Polar Separable Transform for Efficient Orthogonal Rotation-Invariant Image Representation

TL;DR

PSepT introduces a polar separable transform that factorizes kernels into independent radial and angular components using a DCT-based radial basis and Fourier angular basis. This tensor-product design achieves exact discrete orthogonality, energy conservation, and rotation covariance, while reducing computational complexity to $\mathcal{O}(N_r N_\theta \log N)$ and conditioning to $O(\sqrt{N})$, enabling stable high-order moment analysis. Empirical results show superior numerical stability and efficiency across MNIST, PneumoniaMNIST, and CIFAR-10, with robust rotation invariance and strong reconstruction fidelity, particularly outperforming classical Zernike/ Pseudo-Zernike methods at high orders. The approach is especially advantageous for rotation-invariant recognition and medical imaging, offering substantial speedups and reliable performance under noise and varying orientations, albeit with some limits in capturing radial-angular couplings in very textured natural images.

Abstract

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.

Figures (10)

• Figure 1: Separable components of PSepT kernels. (a) shows radial basis functions $\Phi_{n}(r)$ and (b) shows angular basis functions $\Psi_{m}(\theta)$ for orders $n,m=0,1,2,3$.
• Figure 2: Comprehensive kernel gallery showing PSepT kernels $K_{n,m}$ for $n,m=0\text{--}4$ . Rows correspond to radial complexity ($n$), and columns correspond to angular complexity ($m$).
• Figure 3: Three-dimensional visualization of kernel $K_{3,2}$ over the unit disk, showing three radial oscillations modulated by two angular cycles.
• Figure 4: Reconstruction of images using different moment methods. The number of features used for reconstruction increases from left to right, ranging from 50 to 6000 features. The range of $C$ values used for all methods is from 0 to 150, with increments of 10.
• Figure 5: Sample images from MNIST dataset

Theorems & Definitions (1)

• proof