Spherical Kernel Functions

A spherical kernel is a scalar function whose value depends only on angular distance from the north pole \(\vec e_3\). It is therefore radial: directions on the same circle around \(\vec e_3\) have the same value.

Kernels set the shape of localized spherical peaks. Their real-space profile controls locality, while their Legendre coefficients control harmonic cost and smoothing behaviour.

Reading a kernel in real space

Begin with a de la Vallee Poussin kernel whose halfwidth is \(10\) degree. The halfwidth is the angle at which the value falls to half of its peak.

psi = S2DeLaValleePoussinKernel('halfwidth',10*degree)

surf(psi,'resolution',2*degree,'EdgeColor','none')
hold on
arrow3d(2.4*zvector,'labeled','arrowwidth',0.01)
hold off
axis off

psi = S2DeLaValleePoussinKernel
  bandwidth: 25
  halfwidth: 10°

The peak is centred on \(\vec e_3\), marked by the arrow. Concentric colour bands show that changing azimuth while keeping angular distance fixed does not change the value.

A meridian profile makes the angular dependence easier to read.

close all
plot(psi,'linewidth',2,'symmetric')

The mirrored curve shows the same radial profile on either side of the north pole. The halfwidth is a half-maximum location, not a hard cutoff: this kernel remains positive beyond \(10\) degree.

Halfwidth and bandwidth answer different questions

Halfwidth describes the visible angular spread of a profile. Bandwidth is the largest Legendre degree stored by the kernel. A narrow or abruptly cut off profile usually needs a higher bandwidth.

The main kernel families emphasize different properties:

MTEX uses these kernels for several tasks. Pass one to calcDensity for directional density estimation or to smooth for spherical smoothing. The Schulz kernel corrects XRD defocusing. The calcGBND method uses a kernel to estimate a habit-plane normal distribution, and fibreODF uses one to set the profile around a fibre.

De la Vallee Poussin: a nonnegative peak

Compare two de la Vallee Poussin kernels. The constructor converts each requested halfwidth to its concentration parameter \(\kappa\).

psi15 = S2DeLaValleePoussinKernel('halfwidth',15*degree)
psi20 = S2DeLaValleePoussinKernel('halfwidth',20*degree)

fprintf('kappa(15 deg) = %.2f; kappa(20 deg) = %.2f\n', ...
  psi15.kappa,psi20.kappa)

psi15 = S2DeLaValleePoussinKernel
  bandwidth: 17
  halfwidth: 15°
 
 
psi20 = S2DeLaValleePoussinKernel
  bandwidth: 13
  halfwidth: 20°
 
kappa(15 deg) = 40.34; kappa(20 deg) = 22.64

The parameters are \(40.34\) and \(22.64\), respectively. Compare each real-space profile with its Fourier coefficients.

figure
subplot(1,2,1)
plot(psi15,'linewidth',2,'symmetric')
hold on
plot(psi20,'linewidth',2,'symmetric')
hold off
legend('halfwidth = 15°','halfwidth = 20°')
subplot(1,2,2)
plotSpektra(psi15,'linewidth',2)
hold on
plotSpektra(psi20,'linewidth',2)
hold off
legend('halfwidth = 15°','halfwidth = 20°')

The \(15\) degree kernel is narrower and taller. Its coefficients remain significant to higher degrees, which is the harmonic cost of resolving the sharper peak. Both profiles remain nonnegative.

Dirichlet: an exact spectral cutoff

The Dirichlet kernel keeps every Fourier coefficient through its requested bandwidth. Compare bandwidths 10 and 5 in real and harmonic space.

dirichlet10 = S2DirichletKernel(10)
dirichlet5 = S2DirichletKernel(5)

figure
subplot(1,2,1)
plot(dirichlet10,'linewidth',2,'symmetric')
hold on
plot(dirichlet5,'linewidth',2,'symmetric')
hold off
legend('bandwidth = 10','bandwidth = 5')
subplot(1,2,2)
plotSpektra(dirichlet10,'linewidth',2)
hold on
plotSpektra(dirichlet5,'linewidth',2)
hold off
legend('bandwidth = 10','bandwidth = 5')

dirichlet10 = S2DirichletKernel
  bandwidth: 10
  halfwidth: 180°
 
 
dirichlet5 = S2DirichletKernel
  bandwidth: 5
  halfwidth: 180°

The bandwidth-10 kernel has a narrower central lobe and more side lobes. It crosses below zero, so it is not a nonnegative density kernel. The spectra verify that every retained Fourier coefficient is exactly one. This exact cutoff is why the family is recommended for calculating physical properties.

Bump: an exact spatial cutoff

The bump kernel is constant inside its halfwidth and zero outside. Compare two cutoffs using bandwidth 128 for their stored Legendre expansions.

bump30 = S2BumpKernel(30*degree,'bandwidth',128)
bump50 = S2BumpKernel(50*degree,'bandwidth',128)

figure
subplot(1,2,1)
plot(bump30,'linewidth',2,'symmetric')
hold on
plot(bump50,'linewidth',2,'symmetric')
hold off
legend('halfwidth = 30°','halfwidth = 50°')
subplot(1,2,2)
plotSpektra(bump30,'linewidth',2)
hold on
plotSpektra(bump50,'linewidth',2)
hold off
legend('halfwidth = 30°','halfwidth = 50°')

bump30 = S2BumpKernel
  bandwidth: 128
  halfwidth: 30°
 
 
bump50 = S2BumpKernel
  bandwidth: 128
  halfwidth: 50°

The flat tops and abrupt cutoffs give exact compact support in real space. The oscillating spectra decay slowly, so many coefficients are needed to represent that discontinuity. This can produce high runtimes.

Earlier documentation stated that the value inside the cap was 1. The current kernel instead divides by the cap's relative area. This gives the kernel mean value one, consistent with the other normalized families.

The maths behind radial kernels

Let \(\theta\) be the angle from \(\vec v\) to \(\vec e_3\) and set \(t=\cos\theta=\vec v\cdot\vec e_3\). A kernel \(\Psi\) on the sphere is then represented by a one-variable function \(\psi\):

\[ \Psi(\vec v)=\psi(t), \qquad t\in[-1,1]. \]

MTEX expands this function in Legendre polynomials \(\mathcal P_n\),

\[ \psi(t)=\sum_{n=0}^{\infty}(2n+1)\,\widehat\psi_n\, \mathcal P_n(t). \]

A general S2FunHarmonic has \(2n+1\) spherical harmonic coefficients at degree \(n\). Radial symmetry leaves only the zonal coefficient. An S2Kernel stores the scaled coefficients \((2n+1)\widehat\psi_n\) in its A property. The plotSpektra method divides by \(2n+1\) and plots \(\widehat\psi_n\).

This degree-by-degree representation makes radial convolution cheap. The spherical Radon transform earlier in this chapter is another radial convolution, with zero coefficients at every odd degree.

Formulae for the three examples

On the full interval \(t\in[-1,1]\), the de la Vallee Poussin profile is

\[ K(t)=(1+\kappa)\left(\frac{1+t}{2}\right)^\kappa. \]

Earlier documentation restricted this formula to \(t\in[0,1]\). The class evaluates it on the full cosine interval. For positive integer \(\kappa\in\mathbb N\setminus\{0\}\), the profile is a polynomial and has an exact finite Legendre expansion. A requested halfwidth generally gives a noninteger \(\kappa\), so its stored finite expansion is a truncation.

With the normalization above, the first coefficients are \(\widehat\psi_0=1\) and \(\widehat\psi_1=\kappa/(\kappa+2)\). The implementation continues them with

\[ (\kappa+l+2)\widehat\psi_{l+1}=-(2l+1)\widehat\psi_l +(\kappa-l+1)\widehat\psi_{l-1}. \]

Earlier documentation instead gave \(\widehat\psi_1=\kappa/(6+3\kappa)\) and the recurrence

\[ (\kappa+l+2)(2l+3)\widehat\psi_{l+1} =-(2l+1)^2\widehat\psi_l +(\kappa-l+1)(2l-1)\widehat\psi_{l-1}. \]

Those factors do not match the coefficients returned by the current constructor.

The bandwidth \(N\) Dirichlet kernel is

\[ \psi_N(t)=\sum_{n=0}^{N}(2n+1)\mathcal P_n(t). \]

Hence \(\widehat\psi_n=1\) for \(0\leq n\leq N\) and zero above \(N\). For bump halfwidth \(r\in(0,\pi)\), the normalized profile is

\[ \psi_r(\cos\theta)= \begin{cases} \displaystyle\frac{2}{1-\cos r}, & 0\leq\theta<r,\\ 0, & r\leq\theta\leq\pi. \end{cases} \]

References

Next

Spherical Harmonics develops the basis functions whose zonal coefficients reduce to one value per degree for a kernel.