Theory figure: covariance and simulated random fields

Gaussian probability distributions and spatial covariance are separate choices. These simulations compare Gaussian, exponential and Matérn covariances using the same refractive-index standard deviation and correlation length. The plotted covariance estimate averages products within each realization, then averages independent realizations. No spatial sample mean is removed.

Discrete spectral predictions include the finite periodic synthesis box and spectral truncation. Continuum covariance curves are reference functions, not exact descriptions of the sampled finite grid.

from bornsim.medium.random_medium import GaussianMedium, ExponentialMedium, WhittleMaternMedium

import matplotlib.pyplot as plt
import numpy as np

from bornsim import EnsembleSampling, Grid
from bornsim.units import ureg

voxel_spacing = 20 * ureg.nanometer

grid = Grid(
    shape=(24, 24, 24),
    spacing=voxel_spacing,
)

ensemble_sampling = EnsembleSampling(
    realizations=32,
    seed=42,
)

correlation_length = 60 * ureg.nanometer

refractive_index_std = 0.01

gaussian_medium = GaussianMedium(
    background_refractive_index=1.33,
    refractive_index_std=refractive_index_std,
    correlation_length=correlation_length,
)

exponential_medium = ExponentialMedium(
    background_refractive_index=1.33,
    refractive_index_std=refractive_index_std,
    correlation_length=correlation_length,
)

whittle_matern_medium = WhittleMaternMedium(
    background_refractive_index=1.33,
    refractive_index_std=refractive_index_std,
    correlation_length=correlation_length,
    smoothness=1.5,
)

covariance_models = (
    (gaussian_medium, "Gaussian", "#0072B2"),
    (exponential_medium, "Exponential", "#D55E00"),
    (whittle_matern_medium, "Matérn, ν = 3/2", "#009E73"),
)

preview_volumes = {}

Sample fields and estimate covariance

slice_figure, slice_axes = plt.subplots(1, 3, figsize=(12, 4), layout="constrained")

covariance_comparisons = []

# Each offset is a voxel count along x; the corresponding distance carries units.
voxel_offsets_x = np.arange(9)

distance_between_points = voxel_offsets_x * voxel_spacing

# Random-field synthesis uses a periodic field on twice the sample size before cropping.
synthesis_grid_shape = tuple(2 * voxel_count for voxel_count in grid.shape)

voxel_spacing_m = voxel_spacing.to("meter").magnitude

axis_wavenumbers_per_meter = [
    2 * np.pi * np.fft.fftfreq(voxel_count, d=voxel_spacing_m) for voxel_count in synthesis_grid_shape
]

squared_wavenumbers = (
    sum(component**2 for component in np.meshgrid(*axis_wavenumbers_per_meter, indexing="ij")) / ureg.meter**2
)

for slice_axis, (random_medium, covariance_label, color) in zip(slice_axes, covariance_models):
    covariance_model = random_medium.correlation

    covariance_estimates_by_realization = []

    for realization_seed in ensemble_sampling.seeds:
        sampled_volume = random_medium.to_volume(
            grid=grid,
            seed=realization_seed,
        )

        refractive_index_fluctuations = sampled_volume.delta_refractive_index

        covariance_estimates_by_offset = [np.mean(refractive_index_fluctuations**2)]

        for voxel_offset_x in voxel_offsets_x[1:]:
            covariance_estimates_by_offset.append(
                np.mean(
                    refractive_index_fluctuations[:-voxel_offset_x] * refractive_index_fluctuations[voxel_offset_x:]
                )
            )

        covariance_estimates_by_realization.append(covariance_estimates_by_offset)

        if realization_seed == ensemble_sampling.seeds[0]:
            preview_volumes[covariance_model] = sampled_volume

            sample_width_nm = (grid.shape[0] * voxel_spacing).to("nanometer").magnitude

            slice_image = slice_axis.imshow(
                refractive_index_fluctuations[:, :, grid.shape[2] // 2].T,
                origin="lower",
                extent=(-sample_width_nm / 2, sample_width_nm / 2, -sample_width_nm / 2, sample_width_nm / 2),
                cmap="RdBu_r",
                vmin=-3 * refractive_index_std,
                vmax=3 * refractive_index_std,
            )

    slice_axis.set(title=covariance_label, xlabel="x (nm)", ylabel="y (nm)")

    spectral_weights = random_medium.spectral_weight(q_squared=squared_wavenumbers)

    normalized_discrete_covariance = np.fft.ifftn(spectral_weights / spectral_weights.mean()).real[
        voxel_offsets_x, 0, 0
    ]

    distance_over_correlation_length = (distance_between_points / correlation_length).to("dimensionless").magnitude

    if covariance_model == "gaussian":
        normalized_continuum_covariance = np.exp(-(distance_over_correlation_length**2) / 2)
    elif covariance_model == "exponential":
        normalized_continuum_covariance = np.exp(-distance_over_correlation_length)
    else:
        normalized_continuum_covariance = (1 + np.sqrt(3) * distance_over_correlation_length) * np.exp(
            -np.sqrt(3) * distance_over_correlation_length
        )

    normalized_covariance_samples = np.asarray(covariance_estimates_by_realization) / refractive_index_std**2

    covariance_comparisons.append(
        (
            covariance_label,
            color,
            normalized_covariance_samples,
            normalized_discrete_covariance,
            normalized_continuum_covariance,
        )
    )

slice_figure.colorbar(slice_image, ax=slice_axes, label="Refractive-index fluctuation δn", shrink=0.8)

slice_figure.suptitle("Seed 42 · same σn = 0.01 and ℓ = 60 nm · 20 nm voxels")

plt.show()

Gaussian medium in 3D

Show one finite input sample with its physical spatial axes.

preview_volume = preview_volumes["gaussian"]

medium_figure = preview_volume.plot_3d(
    backend="matplotlib",
    mode="voxels",
    field="refractive_index",
)

medium_figure.suptitle(f"Gaussian medium · seed {ensemble_sampling.seeds[0]}")

plt.show()

Exponential medium in 3D

Show one finite input sample with its physical spatial axes.

preview_volume = preview_volumes["exponential"]

medium_figure = preview_volume.plot_3d(
    backend="matplotlib",
    mode="voxels",
    field="refractive_index",
)

medium_figure.suptitle(f"Exponential medium · seed {ensemble_sampling.seeds[0]}")

plt.show()

Whittle–Matérn medium in 3D

Show one finite input sample with its physical spatial axes.

preview_volume = preview_volumes["matern"]

medium_figure = preview_volume.plot_3d(
    backend="matplotlib",
    mode="voxels",
    field="refractive_index",
)

medium_figure.suptitle(f"Whittle–Matérn medium · seed {ensemble_sampling.seeds[0]}")

plt.show()

Compare sampled and predicted covariances

Reuse the ensemble estimates above. Error bars describe realization sampling; dashed curves use the discrete synthesis and dotted curves the continuum covariance, which need not match on a finite grid.

covariance_figure, covariance_axis = plt.subplots(figsize=(8, 5), layout="constrained")

for (
    covariance_label,
    color,
    normalized_covariance_samples,
    normalized_discrete_covariance,
    normalized_continuum_covariance,
) in covariance_comparisons:
    covariance_axis.errorbar(
        distance_between_points.to("nanometer").magnitude,
        normalized_covariance_samples.mean(axis=0),
        yerr=normalized_covariance_samples.std(axis=0, ddof=1) / np.sqrt(ensemble_sampling.realizations),
        fmt="o",
        capsize=2,
        color=color,
        label=f"{covariance_label}: simulation",
    )

    covariance_axis.plot(
        distance_between_points.to("nanometer").magnitude,
        normalized_discrete_covariance,
        color=color,
        linestyle="--",
    )

    covariance_axis.plot(
        distance_between_points.to("nanometer").magnitude,
        normalized_continuum_covariance,
        color=color,
        linestyle=":",
    )

covariance_axis.set(
    xlabel="Distance between points along x (nm)",
    ylabel="Covariance / σn²",
    title="32 realizations: dashed = discrete synthesis; dotted = continuum",
)

covariance_axis.legend(frameon=False)

covariance_axis.grid(alpha=0.25)

plt.show()