Polar Separable Transform for Efficient Orthogonal Rotation-Invariant Image Representation
Orthogonal moment-based image representations are fundamental in computer vision, but classical methods suffer from high computational complexity and numerical instability at large orders. Zernike and pseudo-Zernike moments, for instance, require coupled radial-angular processing that precludes efficient factorization, resulting in $\mathcal{O}(n^3N^2)$ to $\mathcal{O}(n^6N^2)$ complexity and $\mathcal{O}(N^4)$ condition number scaling for the $n$th-order moments on an $N\times N$ image. We introduce \textbf{PSepT} (Polar Separable Transform), a separable orthogonal transform that overcomes the non-separability barrier in polar coordinates. PSepT achieves complete kernel factorization via tensor-product construction of Discrete Cosine Transform (DCT) radial bases and Fourier harmonic angular bases, enabling independent radial and angular processing. This separable design reduces computational complexity to $\mathcal{O}(N^2 \log N)$, memory requirements to $\mathcal{O}(N^2)$, and condition number scaling to $\mathcal{O}(\sqrt{N})$, representing exponential improvements over polynomial approaches. PSepT exhibits orthogonality, completeness, energy conservation, and rotation-covariance properties. Experimental results demonstrate better numerical stability, computational efficiency, and competitive classification performance on structured datasets, while preserving exact reconstruction. The separable framework enables high-order moment analysis previously infeasible with classical methods, opening new possibilities for robust image analysis applications.