aspcol.kernelinterpolation_jax.directional_kernel_vonmises

aspcol.kernelinterpolation_jax.directional_kernel_vonmises(pos1, pos2, wave_num, direction, beta)

Directional sound field kernel.

For sound field estimation, direction should be chosen as the primary propagation direction. This means that for a receiver at [0,0,0] and a source at [10,0,0], the direction should be [-1,0,0].

Parameters:

• pos1 (np.ndarray of shape (num_points1, 3)) – Position of the first set of points.

• pos2 (np.ndarray of shape (num_points2, 3)) – Position of the second set of points.

• wave_num (np.ndarray of shape (num_real_freqs,)) – Wave number, defined as 2*pi*f/c, where f is the frequency and c is the speed of sound.

• direction (np.ndarray of shape (3,) or (num_dirs,3)) – The direction of the directional weighting.

• beta (float) – The strength of the directional weighting. A larger value will give more regularization.

Returns:

The kernel matrix. If only one direction is given, the axis is removed.

Return type:

np.ndarray of shape (num_real_freqs, num_points1, num_points2) or (num_real_freqs, num_dirs, num_points1, num_points2)

Notes

This function implements the kernel k(r, r’) = j_0(sqrt{xi^T xi}) where xi = k * (r - r’) - 1j * beta * direction and k is the wave number.

This kernel is obtained from the inner product langle u, v rangle_H = int_{S^2} u(d) conj(v(d)) / gamma(d) ds(d) where the weighting function gamma(d) = e^{-beta direction^T d}

The norm is therefore higher for u(eta), and u(eta) is defined as the plane wave coefficient for a plane wave incoming from eta. Therefore such a wave is less preferred. -> A wave incoming from -eta is more preferred, -> a wave propagating towards eta is preferred.

Another expression for the kernel here is k(r, r’) = int_{S^2} e^{-i k (r-r’)^T d} gamma(d) ds(d), which if evaluated gives the closed form solution above.