Example 3: controlled adaptive compression

To test adaptive truncation cleanly, use a field with a known, smooth angular spectrum:

\[ f(r,\theta)=g(r)\left[ 1+0.4e^{2i\theta}+0.2e^{-4i\theta} \right], \qquad g(r)=r^4e^{-0.7r^2}. \]

The common \(r^4\) factor makes every angular contribution regular at the origin. In Cartesian language, each term is a smooth polynomial times a Gaussian:

• \(r^4\) is \((x^2+y^2)^2\)

• \(r^4e^{2i\theta}=r^2(x+iy)^2\)

• \(r^4e^{-4i\theta}=(x-iy)^4\).

Because all three channels share the same radial envelope, their powers are proportional to

\[ 1^2=1,\qquad 0.4^2=0.16,\qquad 0.2^2=0.04. \]

The normalized power fractions are therefore

\[ \frac{1}{1.20},\qquad \frac{0.16}{1.20},\qquad \frac{0.04}{1.20}. \]

If only the first two channels are kept, the exact relative \(L^2\) error is

\[ \sqrt{\frac{0.04}{1.20}}\approx18.2574\%. \]

A complete minimal version is

import numpy as np
from petal2d import PolarDecomposition

x = np.linspace(-6.0, 6.0, 161)
y = np.linspace(-6.0, 6.0, 161)

def field(x, y):
    r = np.hypot(x, y)
    theta = np.arctan2(y, x)
    g = r**4 * np.exp(-0.7 * r**2)
    return g * (
        1.0
        + 0.4 * np.exp(2j * theta)
        + 0.2 * np.exp(-4j * theta)
    )

dec = PolarDecomposition(
    field,
    x,
    y,
    Ntheta=256,
    recon_err_tol=20.0,
    origin=(0.0, 0.0),
)

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

[0, 2]
18.257418583505537
18.257418583505537

Run the complete example:

python examples/adaptive_truncation.py

=== Adaptive truncation of a smooth known mixed spectrum ===
domain_consistency: 1.00000000
selected_pairs: [(0,), (2,)]
retained_modes: [0, 2]
reconstruction_error_percent: 18.2574
target_reached: True
Parseval-predicted error (%): 18.257419
directly measured error (%): 18.257419
     m           power      fraction   r_power_support   r_amplitude_support      r_cutoff  retained
----------------------------------------------------------------------------------------------------
     0    2.231213e+00  8.333333e-01           4.09288               4.24065       4.24065      True
     2    3.569941e-01  1.333333e-01           4.09288               4.24065       4.24065      True

The \(m=-4\) channel is omitted because the first two channels already meet the requested 20% error. The predicted and directly measured errors agree to numerical precision. This is not a heuristic compression score: for the computed discrete spectrum it follows directly from orthogonality and Parseval’s identity.

Only the two channels required by the requested error are retained.

The visible difference is the omitted \(m=-4\) contribution.