Rotation-Invariant Spherical Watermarking via Third-Order SO(3) Representation Coupling

2026-05-26

AI Summary

This work addresses the challenge of achieving robust watermark embedding and extraction in panoramic images under arbitrary 3D rotations, a task for which existing planar-based methods lack theoretical guarantees. The authors model panoramic images as spherical signals and leverage representation theory of the rotation group SO(3) to construct the first strictly rotation-invariant third-order spherical bispectral descriptor. This is achieved by coupling irreducible representations via tensor products and projecting onto the trivial representation, thereby preserving both phase and directional information while enabling high-capacity watermarking. Theoretical analysis establishes the SO(3) invariance of the proposed mechanism, and experiments demonstrate near-perfect robustness against continuous arbitrary rotations while maintaining high visual fidelity.

Abstract

Reliable watermarking of panoramic imagery is fundamentally challenged by arbitrary 3D rotations. As panoramas are defined on the sphere, they naturally transform under the action of $SO(3)$, rendering conventional planar representations and augmentation-based robustness strategies inadequate and devoid of theoretical guarantees. To address this, we formulate panoramas as spherical signals and leverage $SO(3)$ representation theory to derive provably rotation-invariant descriptors. While spherical harmonic coefficients transform equivariantly under rotations, the natural invariant constructions are typically limited to zeroth-order statistics which eliminate directional information and severely constrain embedding capacity. In this work, we introduce a principled third-order invariant construction by coupling higher-order $SO(3)$ irreducible representations via tensor products and projecting onto the trivial representation. This yields a spherical invariant bispectrum that preserves phase information while remaining strictly rotation-invariant. Leveraging this property, we embed watermarks into higher-order spherical harmonic coefficients and recover them from invariant bispectral scalars, enabling reliable extraction under arbitrary 3D rotations. We provide a theoretical proof of $SO(3)$ invariance for it and demonstrate experimentally its near-perfect robustness to continuous rotations while maintaining high visual fidelity.

Problem

• rotation-invariant
• spherical watermarking
• SO(3)
• panoramic imagery
• 3D rotations

Innovation

• rotation-invariant watermarking
• spherical harmonics
• SO(3) representation theory
• bispectrum
• panoramic imagery