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current_density — NV-magnetometry current imaging

At a glance
Problem A thin-film sheet current from the out-of-plane stray field imaged by a widefield NV-diamond magnetometer
Unknown stream function g with K = ∇×(g ẑ), so ∇·K = 0 holds by construction; 64²
Physics planar magnetostatics at a standoff z0: B̂z = (μ0/2) k e−k z0 ĝ, zero-padded FFT
Measurement Bz image + 1 % noise; data by a direct real-space Biot–Savart sum on a 2× grid in float64
Difficulty classes wires, loops, branching
Baselines grid, fourier (regularized k-space inversion)
Metrics current PSNR · current relative error (+ Bz data-space metrics)
Run nefi run current_density --smoke · full: nefi run configs/current_density_full.yaml

current_density: measurement, ground truth, reconstruction and error from the gallery run

Gallery run (smoke preset: 32², 4-pixel standoff; 600 steps, 0.7 s on a CPU): current PSNR 29.7 dB.

Recover a thin-film sheet current K(x, y) from the out-of-plane stray field B_z measured by a wide-field NV-diamond microscope at standoff z0 (Roth, Sepulveda & Wikswo 1989; Tetienne et al. 2019; Broadway et al. 2020; Midha et al. 2024 — the static-field sibling of NV noise sensing cited by NeTMY). The unknown is the stream function g with K = ∇×(g ẑ) = (∂_y g, −∂_x g), so charge conservation ∇·K = 0 holds by construction:

B̂_z(k) = (μ0/2) k e^{−k z0} ĝ(k)        (zero-padded FFT, f̂ = Σ f e^{−ik·ρ})


from nefi.instances.current_density import CurrentDensity
out = CurrentDensity(n=32, pixel=0.25, z0=1.0, steps=(200, 400), lr=1e-2, n_octaves=4, tv=1e-3,
                     hidden=64, depth=3).run(seed=0)                     # the smoke preset
out.metrics      # {"j_psnr", "j_relative_error", "bz_psnr", "bz_relative_error"}
CLI: nefi run current_density --smoke, nefi run configs/current_density_smoke.yaml --baseline fourier. Example: python examples/current_density.py --config configs/current_density_smoke.yaml.

Physics (nefi.physics.magnetostatics)

CurrentDensityOperator(domain, z0, mu0, pad_factor=2) — derivation, sign conventions and the in-plane / NV-axis / sensor-layer variants are in physics_elliptic.md §6. Units default to μm, mA and μT (μ0 = magnetic_constant("um", "mA", "uT") = 400π). Linear, homogeneity = 1, rebuilt per curriculum resolution (spacing re-derived, k-grid cached).

Data (inverse-crime guard)

BiotSavartDataGenerator with BiotSavartOperator: the scene's g is rendered on a supersample = 2× finer grid, K is formed with central differences there, and B_z is summed directly in real space (Biot–Savart midpoint rule, float64, chunked) at the native pixel centres — no FFT, no padding, no periodic images; fidelity_tag = "biot-savart-direct-float64" vs "magnetostatics-fft". Noise: noise_std × max|B_z|. The FFT operator at the true g differs from these data by 1.3–2.7 % (model mismatch). The noise-free field is kept as gt["bz"] for data-space metrics.

Scenes (CurrentScenes): g = Σ_s I_s Φ(−d_s/w_s) (smoothed indicators; d_s signed distance, Φ normal CDF) → currents with a Gaussian cross-section of std w_s along region boundaries:

class content
loops 1–3 elliptical current loops
wires 1–2 hairpin circuits (go-and-return wire pairs along a random polyline)
branching trunk + 2–3 branches from a junction with different currents (current splits/merges)

All sources keep margin from the edges so g → 0 there (no hidden return currents).

Inversion and baselines

  • NeuralField (tanh; ReLU was 2–4 dB worse here) with an Identity head for g; RelativeMSE data term (scale-free in μT) + isotropic TV(g) (piecewise-constant g ⇔ currents on thin paths); two-stage curriculum n/2 → n.
  • grid: free pixels + the same losses and curriculum.
  • fourier: the classical regularized k-space inversion fourier_inversion (Tikhonov fourier_reg relative to the transfer peak, Hanning window with cutoff k_c z0 = 3 unless fourier_cutoff), wrapped as a GridField initialized at the reconstruction with a single zero-learning-rate step, so it is reported through the same metrics and benchmark protocol.

Metrics

On K = ∇×g (central differences, both components stacked): j_psnr, j_relative_error; on the predicted field vs the noise-free data: bz_psnr, bz_relative_error.

Configuration (CurrentDensityConfig)

field smoke full meaning
n, pixel, z0 32, 0.25, 1.0 (4 px) 128, 0.1, 0.2 (2 px) pixels, pixel size, standoff (μm)
length_unit, current_unit, field_unit um, mA, uT same unit system of μ0
nv_layer, pad_factor 0, 2 0, 2 sensor-layer averaging, FFT padding
scene, current, width_px, margin wires, (0.5, 1.5), (1, 1.5), 0.15 branching, same scenes
loop_radius, wire_separation, branch_width (0.08, 0.2), (0.08, 0.14), (0.04, 0.07) same scene geometry (fractions of the FOV)
noise_std, supersample 0.01, 2 0.01, 2 data
hidden, depth, n_octaves, activation 64, 3, 4, tanh 256, 6, 8, tanh field
steps, lr, tv, data_loss (200, 400), 1e-2, 1e-3, relative_mse (1000, 3000), 3e-3, 1e-4, relative_mse optimization
fourier_reg, fourier_window, fourier_cutoff 1e-3, hanning, None same Fourier baseline

Smoke preset = gallery preset: the standoff is 4 pixels (1 μm at 0.25 μm pixels), so the measured B_z is a smooth, visibly smeared signed map of the thin current paths (at 2 px it looked like the current itself), and the prior is a piecewise-constant stream function (tv = 1e-3). Smoke result (seed 0, wires, 600 steps, CPU ≈ 1 s): neural j_psnr 29.7 dB / rel. error 0.32 (seeds 0-3: 28.9-29.7 dB) vs zero field 19.7 dB / 1.0; grid (same objective and budget) 27.9 dB / 0.39; Fourier 21.0 dB / 0.86 — the classical k-space inversion only keeps wavelengths above 2π z0 / 3 ≈ 8 px at this standoff. At a 5 px standoff the neural field still reaches 27-28 dB.

Display (CurrentDensity.viz_hints, nefi.viz.hints): B_z is drawn with a diverging map centred at 0 and the ground-truth / reconstruction panels show the sheet-current magnitude |K| = |∇×(g ẑ)|, the quantity the j_* metrics score.