Circular Harmonic Expansion (CHE)

• Circular Harmonic Expansion (CHE) is an orthogonal representation that decomposes scalar or vector fields in polar coordinates into angular harmonics and radial functions.
• CHE is widely applied in acoustics, image analysis, and fluid dynamics, facilitating efficient field reconstruction and feature detection in circular geometries.
• Practical implementations of CHE involve regularized inversions and tailored sampling strategies, though high angular orders can introduce computational and accuracy challenges.

Circular Harmonic Expansion (CHE) provides an orthogonal representation of scalar or vector fields on the plane, particularly useful in polar or circular geometries. This expansion decomposes a field into modal components with quantized angular momentum, yielding a compact, physically meaningful, and mathematically tractable basis for both analysis and reconstruction. CHE has found applications in acoustics, fluid dynamics, and image processing, particularly in contexts where circular or rotational symmetries are dominant (Nguyen et al., 2023, Kennedy, 2019, Rendón et al., 2014).

1. Mathematical Foundations of Circular Harmonic Expansion

CHE expresses a function $f(r,\theta)$ defined in polar coordinates as a sum over angular harmonics modulated by suitable radial functions. For scalar fields such as acoustic pressure or image intensity, the general expansion is: $$f(r,\theta) = \sum_{n=-\infty}^{\infty} C_n(r) e^{i n \theta}$$ where $C_n(r)$ denotes the radial profile of the $n$-th angular order. In two-dimensional exterior Helmholtz problems, an acoustic field $p(r, \phi; \omega)$ is expanded as (Nguyen et al., 2023): $$p(r, \phi; \omega) = \sum_{n=-\infty}^\infty a_n(\omega) H^{(2)}_n(k r) e^{j n \phi}$$ with $H^{(2)}_n$ the Hankel function of the second kind, $k = \omega/c$ the wavenumber, and $a_n(\omega)$ the mode amplitudes. In the context of incompressible Euler flows (Rendón et al., 2014), CHE encodes the scalar stream-function or the vector potential as: $$A_z(r, \theta) = A_0 \ln r_> + \sum_{m=1}^\infty A_m (r_{<}^m / r_{>}^m) \cos(m \theta) + B_m (r_{<}^m / r_{>}^m) \sin(m \theta)$$ where $$f(r,\theta) = \sum_{n=-\infty}^{\infty} C_n(r) e^{i n \theta}$$0 and $$f(r,\theta) = \sum_{n=-\infty}^{\infty} C_n(r) e^{i n \theta}$$1 for a reference radius $$f(r,\theta) = \sum_{n=-\infty}^{\infty} C_n(r) e^{i n \theta}$$2.

CHE’s orthogonality arises from the integral: $$f(r,\theta) = \sum_{n=-\infty}^{\infty} C_n(r) e^{i n \theta}$$3 guaranteeing mutual independence of the expansion modes.

2. CHE in Acoustic and Sound Field Reconstruction

A principal application of CHE is in reconstructing two-dimensional exterior acoustic fields, especially in acousto-optic sensing configurations employing concentric-circle sampling (Nguyen et al., 2023). Here, the acoustic field is recovered from line-integral measurements (e.g., via LDV), formalized as: $$f(r,\theta) = \sum_{n=-\infty}^{\infty} C_n(r) e^{i n \theta}$$4 where $$f(r,\theta) = \sum_{n=-\infty}^{\infty} C_n(r) e^{i n \theta}$$5 is the measurement along the $$f(r,\theta) = \sum_{n=-\infty}^{\infty} C_n(r) e^{i n \theta}$$6-th path. The reconstruction proceeds by solving the linear system

$$f(r,\theta) = \sum_{n=-\infty}^{\infty} C_n(r) e^{i n \theta}$$7

where $$f(r,\theta) = \sum_{n=-\infty}^{\infty} C_n(r) e^{i n \theta}$$8 collects the pathwise integrals for all sampled modes, and $$f(r,\theta) = \sum_{n=-\infty}^{\infty} C_n(r) e^{i n \theta}$$9 contains the modal amplitudes. Regularized least-squares inversion is employed for stability: $C_n(r)$0 Choice and distribution of concentric sampling radii directly affect stability and reconstruction fidelity. For effective recovery, the sampled data must densely cover both angular and radial domains, with $C_n(r)$1 where $C_n(r)$2 is the total number of projections and $C_n(r)$3 the angular order considered (Nguyen et al., 2023).

3. CHE in Image Analysis and Feature Detection

CHE supports compact representation and detection of oriented image features, leveraging the angular Fourier structure to characterize local shape and symmetry (Kennedy, 2019). For a local intensity patch $C_n(r)$4, CHE coefficients quantify orientation and curvature information: $C_n(r)$5 For discrete images, implementation proceeds via FIR filter banks, using polar-separable filters

$C_n(r)$6

with radial apodization $C_n(r)$7 based on Laguerre-Gaussian functions. This formulation, linked to the Laguerre/Hermite correspondence, allows efficient computation: 2D convolutions decompose into cascaded 1D convolutions in Cartesian coordinates.

Mitigation of angular aliasing is achieved through apodization (e.g., Gaussian taper) and appropriate boundary handling. In corner (wedge) detection, the local spectrum $C_n(r)$8 is exploited to test for high-contrast angular patterns under noise and sampling uncertainty, outperforming classical detectors in area-under-curve metrics across wedge widths (Kennedy, 2019).

4. CHE in Fluid Dynamics and Green’s Function Expansions

CHE underpins analytic solutions for incompressible, steady 2D Euler flows with circular symmetry (Rendón et al., 2014). The vector potential or streamfunction is expanded via CHE to ensure both harmonicity and appropriate continuity/jump conditions at boundaries. In the exterior (outside a reference circle), all physically admissible harmonic flows are reconstructible from the CHE basis.

This framework enables closed-form expressions for streamlines, vorticity distributions, and velocity fields for vortex systems of arbitrary angular multipolarity: $C_n(r)$9 Expansions of the 2D Green’s function, as required in potential theory and flow reconstruction, naturally align with the structure of CHE, facilitating analytic treatment of multipolar vortex sheet distributions and their resultant velocity fields (Rendón et al., 2014).

5. Algorithmic and Practical Considerations

For practical CHE-based field reconstruction, several factors govern performance and computational cost (Nguyen et al., 2023, Kennedy, 2019):

• Expansion order $n$0: Sufficiently large to capture all significant modal content; undersized $n$1 leads to underfitting (loss of detail), while oversized $n$2 induces overfitting (spurious oscillations).
• Sampling scheme: Dense sampling in both angle and radius is essential. Efficacy improves with increased radial diversity; however, even two well-chosen radii can yield near-optimal results under certain conditions.
• Regularization: Inversion stability is maintained via Tikhonov or discrepancy-constrained regularization, crucial under noisy measurements or ill-conditioned sampling geometries.
• Computation: For acoustic field inversion, complexity scales as $n$3. In image analysis, the use of separable filter banks with precomputed Hermite-Gaussian kernels accelerates coefficient extraction.
• Noise and artefacts: CHE exhibits pronounced robustness to additive Gaussian noise compared to filtered back‐projection (FBP) and plane‐wave expansion (PWE). However, discretization, finite windowing, and border effects may introduce edge artefacts in image processing (Kennedy, 2019).

6. Comparative Performance and Limitations

CHE-based schemes provide clear improvements over alternative methods, particularly in “exterior” inverse problems where conventional algorithms (FBP, PWE) degrade in accuracy. Across simulations and experiments in acoustics, CHE consistently achieves lower normalized mean-squared error (NMSE), improved spatial fidelity, and better noise robustness (Nguyen et al., 2023).

In image analysis, CHE formulations outperform classical corner detectors (e.g., Harris, Kitchen–Rosenfeld) on synthetic wedge detection and preserve rotational invariance. Practical limitations include orthogonality degradation at high angular orders, challenges in optimal sampling/adaptive circle placement, computational cost at high frequencies or spatial resolutions, and the need for precomputed filter-bank components in discrete settings.

Plausible implications are that extension to three-dimensional volumetric data, physically optimized sampling strategies, and real-time algorithmic realizations remain open areas for research and further CHE-based methodological innovation (Nguyen et al., 2023, Kennedy, 2019).

7. Summary Table: CHE Modal Expansions and Application Domains

Domain	Basis Functions	Expansion Formula
Acoustic Fields	$n$4	$n$5
Image Analysis	$n$6 (Laguerre/Hermite)	$n$7
Fluid Dynamics	$n$8	$n$9

CHE thus provides a generalizable, orthogonal framework for modal analysis, efficient reconstruction, and feature extraction in systems governed by circular or rotational symmetry, with demonstrable advantages over conventional projection and Fourier-based methods in both forward and inverse settings (Nguyen et al., 2023, Kennedy, 2019, Rendón et al., 2014).