Polar Separable Transform (PSepT)

• Polar Separable Transform (PSepT) is an orthogonal, separable transform for representing images on the unit disk with strict rotation covariance and numerical stability.
• It employs DCT-based radial bases and Fourier angular harmonics to decouple analysis along radial and angular coordinates, enabling efficient computation.
• PSepT reduces computational complexity to O(N² log N) and mitigates interpolation artifacts, outperforming traditional moment-based methods in speed and robustness.

The Polar Separable Transform (PSepT) is an orthogonal, separable transform providing an explicit basis for image representation on the unit disk with strict rotation covariance, efficient computational scaling, and improved numerical stability. Unlike classical moment-based representations—such as Zernike and pseudo-Zernike moments—that are limited by polynomial coupling in radial and angular variables, PSepT achieves full kernel separability, enabling independent and efficient analysis along radial and angular coordinates. PSepT exists both as a continuous transform constructed from Discrete Cosine Transform (DCT) radial bases and Fourier angular harmonics and as a discrete unitary map between Cartesian and polar grids rooted in the finite oscillator algebra framework.

1. Mathematical Foundations and Kernel Construction

The PSepT is defined for functions $f(r,\theta)$ on the unit disk $D=\{0\le r\le 1, -\pi\le\theta<\pi\}$ using an explicitly separable kernel: $$F_{n,m} = \int_{0}^{2\pi} \int_{0}^{1} f(r,\theta)K_{n,m}(r,\theta)\, r\, dr\, d\theta$$ with reconstruction

$$f(r,\theta) = \sum_{n=0}^{N_r-1} \sum_{m=-M}^{M} F_{n,m}K_{n,m}(r,\theta)$$

where the kernel $K_{n,m}(r,\theta)$ factorizes as $$K_{n,m}(r_k,\theta_j) = \Phi_n(r_k)\Psi_m(\theta_j)$$.

The radial basis $\Phi_n(r_k)$ is the DCT-II: $$\Phi_n(r_k) = \sqrt{\frac{2}{N_r}} \cos\left(\frac{n\pi(k+\frac12)}{N_r}\right), \quad n=0,\ldots,N_r-1$$ and the angular basis $\Psi_m(\theta_j)$ comprises normalized complex Fourier harmonics: $$\Psi_m(\theta_j) = \frac{1}{\sqrt{N_\theta}}e^{im\theta_j}, \quad m = -\frac{N_\theta}{2},\ldots,\frac{N_\theta}{2}-1$$ ensuring orthonormality over the polar grid. This design achieves complete decoupling across the polar coordinates.

Discrete PSepT on an $D=\{0\le r\le 1, -\pi\le\theta<\pi\}$0 Cartesian image involves:

• Resampling the image onto a polar grid $D=\{0\le r\le 1, -\pi\le\theta<\pi\}$1,
• Applying 1D DCT in the radial direction for each fixed angle,
• Applying 1D FFT in the angular direction for each fixed radial order,
• Optional coefficient truncation.

The inverse transform comprises IFFT and IDCT in the reverse order, restoring the image, followed by mapping to Cartesian coordinates if required (Singh et al., 10 Oct 2025).

2. Algebraic and Group-Theoretic Context

The finite oscillator model based on the $D=\{0\le r\le 1, -\pi\le\theta<\pi\}$2 algebra allows an exact, constructive approach to PSepT within discrete signal domains. Two subalgebra chains define natural coordinate systems: the Cartesian chain ($D=\{0\le r\le 1, -\pi\le\theta<\pi\}$3) for square pixel arrays, and the polar chain ($D=\{0\le r\le 1, -\pi\le\theta<\pi\}$4), where radial and angular quantum numbers emerge natively.

The unitary kernel $D=\{0\le r\le 1, -\pi\le\theta<\pi\}$5 maps images from the square lattice to a true discrete polar grid, exploiting Clebsch–Gordan coefficients, Kravchuk polynomials, and block-circulant structures for algorithmic efficiency. This formalism permits exact orthonormal transforms without interpolation artifacts and enables physically meaningful radial–angular decompositions in finite dimensions (Wolf et al., 2011).

3. Theoretical Properties

PSepT satisfies several foundational properties:

• Orthogonality: $D=\{0\le r\le 1, -\pi\le\theta<\pi\}$6, supporting Parseval’s theorem and energy conservation.
• Completeness: The radial and angular bases are separately complete in $D=\{0\le r\le 1, -\pi\le\theta<\pi\}$7, and their tensor product yields full completeness over $D=\{0\le r\le 1, -\pi\le\theta<\pi\}$8.
• Rotation covariance: Rotation by $D=\{0\le r\le 1, -\pi\le\theta<\pi\}$9 multiplies coefficients by $$F_{n,m} = \int_{0}^{2\pi} \int_{0}^{1} f(r,\theta)K_{n,m}(r,\theta)\, r\, dr\, d\theta$$0, making $$F_{n,m} = \int_{0}^{2\pi} \int_{0}^{1} f(r,\theta)K_{n,m}(r,\theta)\, r\, dr\, d\theta$$1 strictly rotation-invariant.
• Numerical conditioning: The condition number is $$F_{n,m} = \int_{0}^{2\pi} \int_{0}^{1} f(r,\theta)K_{n,m}(r,\theta)\, r\, dr\, d\theta$$2, representing an exponential improvement over the $$F_{n,m} = \int_{0}^{2\pi} \int_{0}^{1} f(r,\theta)K_{n,m}(r,\theta)\, r\, dr\, d\theta$$3 scaling of classical coupled polynomial moments.
• Unitarity: The discrete PSepT kernel $$F_{n,m} = \int_{0}^{2\pi} \int_{0}^{1} f(r,\theta)K_{n,m}(r,\theta)\, r\, dr\, d\theta$$4 satisfies $$F_{n,m} = \int_{0}^{2\pi} \int_{0}^{1} f(r,\theta)K_{n,m}(r,\theta)\, r\, dr\, d\theta$$5; the transform is exactly invertible with no information loss (Singh et al., 10 Oct 2025, Wolf et al., 2011).

4. Computational Complexity and Stability

Classical polar moment methods require joint evaluation of high-order polynomials, resulting in computational complexity between $$F_{n,m} = \int_{0}^{2\pi} \int_{0}^{1} f(r,\theta)K_{n,m}(r,\theta)\, r\, dr\, d\theta$$6 and $$F_{n,m} = \int_{0}^{2\pi} \int_{0}^{1} f(r,\theta)K_{n,m}(r,\theta)\, r\, dr\, d\theta$$7 and severe numerical instability at high order due to ill-conditioning. PSepT, by virtue of kernel separability, reduces the forward/inverse transform to $$F_{n,m} = \int_{0}^{2\pi} \int_{0}^{1} f(r,\theta)K_{n,m}(r,\theta)\, r\, dr\, d\theta$$8 total complexity—$$F_{n,m} = \int_{0}^{2\pi} \int_{0}^{1} f(r,\theta)K_{n,m}(r,\theta)\, r\, dr\, d\theta$$9 for DCT and $$f(r,\theta) = \sum_{n=0}^{N_r-1} \sum_{m=-M}^{M} F_{n,m}K_{n,m}(r,\theta)$$0 for FFT, with $$f(r,\theta) = \sum_{n=0}^{N_r-1} \sum_{m=-M}^{M} F_{n,m}K_{n,m}(r,\theta)$$1 memory usage. Its improved condition number ($$f(r,\theta) = \sum_{n=0}^{N_r-1} \sum_{m=-M}^{M} F_{n,m}K_{n,m}(r,\theta)$$2) enables stable high-order expansions and analysis (Singh et al., 10 Oct 2025).

5. Empirical Performance

Experimental results highlight PSepT’s performance advantages:

• Reconstruction stability: Maintains convergence and stability even for large feature counts, where classical moments exhibit catastrophic artifacts.
• Classification accuracy: On MNIST, rotation-invariant classification accuracy remains 92.0–92.4% across 0°–90°, with runtimes below 65 s, outperforming pseudo-Zernike and other competing methods, which require around 1500 s and display significant accuracy drops under rotation and noise.
• Noise tolerance: Under 10% Gaussian noise, accuracy degrades by only 1.6% (from 98.4% to 96.8%), while Zernike moments degrade by ~19%.
• Generalization: On CIFAR-10, test accuracy of 46.15% in under 520 s exceeds both separable and non-separable techniques by a factor of 2–3 in runtime.
• Rotation robustness in medical images: PneumoniaMNIST classification accuracy drop is capped at 3.3% under rotation, contrasting with 15%+ for competing approaches (Singh et al., 10 Oct 2025).

6. Applications, Limitations, and Extensions

Applications

PSepT is suited for:

• Rotation-robust feature extraction in image classification (e.g., handwritten digits, medical imaging, remote sensing).
• Real-time or large-scale systems where computational efficiency and numerical stability are critical.
• Tasks requiring exact radial–angular decompositions, such as radial-angular filtering and modal analysis.

Limitations

• The separable kernel cannot capture complex radial-angular coupling inherent in highly textured natural scenes; a performance plateau appears at high spatial resolution.
• Interpolation artifacts may occur at cardinal orientations due to the resampling between Cartesian and polar grids.

Prospective Extensions

• Adaptive or warped DCT bases could better represent domains where radial-angular coupling is important.
• Incorporating multi-resolution or wavelet bases would facilitate localized texture analysis.
• Extension to 3D spherical settings could generalize PSepT to volumetric data (Singh et al., 10 Oct 2025).

7. Comparison with Other Polar and Cartesian Transforms

Unlike the discrete Fourier transform (DFT) on the Cartesian grid, which does not provide orthogonality or separation in radius and angle on the disk, PSepT’s kernel yields a true orthonormal, unitarily invertible mapping onto the circular (polar) domain. This avoids the need for interpolation and introduces natural coordinates for radial and angular processing. The algebraic perspective provided by the finite $$f(r,\theta) = \sum_{n=0}^{N_r-1} \sum_{m=-M}^{M} F_{n,m}K_{n,m}(r,\theta)$$3 model and the block-circulant structure of the kernel supports not only efficient computation but also the direct importation of additional symmetries, such as anisotropic FTs and gyrations, into the polar domain (Wolf et al., 2011).