sleplet 1.2.1

Slepian scale-discretised wavelets in Python

SLEPLET

SLEPLET is a Python package for the construction of Slepian wavelets in the spherical and manifold (via meshes) settings. The API of SLEPLET has been designed in an object-orientated manner and is easily extendible. Upon installation, SLEPLET comes with two command line interfaces - sphere and mesh - which allows one to easily generate plots on the sphere and a set of meshes using plotly.

Installation

The recommended way to install SLEPLET is via pip

pip install sleplet

To install the latest development version of SLEPLET clone this repository and run

pip install -e .

This will install two scripts sphere and mesh which can be used to generate the figures in the figure section.

Bandlimit

The bandlimit is set as L throughout the code and the CLIs. The default value is set to L=16 and the figures created in the figure section all use L=128. The pre-computed data exists on Zenodo for powers of two up to L=128. Other values will be computed when running the appropriate code (and saved for future use). Note that beyond L=32 the code can be slow due to the difficulties of computing the Slepian matrix prior to the eigendecomposition, as such it is recommended to stick to the powers of two up to L=128.

Environment Variables

• NCPU: sets the number of cores to use

When it comes to selecting a Slepian region the order precedence is polar cap region > limited latitude longitude region > arbitrary region, as seen in the code. The default region is the south_america arbitrary region.

• POLAR_GAP
  • for a Slepian polar cap region, when set in conjunction with THETA_MAX but without the other PHI/THETA variables
• THETA_MAX
  • for a Slepian polar cap region, when set without the other PHI/THETA variables
  • for a Slepian limited latitude longitude region
• THETA_MIN
  • for a Slepian limited latitude longitude region
• PHI_MAX
  • for a Slepian limited latitude longitude region
• PHI_MIN
  • for a Slepian limited latitude longitude region
• SLEPIAN_MASK
  • for an arbitrary Slepian region, currently africa/south_america supported

Paper Figures

To recreate the figures from the below papers, one may use the CLI or the API. For those which don't use the mesh or sphere CLIs, the relevant API code isn't provided as it is contained within the examples folder.

Sifting Convolution on the Sphere

Sifting Convolution on the Sphere: Fig. 1

for ell in $(seq 2 -1 1); do
    sphere harmonic_gaussian -a 0.75 -b 0.125 -e ${ell} 1 -L 128 -m translate -o
done

import numpy as np
import pyssht as ssht
import sleplet

for ell in range(2, 0, -1):
    f = sleplet.functions.HarmonicGaussian(128, l_sigma=10**ell, m_sigma=10)
    flm = f.translate(alpha=0.75 * np.pi, beta=0.125 * np.pi)
    f_sphere = ssht.inverse(flm, f.L, Method="MWSS")
    sleplet.plotting.PlotSphere(
        f_sphere,
        f.L,
        f"fig_1_ell_{ell}",
        annotations=[],
    ).execute()

Sifting Convolution on the Sphere: Fig. 2

sphere earth -L 128

import pyssht as ssht
import sleplet

f = sleplet.functions.Earth(128)
flm = sleplet.harmonic_methods.rotate_earth_to_south_america(f.coefficients, f.L)
f_sphere = ssht.inverse(flm, f.L, Method="MWSS")
sleplet.plotting.PlotSphere(f_sphere, f.L, "fig_2").execute()

Sifting Convolution on the Sphere: Fig. 3

for ell in $(seq 2 -1 1); do
    sphere harmonic_gaussian -c earth -e ${ell} 1 -L 128
done

import pyssht as ssht
import sleplet

for ell in range(2, 0, -1):
    f = sleplet.functions.HarmonicGaussian(128, l_sigma=10**ell, m_sigma=10)
    g = sleplet.functions.Earth(128)
    flm = f.convolve(f.coefficients, g.coefficients.conj())
    flm_rot = sleplet.harmonic_methods.rotate_earth_to_south_america(flm, f.L)
    f_sphere = ssht.inverse(flm_rot, f.L, Method="MWSS")
    sleplet.plotting.PlotSphere(f_sphere, f.L, f"fig_3_ell_{ell}").execute()

Slepian Scale-Discretised Wavelets on the Sphere

Slepian Scale-Discretised Wavelets on the Sphere: Fig. 2

python -m examples.arbitrary.south_america.tiling_south_america

Slepian Scale-Discretised Wavelets on the Sphere: Fig. 3

export SLEPIAN_MASK="south_america"
# a
sphere earth -L 128 -s 2 -u
# b
sphere slepian_south_america -L 128 -s 2 -u

import pyssht as ssht
import sleplet

# a
f = sleplet.functions.Earth(128, smoothing=2)
flm = sleplet.harmonic_methods.rotate_earth_to_south_america(f.coefficients, f.L)
f_sphere = ssht.inverse(flm, f.L, Method="MWSS")
sleplet.plotting.PlotSphere(f_sphere, f.L, "fig_3_a", normalise=False).execute()
# b
region = sleplet.slepian.Region(mask_name="south_america")
g = sleplet.functions.SlepianSouthAmerica(128, region=region, smoothing=2)
g_sphere = sleplet.slepian_methods.slepian_inverse(g.coefficients, g.L, g.slepian)
sleplet.plotting.PlotSphere(
    g_sphere,
    g.L,
    "fig_3_b",
    normalise=False,
    region=g.region,
).execute()

Slepian Scale-Discretised Wavelets on the Sphere: Fig. 4

export SLEPIAN_MASK="south_america"
for p in 0 9 24 49 99 199; do
    sphere slepian -e ${p} -L 128 -u
done

import sleplet

region = sleplet.slepian.Region(mask_name="south_america")
for p in [0, 9, 24, 49, 99, 199]:
    f = sleplet.functions.Slepian(128, region=region, rank=p)
    f_sphere = sleplet.slepian_methods.slepian_inverse(f.coefficients, f.L, f.slepian)
    sleplet.plotting.PlotSphere(
        f_sphere,
        f.L,
        f"fig_4_p_{p}",
        normalise=False,
        region=f.region,
    ).execute()

Slepian Scale-Discretised Wavelets on the Sphere: Fig. 5

python -m examples.arbitrary.south_america.eigenvalues_south_america

Slepian Scale-Discretised Wavelets on the Sphere: Fig. 6

export SLEPIAN_MASK="south_america"
# a
sphere slepian_wavelets -L 128 -u
# b-f
for j in $(seq 0 4); do
    sphere slepian_wavelets -e 3 2 ${j} -L 128 -u
done

import sleplet

region = sleplet.slepian.Region(mask_name="south_america")
for j in [None, *list(range(5))]:
    f = sleplet.functions.SlepianWavelets(128, region=region, B=3, j_min=2, j=j)
    f_sphere = sleplet.slepian_methods.slepian_inverse(f.coefficients, f.L, f.slepian)
    sleplet.plotting.PlotSphere(
        f_sphere,
        f.L,
        f"fig_6_j_{j}",
        normalise=False,
        region=f.region,
    ).execute()

Slepian Scale-Discretised Wavelets on the Sphere: Fig. 7

export SLEPIAN_MASK="south_america"
# a
sphere slepian_wavelet_coefficients_south_america -L 128 -s 2 -u
# b-f
for j in $(seq 0 4); do
    sphere slepian_wavelet_coefficients_south_america -e 3 2 ${j} -L 128 -s 2 -u
done

import sleplet

region = sleplet.slepian.Region(mask_name="south_america")
for j in [None, *list(range(5))]:
    f = sleplet.functions.SlepianWaveletCoefficientsSouthAmerica(
        128,
        region=region,
        B=3,
        j_min=2,
        j=j,
        smoothing=2,
    )
    f_sphere = sleplet.slepian_methods.slepian_inverse(f.coefficients, f.L, f.slepian)
    sleplet.plotting.PlotSphere(
        f_sphere,
        f.L,
        f"fig_7_j_{j}",
        normalise=False,
        region=f.region,
    ).execute()

Slepian Scale-Discretised Wavelets on the Sphere: Fig. 8

export SLEPIAN_MASK="south_america"
# a
sphere slepian_south_america -L 128 -n -10 -s 2 -u
# b-d
for s in 2 3 5; do
    python -m examples.arbitrary.south_america.denoising_slepian_south_america -n -10 -s ${s}
done

import sleplet

# a
region = sleplet.slepian.Region(mask_name="south_america")
f = sleplet.functions.SlepianSouthAmerica(128, region=region, noise=-10, smoothing=2)
f_sphere = sleplet.slepian_methods.slepian_inverse(f.coefficients, f.L, f.slepian)
amplitude = sleplet.plot_methods.compute_amplitude_for_noisy_sphere_plots(f)
sleplet.plotting.PlotSphere(
    f_sphere,
    f.L,
    "fig_8_a",
    amplitude=amplitude,
    normalise=False,
    region=f.region,
).execute()

Slepian Scale-Discretised Wavelets on the Sphere: Fig. 9

export SLEPIAN_MASK="africa"
# a
sphere earth -L 128 -p africa -s 2 -u
# b
sphere slepian_africa -L 128 -s 2 -u

import pyssht as ssht
import sleplet

# a
f = sleplet.functions.Earth(128, smoothing=2)
flm = sleplet.harmonic_methods.rotate_earth_to_africa(f.coefficients, f.L)
f_sphere = ssht.inverse(flm, f.L, Method="MWSS")
sleplet.plotting.PlotSphere(f_sphere, f.L, "fig_9_a", normalise=False).execute()
# b
region = sleplet.slepian.Region(mask_name="africa")
g = sleplet.functions.SlepianAfrica(128, region=region, smoothing=2)
g_sphere = sleplet.slepian_methods.slepian_inverse(g.coefficients, g.L, g.slepian)
sleplet.plotting.PlotSphere(
    g_sphere,
    g.L,
    "fig_9_b",
    normalise=False,
    region=g.region,
).execute()

Slepian Scale-Discretised Wavelets on the Sphere: Fig. 10

python -m examples.arbitrary.africa.eigenvalues_africa

Slepian Scale-Discretised Wavelets on the Sphere: Fig. 11

export SLEPIAN_MASK="africa"
for p in 0 9 24 49 99 199; do
    sphere slepian -e ${p} -L 128 -u
done

import sleplet

region = sleplet.slepian.Region(mask_name="africa")
for p in [0, 9, 24, 49, 99, 199]:
    f = sleplet.functions.Slepian(128, region=region, rank=p)
    f_sphere = sleplet.slepian_methods.slepian_inverse(f.coefficients, f.L, f.slepian)
    sleplet.plotting.PlotSphere(
        f_sphere,
        f.L,
        f"fig_11_p{p}",
        normalise=False,
        region=f.region,
    ).execute()

Slepian Scale-Discretised Wavelets on the Sphere: Fig. 12

export SLEPIAN_MASK="africa"
# a
sphere slepian_wavelets -L 128 -u
# b
for j in $(seq 0 5); do
    sphere slepian_wavelets -e 3 2 ${j} -L 128 -u
done

import sleplet

region = sleplet.slepian.Region(mask_name="africa")
for j in [None, *list(range(6))]:
    f = sleplet.functions.SlepianWavelets(128, region=region, B=3, j_min=2, j=j)
    f_sphere = sleplet.slepian_methods.slepian_inverse(f.coefficients, f.L, f.slepian)
    sleplet.plotting.PlotSphere(
        f_sphere,
        f.L,
        f"fig_12_j_{j}",
        normalise=False,
        region=f.region,
    ).execute()

Slepian Scale-Discretised Wavelets on the Sphere: Fig. 13

export SLEPIAN_MASK="africa"
# a
sphere slepian_wavelet_coefficients_africa -L 128 -s 2 -u
# b
for j in $(seq 0 5); do
    sphere slepian_wavelet_coefficients_africa -e 3 2 ${j} -L 128 -s 2 -u
done

import sleplet

region = sleplet.slepian.Region(mask_name="africa")
for j in [None, *list(range(6))]:
    f = sleplet.functions.SlepianWaveletCoefficientsAfrica(
        128,
        region=region,
        B=3,
        j_min=2,
        j=j,
        smoothing=2,
    )
    f_sphere = sleplet.slepian_methods.slepian_inverse(f.coefficients, f.L, f.slepian)
    sleplet.plotting.PlotSphere(
        f_sphere,
        f.L,
        f"fig_13_j_{j}",
        normalise=False,
        region=f.region,
    ).execute()

Slepian Scale-Discretised Wavelets on the Sphere: Fig. 14

export SLEPIAN_MASK="africa"
# a
sphere slepian_africa -L 128 -n -10 -s 2 -u
# b-d
for s in 2 3 5; do
    python -m examples.arbitrary.africa.denoising_slepian_africa -n -10 -s ${s}
done

import sleplet

# a
region = sleplet.slepian.Region(mask_name="africa")
f = sleplet.functions.SlepianAfrica(128, region=region, noise=-10, smoothing=2)
f_sphere = sleplet.slepian_methods.slepian_inverse(f.coefficients, f.L, f.slepian)
amplitude = sleplet.plot_methods.compute_amplitude_for_noisy_sphere_plots(f)
sleplet.plotting.PlotSphere(
    f_sphere,
    f.L,
    "fig_14_a",
    amplitude=amplitude,
    normalise=False,
    region=f.region,
).execute()

Slepian Scale-Discretised Wavelets on Manifolds

Slepian Scale-Discretised Wavelets on Manifolds: Fig. 2

for r in $(seq 2 9); do
    mesh homer -e ${r} -u
done

import sleplet

mesh = sleplet.meshes.Mesh("homer")
for r in range(2, 10):
    f = sleplet.meshes.MeshBasisFunctions(mesh, rank=r)
    f_mesh = sleplet.harmonic_methods.mesh_inverse(f.mesh, f.coefficients)
    sleplet.plotting.PlotMesh(mesh, f"fig_2_r_{r}", f_mesh, normalise=False).execute()

Slepian Scale-Discretised Wavelets on Manifolds: Fig. 4

python -m examples.mesh.mesh_tiling homer

Slepian Scale-Discretised Wavelets on Manifolds: Fig. 5

python -m examples.mesh.mesh_region homer

Slepian Scale-Discretised Wavelets on Manifolds: Fig. 6

for p in 0 9 24 49 99 199; do
    mesh homer -m slepian_functions -e ${p} -u -z
done

import sleplet

mesh = sleplet.meshes.Mesh("homer", zoom=True)
for p in [0, 9, 24, 49, 99, 199]:
    f = sleplet.meshes.MeshSlepianFunctions(mesh, rank=p)
    f_mesh = sleplet.slepian_methods.slepian_mesh_inverse(
        f.mesh_slepian,
        f.coefficients,
    )
    sleplet.plotting.PlotMesh(
        mesh,
        f"fig_6_p_{p}",
        f_mesh,
        normalise=False,
        region=True,
    ).execute()

Slepian Scale-Discretised Wavelets on Manifolds: Fig. 7

python -m examples.mesh.mesh_slepian_eigenvalues homer

Slepian Scale-Discretised Wavelets on Manifolds: Fig. 8

# a
mesh homer -m slepian_wavelets -u -z
# b-f
for j in $(seq 0 4); do
    mesh homer -e 3 2 ${j} -m slepian_wavelets -u -z
done

import sleplet

mesh = sleplet.meshes.Mesh("homer", zoom=True)
for j in [None, *list(range(5))]:
    f = sleplet.meshes.MeshSlepianWavelets(mesh, B=3, j_min=2, j=j)
    f_mesh = sleplet.slepian_methods.slepian_mesh_inverse(
        f.mesh_slepian,
        f.coefficients,
    )
    sleplet.plotting.PlotMesh(
        mesh,
        f"fig_8_j_{j}",
        f_mesh,
        normalise=False,
        region=True,
    ).execute()

Slepian Scale-Discretised Wavelets on Manifolds: Fig. 9

mesh homer -m field -u

import sleplet

mesh = sleplet.meshes.Mesh("homer")
f = sleplet.meshes.MeshField(mesh)
f_mesh = sleplet.harmonic_methods.mesh_inverse(f.mesh, f.coefficients)
sleplet.plotting.PlotMesh(mesh, "fig_9", f_mesh, normalise=False).execute()

Slepian Scale-Discretised Wavelets on Manifolds: Fig. 10

# a
mesh homer -m slepian_wavelet_coefficients -u -z
# b-f
for j in $(seq 0 4); do
    mesh homer -e 3 2 ${j} -m slepian_wavelet_coefficients -u -z
done

import sleplet

mesh = sleplet.meshes.Mesh("homer", zoom=True)
for j in [None, *list(range(5))]:
    f = sleplet.meshes.MeshSlepianWaveletCoefficients(mesh, B=3, j_min=2, j=j)
    f_mesh = sleplet.slepian_methods.slepian_mesh_inverse(
        f.mesh_slepian,
        f.coefficients,
    )
    sleplet.plotting.PlotMesh(
        mesh,
        f"fig_10_j_{j}",
        f_mesh,
        normalise=False,
        region=True,
    ).execute()

Slepian Scale-Discretised Wavelets on Manifolds: Fig. 11

# a
mesh homer -m slepian_field -u -z
# b
mesh homer -m slepian_field -n -5 -u -z
# c
python -m examples.mesh.denoising_slepian_mesh homer -n -5 -s 2

import sleplet

mesh = sleplet.meshes.Mesh("homer", zoom=True)
# a
f = sleplet.meshes.MeshSlepianField(mesh)
f_mesh = sleplet.slepian_methods.slepian_mesh_inverse(f.mesh_slepian, f.coefficients)
sleplet.plotting.PlotMesh(
    mesh,
    "fig_11_a",
    f_mesh,
    normalise=False,
    region=True,
).execute()
# b
g = sleplet.meshes.MeshSlepianField(mesh, noise=-5)
g_mesh = sleplet.slepian_methods.slepian_mesh_inverse(g.mesh_slepian, g.coefficients)
amplitude = sleplet.plot_methods.compute_amplitude_for_noisy_mesh_plots(g)
sleplet.plotting.PlotMesh(
    mesh,
    "fig_11_b",
    g_mesh,
    amplitude=amplitude,
    normalise=False,
    region=True,
).execute()

Slepian Scale-Discretised Wavelets on Manifolds: Fig. 12

for f in cheetah dragon bird teapot cube; do
    python -m examples.mesh.mesh_region ${f}
done

Slepian Scale-Discretised Wavelets on Manifolds: Tab. 1

python -m examples.mesh.produce_table

Slepian Wavelets for the Analysis of Incomplete Data on Manifolds

Chapter 2

Fig. 2.1

for ell in $(seq 0 4); do
    for m in $(seq 0 ${ell}); do
        sphere spherical_harmonic -e ${ell} ${m} -L 128 -u -z
    done
done

import pyssht as ssht
import sleplet

for ell in range(5):
    for m in range(ell + 1):
        f = sleplet.functions.SphericalHarmonic(128, ell=ell, m=m)
        f_sphere = ssht.inverse(f.coefficients, f.L, Method="MWSS")
        sleplet.plotting.PlotSphere(
            f_sphere,
            f.L,
            f"fig_2_1_ell_{ell}_m_{m}",
            normalise=False,
            unzeropad=True,
        ).execute()

Fig. 2.2

# a
sphere elongated_gaussian -e -1 -1 -L 128
# b
sphere elongated_gaussian -e -1 -1 -L 128 -m rotate -a 0 -b 0 -g 0.25
# c
sphere elongated_gaussian -e -1 -1 -L 128 -m rotate -a 0 -b 0.25 -g 0.25
# d
sphere elongated_gaussian -e -1 -1 -L 128 -m rotate -a 0.25 -b 0.25 -g 0.25

import numpy as np
import pyssht as ssht
import sleplet

# a
f = sleplet.functions.ElongatedGaussian(128, p_sigma=0.1, t_sigma=0.1)
f_sphere = ssht.inverse(f.coefficients, f.L, Method="MWSS")
sleplet.plotting.PlotSphere(f_sphere, f.L, "fig_2_2_a", annotations=[]).execute()
# b-d
for a, b, g in [(0, 0, 0.25), (0, 0.25, 0.25), (0.25, 0.25, 0.25)]:
    glm_rot = f.rotate(alpha=a * np.pi, beta=b * np.pi, gamma=g * np.pi)
    g_sphere = ssht.inverse(glm_rot, f.L, Method="MWSS")
    sleplet.plotting.PlotSphere(
        g_sphere,
        f.L,
        f"fig_2_2_a_{a}_b_{b}_g_{g}",
        annotations=[],
    ).execute()

Fig. 2.3

python -m examples.misc.wavelet_transform

Fig. 2.4

python -m examples.wavelets.axisymmetric_tiling

Fig. 2.5

# a
sphere axisymmetric_wavelets -L 128 -u
# b-e
for j in $(seq 0 3); do
    sphere axisymmetric_wavelets -e 3 2 ${j} -L 128 -u
done

import pyssht as ssht
import sleplet

for j in [None, *list(range(4))]:
    f = sleplet.functions.AxisymmetricWavelets(128, B=3, j_min=2, j=j)
    f_sphere = ssht.inverse(f.coefficients, f.L, Method="MWSS")
    sleplet.plotting.PlotSphere(
        f_sphere,
        f.L,
        f"fig_2_5_j_{j}",
        normalise=False,
    ).execute()

Fig. 2.6

python -m examples.polar_cap.eigenvalues

Fig. 2.7

python -m examples.polar_cap.fried_egg

Fig. 2.8

python -m examples.polar_cap.eigenfunctions

Chapter 3

Fig. 3.1

# a
sphere gaussian -L 128
# b
sphere gaussian -a 0.75 -b 0.125 -L 128 -m translate -o

import numpy as np
import pyssht as ssht
import sleplet

# a
f = sleplet.functions.Gaussian(128)
f_sphere = ssht.inverse(f.coefficients, f.L, Method="MWSS")
sleplet.plotting.PlotSphere(f_sphere, f.L, "fig_3_1_a", annotations=[]).execute()
# b
glm_trans = f.translate(alpha=0.75 * np.pi, beta=0.125 * np.pi)
g_sphere = ssht.inverse(glm_trans, f.L, Method="MWSS")
sleplet.plotting.PlotSphere(g_sphere, f.L, "fig_3_1_b", annotations=[]).execute()

Fig. 3.2

# a
sphere squashed_gaussian -L 128
# b
sphere squashed_gaussian -a 0.75 -b 0.125 -L 128 -m translate -o

import numpy as np
import pyssht as ssht
import sleplet

# a
f = sleplet.functions.SquashedGaussian(128)
f_sphere = ssht.inverse(f.coefficients, f.L, Method="MWSS")
sleplet.plotting.PlotSphere(f_sphere, f.L, "fig_3_2_a", annotations=[]).execute()
# b
glm_trans = f.translate(alpha=0.75 * np.pi, beta=0.125 * np.pi)
g_sphere = ssht.inverse(glm_trans, f.L, Method="MWSS")
sleplet.plotting.PlotSphere(g_sphere, f.L, "fig_3_2_b", annotations=[]).execute()

Fig. 3.3

# a
sphere elongated_gaussian -L 128
# b
sphere elongated_gaussian -a 0.75 -b 0.125 -L 128 -m translate -o

import numpy as np
import pyssht as ssht
import sleplet

# a
f = sleplet.functions.ElongatedGaussian(128)
f_sphere = ssht.inverse(f.coefficients, f.L, Method="MWSS")
sleplet.plotting.PlotSphere(f_sphere, f.L, "fig_3_3_a", annotations=[]).execute()
# b
glm_trans = f.translate(alpha=0.75 * np.pi, beta=0.125 * np.pi)
g_sphere = ssht.inverse(glm_trans, f.L, Method="MWSS")
sleplet.plotting.PlotSphere(g_sphere, f.L, "fig_3_3_b", annotations=[]).execute()

Fig. 3.4

Figs. (c-d) correspond to (a-b) in Fig. 1 of the Sifting Convolution on the Sphere paper. The following creates Figs. (a-b).

for ell in $(seq 2 -1 1); do
    sphere harmonic_gaussian -e ${ell} 1 -L 128
done

import pyssht as ssht
import sleplet

for ell in range(2, 0, -1):
    f = sleplet.functions.HarmonicGaussian(128, l_sigma=10**ell, m_sigma=10)
    f_sphere = ssht.inverse(f.coefficients, f.L, Method="MWSS")
    sleplet.plotting.PlotSphere(
        f_sphere,
        f.L,
        f"fig_3_4_ell_{ell}",
        annotations=[],
    ).execute()

Fig. 3.5

The same as Fig. 2 of the Sifting Convolution on the Sphere paper.

Fig. 3.6

The same as Fig. 3 of the Sifting Convolution on the Sphere paper.

Chapter 4

The plots here are the same as the Slepian Scale-Discretised Wavelets on the Sphere paper without the Africa examples, i.e. Fig. 10 onwards.

Chapter 5

The plots here are the same as the Slepian Scale-Discretised Wavelets on Manifolds paper.