Interpreting results#

This page uses the sampled \(C_3\) field from Example 4: sampled numerical data as a concrete reference.

Full spectrum versus retained spectrum#

PETAL2D computes the complete discrete angular FFT before adaptive truncation. The complete spectrum is available as

dec.m_all
dec.rho_all
dec.powers_all
dec.power_fracs_all

For the sampled \(C_3\) example:

print("m_all shape:", dec.m_all.shape)
print("rho_all shape:", dec.rho_all.shape)
print("powers_all shape:", dec.powers_all.shape)
print("power_fracs_all shape:", dec.power_fracs_all.shape)

m_all shape: (512,)
rho_all shape: (121, 512)
powers_all shape: (512,)
power_fracs_all shape: (512,)

The adaptively retained subset is available through

dec.m_sorted
dec.rho
dec.power_fracs
dec.selected_pairs

and the same run gives

m_sorted: [0, 3, -3]
selected_pairs: [(0,), (3, -3)]
rho keys: [0, 3, -3]
power_fracs: {0: 0.9956769952656367, 3: 0.0021615023671450488, -3: 0.0021615023671450496}

The full spectrum includes floating-point and interpolation-level leakage. The retained spectrum is the compressed representation chosen by recon_err_tol.

Radial coefficients rho[m]#

dec.rho[m] is \(\rho_m(r)\) sampled on dec.r. It is generally complex even when the original field is real. For a real field,

\[ \rho_{-m}(r)=\rho_m(r)^*, \]

so the two channels carry equal integrated power.

Mode power#

PETAL2D defines

\[ P_m=\int_0^{r_{\max}}|\rho_m(r)|^2r\,dr. \]

power_fracs_all is powers_all normalized by \(\sum_mP_m\). The factor \(2\pi\) relating this convention to the physical polar \(L^2\) norm cancels in every fraction.

A large power fraction means that a harmonic accounts for a large fraction of the field’s polar \(L^2\) weight. Radial location is described by rho[m] and the cutoff diagnostics.

selected_pairs#

For real input, selected_pairs lists the adaptive units actually retained. The sampled \(C_3\) run gives

[(0,), (3, -3)]

The singleton (0,) is selected independently. The nonzero real-field harmonics are selected as the conjugate pair (3, -3) so that automatic adaptive reconstruction preserves real-valuedness. For genuinely complex input, each selected channel is a singleton.

Reconstruction error#

recon_error is the Parseval-predicted error from omitted spectral power. recon_error_measured independently evaluates the reconstructed field in the weighted polar norm.

print(dec.recon_error)
print(dec.recon_error_measured)

2.7069303579009995e-05
2.7016812777500244e-05

Both values are percentages. Their agreement is an internal consistency check. Small differences can appear for sampled data because the two errors are evaluated through different numerical routes.

domain_consistency#

PETAL2D defines

\[ C_D= \frac{\int_{\Omega_{\rm polar}}|f|^2d^2r} {\int_{\Omega_{\rm Cartesian}}|f|^2d^2r}. \]

For the sampled \(C_3\) run,

domain_consistency = 0.9996539035424497

Interpretation:

  • \(C_D\approx1\): the polar disk and Cartesian rectangle contain essentially the same field weight

  • \(C_D<1\): appreciable weight lies in parts of the rectangle outside the analyzed disk, or numerical discretization makes the polar estimate smaller

  • \(C_D>1\): possible for callable input with an explicitly enlarged polar domain, or from small quadrature/interpolation overshoot.

A useful limiting example is a uniform field on a square: the largest inscribed disk contains area fraction \(\pi/4\), so its exact domain consistency is \(\pi/4\), not one. A localized Gaussian can have domain_consistency very close to one even though the disk occupies less geometric area, because the omitted corners contain negligible field weight.

domain_consistency is a weight diagnostic, not a pointwise interpolation-error metric.

Cutoff radii#

For every retained mode PETAL2D reports

dec.radial_power_support_radius[m]
dec.radial_amplitude_support_radius[m]
dec.cutoff_radius[m]

with

\[ r_{\rm cutoff,m} = \max(r_{\rm power,m},r_{\rm amplitude,m}). \]

For the sampled \(C_3\) example, the actual values are

m= 0: r_power=3.395936536178625, r_amplitude=3.3938767167529695, r_cutoff=3.395936536178625
m= 3: r_power=4.220679790622508, r_amplitude=4.370270544732886,  r_cutoff=4.370270544732886
m=-3: r_power=4.22067979060771,  r_amplitude=4.370270544732886,  r_cutoff=4.370270544732886

cutoff_radius describes where the remaining sampled radial tail is negligible according to both configured criteria. It never changes f_recon. Reconstruction is performed on the complete analyzed polar grid.