Operations on Rotational Functions edit page

The idea of variables of type SO3Fun is to calculate with rotational functions similarly as MATLAB does with vectors and matrices. In order to illustrate this we consider the following two rotational functions

An ODF determined from XRD data

SO3F1 = SO3Fun.dubna

plot(SO3F1,'sigma')
SO3F1 = SO3FunRBF (Quartz → xyz)
 
  multimodal components
  kernel: de la Vallee Poussin, halfwidth 5°
  center: 19848 orientations, resolution: 5°
  weight: 1

and an unimodal distributed ODF

R = orientation.byAxisAngle(vector3d.Y,pi/4,SO3F1.CS);
SO3F2 = SO3FunRBF(R,SO3DeLaValleePoussinKernel)

plot(SO3F2,'sigma')
SO3F2 = SO3FunRBF (Quartz → xyz)
 
  unimodal component
  kernel: de la Vallee Poussin, halfwidth 10°
  center: 1 orientations
 
  Bunge Euler angles in degree
  phi1    Phi   phi2 weight
    90     45    270      1

Basic arithmetic operations

Now the sum of these two rotational functions is again a rotational function, i.e., a function of type SO3Fun 

1 + 2 * SO3F1 + SO3F2

plot(2 * SO3F1 + SO3F2,'sigma')
ans = SO3FunComposition (Quartz → xyz)
 
  uniform component
  weight: 1
 
  multimodal components
  kernel: de la Vallee Poussin, halfwidth 5°
  center: 19848 orientations, resolution: 5°
  weight: 2
 
  unimodal component
  kernel: de la Vallee Poussin, halfwidth 10°
  center: 1 orientations
 
  Bunge Euler angles in degree
  phi1    Phi   phi2 weight
    90     45    270      1

Accordingly, one can use all basic operations like -, *, ^, /, min, max, abs, sqrt to calculate with variables of type SO3Fun.

% the maximum between two functions
plot(max(2*SO3F1,SO3F2),'sigma');
% the minimum between two functions
plot(min(2*SO3F1,SO3F2),'sigma');

We also can work with the pointwise conj, exp or log of an SO3Fun.

For a given function \(f\colon SO(3) \to \mathbb C\) we get a second function \(g\colon SO(3) \to \mathbb C\) where \(g( {\bf R}) = f( {\bf R}^{-1})\) by the method inv, i.e.

g = inv(SO3F1)

SO3F1.eval(R)
g.eval(inv(R))
g = SO3FunRBF (xyz → Quartz)
 
  multimodal components
  kernel: de la Vallee Poussin, halfwidth 5°
  center: 19848 orientations, resolution: 5°
  weight: 1
 
ans =
    2.3745
ans =
    4.2858

Local Extrema

The above mentioned functions min and max have very different use cases

  • if a single rotational function is provided the global maximum / minimum of the function is computed
  • if two rotational functions are provided, a rotational function defined as the pointwise min/max between these two functions is computed
  • if a rotational function and a single number are passed as arguments a rotational function defined as the pointwise min/max between the function and the value is computed
% * if additionally the option 'numLocal' is provided the certain number of
% local minima / maxima is computed

plot(2 * SO3F1 + SO3F2,'phi2',(0:3)*30*degree)

% compute and mark the global maximum
[maxvalue, maxnodes] = max(2 * SO3F1 + SO3F2,'numLocal',2);
annotate(maxnodes)

Integration

The surface integral of a spherical function can be computed by either mean or sum. The difference between both commands is that sum normalizes the integral of the identical function on the rotation group to \(8 \pi^2\), the command mean normalizes it to one. Compare

mean(SO3F1)

sum(SO3F1) / ( 8 * pi^2 )
ans =
     1
ans =
     1

A practical application of integration is the computation of the \(L^2\)-norm which is defined for a \(SO(3)\) function \(f\) by

\[ \| f\|_2 = \left( \frac{1}{8\pi^2} \int_{SO(3)} \lvert f({\bf R}) \rvert^2 \,\mathrm d {\bf R} \right)^{1/2} \]

accordingly we can compute it by

sqrt(mean(abs(SO3F1).^2))
ans =
    3.7736

or more efficiently by the command norm 

norm(SO3F1)
ans =
    3.7730

Differentiation

The gradient of a \(SO(3)\) function in a specific point can be described by a three-dimensional vector which can be computed by the command grad 

grad(SO3F1,R)
ans = SO3TangentVector
 TagentSpace: leftVector
        x        y        z
  10.1343 -26.5867 -3.26427

The gradients of a \(SO(3)\) function in all points form a \(SO(3)\) vector field and are returned by the function grad as a variable of type SO3VectorFieldHarmonic.

% compute the gradient as a vector field
G = grad(SO3F1)

% plot the gradient on top of the function
plot(SO3F1,'sigma')
hold on
plot(G,'color','black','linewidth',2,'resolution',5*degree)
hold off
G = SO3VectorFieldHarmonic (Quartz → xyz)
  bandwidth: 48
  tangent space: leftVector

We observe long arrows at the positions of big changes in intensity and almost invisible arrows in regions of constant intensity.

Rotating rotational functions

Rotating a \(SO(3)\) function works with the command rotate 

% define a rotation
rot = rotation.byEuler(30*degree,0*degree,90*degree,'Bunge');

% rotate the ODF
SO3F = rotate(SO3FunHarmonic(2 * SO3F1 + SO3F2),rot)

% and plot it
plot(SO3F,'sigma')
SO3F = SO3FunHarmonic (Quartz → xyz)
  bandwidth: 48
  weight: 3