conv

convolution of an SO3FunHarmonic with a function or a kernel on SO(3)

1) SO3Fun * SO3Fun There are two SO3Funs \(f: {S_f^L\backslash}SO(3){/S_f^R} \to \mathbb{C}\) where \(S_f^L\) is the Left symmetry and \(S_f^R\) is the Right symmetry and \(g: {S_g^L\backslash}SO(3){/S_g^R} \to \mathbb{C}\) given. Then the convolution \( f *L g : _{S_f^L\backslash}SO(3){/S_g^R} \to \mathbb{C}\) is defined by

\[ (f *L g)(R) = \frac1{8\pi^2} \int{SO(3)} f(q) \cdot g(q^{-1}\,R) \, dq \]

and the convolution \( f *R g : _{S_g^L\backslash}SO(3){/S_f^R} \to \mathbb{C}\) is defined by

\[ (f *R g)(R) = \frac1{8\pi^2} \int{SO(3)} f(q) \cdot g(R\,q^{-1}) \, dq \].

with \(vol(SO(3)) = \int_{SO(3)} 1 \, dR = 8\pi^2\). The convolution \(*_L\) is used as default. The convolution of matrices of SO3Functions with matrices of SO3Functions works elementwise.

2) SO3Fun * S2Fun The convolution of an SO3Fun \(f: {S_f^L\backslash}SO(3){/S_f^R} \to \mathbb{C}\) with an S2Fun \(h: \mathbb S^2_{/S_h} \to \mathbb{C}\) yields \(f*h:\mathbb S^2_{/S_f^L} \to \mathbb{C}\) with

\[ (f * h)(\xi) = \frac1{8\pi^2} \int_{SO(3)} f(q) \cdot h(q^{-1}\,\xi) \, dq \].

3) In particular we convolute an SO3Fun with an SO3Kernel similar to the first case. Therefore the Right and Left sided convolution are equivalent. The convolution of an SO3Fun with an S2Kernel works analogue to case 2.

Syntax

SO3F = conv(SO3F1,SO3F2)
SO3F = conv(SO3F1,SO3F2,'Right')
SO3F = conv(SO3F1,psi)
sF2 = conv(SO3F1,sF1)
sF2 = conv(SO3F1,phi)

Input

Output

See also

SO3Kernel.conv SO3FunHarmonic.conv SO3FunRBF.calcFourier S2FunHarmonic.conv S2Kernel.conv