Generalized Superellipse Spectral Filtering

• Generalized superellipse spectral filtering is a framework that uses superelliptic contours in the Fourier domain to independently control hyperuniformity, anisotropy, and band-shape.
• It employs an analytic mask for direct filtering via FFT, significantly speeding up simulations without the need for iterative optimizations.
• The technique is pivotal for designing advanced photonic materials, porous media, and complex random fields through precise spectral and morphological tuning.

Generalized superellipse spectral filtering is a parametric framework for constructing spectral filters whose frequency (or wavenumber) response is shaped by superelliptic contours, enabling precise manipulation of both the radial scaling and angular (anisotropic) profile of the spectral envelope. This formalism generalizes conventional isotropic and elliptical spectral masks (such as those found in standard Gaussian or elliptic filters) to a broader class defined by the superellipse equation, facilitating independent control over hyperuniformity, band-shape, anisotropy, and angular selectivity. The technology finds particular utility in the large-scale simulation and synthetic design of hyperuniform materials, random fields, and advanced image-processing algorithms.

1. Mathematical Formulation of Superellipse Spectral Masks

The generalized superellipse mask is defined in the Fourier domain by the norm:

$$K_{p,a,b}(\mathbf{k}) = \left( \left| \frac{k_x}{a} \right|^p + \left| \frac{k_y}{b} \right|^p \right)^{1/p}$$

Here, $p > 0$ governs angular shape: $p = 2$ yields ellipsoidal/circular windows; $p = 1$ produces diamond-shaped masks; higher $p$ approach square-like profiles. Scale factors $a$, $b$ dictate anisotropy along $k_x$ and $k_y$ directions.

The target spectral density for each Fourier mode is:

$$\tilde{\chi}_k(\mathbf{k}) = C \cdot \exp\big[\alpha \ln(K_{p,a,b}(\mathbf{k})) - K_{p,a,b}^2(\mathbf{k})/(2 \sigma^2)\big]$$

where $\alpha$ sets the small-$\mathbf{k}$ power-law scaling, controlling hyperuniformity degree, $\sigma$ modulates the shell bandwidth, and $C$ normalizes the density ($\max_{\mathbf{k}} \tilde{\chi}_k(\mathbf{k}) = 1$) (Zhong et al., 10 Sep 2025). In three dimensions, a third term is added in the norm.

The superellipse mask thus generalizes from basic circular (isotropic) shells to star-like, diamond, squircular, elliptical, and rectilinear Fourier space shapes. The exponent $\alpha$ determines the order of spectral depletion (hyperuniform exponent), and the anisotropy parameters ($a$, $b$) set the directionality.

2. Filter Construction and Computational Workflow

Generalized superellipse spectral filtering harnesses the analytic mask for direct filtering in the frequency domain. The workflow is:

1. Sample i.i.d. complex Gaussian white noise in Fourier space on the desired grid.
2. Compute the amplitude modulation via $A(\mathbf{k}) = \sqrt{\tilde{\chi}_k(\mathbf{k})}$
3. Multiply noise by $A(\mathbf{k})$ for all Fourier modes.
4. Transform filtered field to real space via the inverse FFT.

This single-shot process avoids iterative energy-minimization routines (e.g., Yeong–Torquato reconstruction), resulting in scaling of $O(N \log N)$, where $N$ is the total number of grid points. No optimization or spectral fitting steps are required, as mask parameters specify the spectral envelope directly.

The approach is compatible with thresholding: after reconstructing the continuous field, a simple sign operation may be used to produce binary (two-phase) microstructures, facilitating morphological and power-spectrum analysis of the resulting materials or fields.

3. Spectral and Morphological Control

Superellipse spectral filtering enables independent and systematic tuning of all key aspects of the spectrum:

• Radial properties (hyperuniformity and bandwidth): Controlled by $\sigma$ and $\alpha$; low-$\mathbf{k}$ scaling as $|\mathbf{k}|^\alpha$ allows direct prescription of hyperuniform class.
• Angular/shape tuning ($p$): Varying $p$ alters the edge shape from diamond ($p=1$), through elliptical ($p=2$), to square-like ($p\gg 2$) and star-like ($p<2$).
• Anisotropy ($a$, $b$): Stretching masks along axes, the envelope becomes elongated or compressed, giving rise to directional statistical correlations or stripe-like features in real-space morphologies.

This multi-parametric design space affords direct tailoring of microstructure for advanced photonic, thermal, mechanical, and biological material systems.

Continuous and Binary Microstructures: After thresholding, the morphology transitions are preserved (with some reduction of strict hyperuniformity), and the spectral shell signature remains prominent in both the continuous and binary cases, with physical property implications traceable to these spectral features.

4. Computational Efficiency and Performance

Performance comparison (Zhong et al., 10 Sep 2025):

Method	Time (2D, 1024×1024)	Time (3D, 256³)	Spectral Control	Iterative Steps
Superellipse Filtering	0.03 s	1.3 s	Full, analytic	None
Iterative Techniques	O(100-1000 s)	Infeasible	Limited/Indirect	1000s+

Two FFTs and pointwise multiplication yield orders-of-magnitude speedup vs. iterative approaches. The analytic expression ensures exact spectral control absent in typical polynomial or rational approximations.

5. Applications in Hyperuniform and Anisotropic Field Synthesis

Generalized superellipse spectral filtering provides a versatile toolkit for the synthesis of hyperuniform continuous random fields with arbitrary spectral band shape (Zhong et al., 10 Sep 2025). Key application domains include:

• Photonic and phononic material design: Controlling spatial scale, orientation, and statistical disorder directly impacts band gaps and wave transport properties.
• Porous media and composites: Precise morphological engineering for permeability, mechanical strength, or optical transparency.
• Simulation of advanced random processes: Study of the impact of hyperuniformity and anisotropy on phase behavior, transport, or emergent phenomena.

Thresholded fields become two-phase microstructures, expanding possible studies to network formation, domain connectivity, and interface physics.

6. Relation to Other Spectral Filter Formalisms

Connection to rational polynomial spectral filters (Patanè, 2020) and generalized-exponent filters (Alkhairy, 5 Apr 2024):

• Rational Filters: While rational polynomial filters approximate various spectral responses, superellipse filtering prescribes the envelope directly, including steep transitions or power-law tails, and is agnostic to eigen-decomposition or fitting (Patanè, 2020).
• Generalized-Exponent Filters: Both paradigms use tunable exponents, but superellipse masks define the global envelope in the Fourier domain for spatial field synthesis, whereas GEFs typically govern the transfer function in the time/frequency domain for signal-processing applications (Alkhairy, 5 Apr 2024).
• Graph Spectral Filtering: Diverse spectral filter architectures in GNNs could, in principle, be extended to superellipse-type spectral shapes, though most current methods employ polynomial bases (Guo et al., 2023).

7. Implications and Future Directions

Generalized superellipse spectral filtering advances both theory and practice in the design and simulation of hyperuniform and anisotropic fields. The analytical prescription of spectral envelopes allows scalable inverse design for materials with complex spatial correlations and fluctuations. The computational efficiency opens avenues for large-scale simulation, optimization, and direct structure–property studies, including the generation of two-phase composites, exploration of anisotropic transport, and fast prototyping of random media for next-generation photonic, thermal, and mechanical applications.

A plausible implication is that further integration of superellipse mask parameterization into adaptive filtering, graph signal processing, or dynamical models will generalize current frameworks such as rational spectral filtering and diverse node-wise graph filters, potentially bringing similar spectral shape control to a wider array of data-driven and adaptive systems.