conv

convolution of a rotational function with a rotational or spherical function

1. Convolution of two rotational functions

If there are two SO3Fun \(f \colon {}_{S_f^L } \backslash SO(3) /_{S_f^R} \to \mathbb{C}\) 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 \(f {*}_L g \colon {}_{S_f^L} \backslash SO(3) /_{S_g^R} \to \mathbb C\) is defined by

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

and the convolution \(f {*}_R g \colon {}_{S_g^L}\backslash SO(3) /_{S_f^R} \to \mathbb{C}\) 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. Convolution of a rotational function with a spherical function

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

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

3. Convolution of a rotational function with a kernel function

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