y↑→x → y↑→x)
  antipodal: true
  size: 2 × 1
  bandwidth: 1
  weights: [0.2 0]

Minima and maxima

With a single argument, min and max find a global extremum of every component separately. They return numeric arrays and require real-valued functions. Marking SO3F4 as real enforces the coefficient symmetry required by this search.

SO3F4.isReal = 1;
min(SO3F4)

ans =
  Columns 1 through 7
    1.0000   -1.7320   -1.2247   -1.7320   -1.2247   -1.7321   -1.2247
  Columns 8 through 10
   -1.7320   -1.2247   -1.7320

Passing a dimension as min(SO3F,[],dim) instead computes a pointwise minimum along that component dimension. The result is again a vector-valued SO3FunHarmonic. Since SO3F4 is \(2\mathbin{\times}5\), the following reduction leaves a \(1\mathbin{\times}5\) function.

min(SO3F4,[],1)

ans = SO3FunHarmonic (y↑→x → y↑→x)
  size: 1 × 5
  bandwidth: 1

Pointwise products

The element-wise product .* is evaluated pointwise in orientation space. It is not the usual matrix product between the two function arrays. Both operands must have compatible symmetries, so the symmetry of SO3F1 is set explicitly before multiplication.

SO3F1.CS = specimenSymmetry;
SO3F1 .* SO3F4

ans = SO3FunHarmonic (y↑→x → y↑→x)
  isReal: false
  size: 2 × 5
  bandwidth: 49

Inspect all components

plotSpektra draws the harmonic power spectrum of every component. The four curves show that components of one function array may have different distributions over harmonic degree.

plotSpektra(SO3F3,'linewidth',2,'figSize','small')

An orientation-space section plot displays only the first component of a vector-valued function. Since SO3F3 is not marked as real, this command also warns that it displays the real part. Select and prepare a component explicitly when another component or its imaginary part is required.

plot(SO3F3)

The three-dimensional plot has the same first-component restriction. The shape therefore describes the real part of the first scalar function, not the magnitude of the four-component array.

plot3d(SO3F3)

A fibre is a one-dimensional family of orientations. In contrast to the section and three-dimensional plots, plotFibre draws every component along the selected beta fibre. Here it draws the real parts, so the four curves can be compared at the same orientations.

plotFibre(SO3F3,fibre.beta,'linewidth',2)

close all

References

Next

Continue with Rotational Vector Fields to attach three components to specimen or crystal coordinates and account for how those vectors transform with orientation.