The spherical de la Vallee Poussin kernel is defined by
\[ K(t) = (1+\kappa)\,(\frac{1+t}{2})^{kappa}\]
for \(t\in[0,1]\). The de la Vallee Poussin kernel additionaly has the unique property that for a given halfwidth it can be described exactly by a finite number of Fourier coefficients. This kernel is recommended for Texture analysis as it is always positive and there is no truncation error in Fourier space.
Hence we can define the de la Vallee Poussin kernel \(\psi_{\kappa}\) depending on a parameter \(\kappa \in \mathbb N \setminus \{0\}\) by its finite Legendre polynomial expansion
\[ \psi_{\kappa}(t) = \sum\limits_{n=0}^{L} a_n(\kappa) \mathcal P_{n}(t)\].
We obtain the Legendre coefficients \(a_n(\kappa)\) by \(a_0=1\), \(a_1=\frac{\kappa}{2+\kappa}\) and the three term recurence relation
\[ (\kappa+l+2) a_{l+1} = -(2l+1)\,a_l + (\kappa-l+1)\,a_{l-1}\].
Syntax
psi = S2DeLaValleePoussinKernel(20)
psi = S2DeLaValleePoussinKernel('halfwidth',10*degree)Input
| kappa | kernel parameter |
Options
| halfwidth | angle at which the kernel function has reduced to half its peak value |
| bandwidth | harmonic degree |