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

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.

As AI tools make it easier to generate realistic 360-degree panoramic images, it becomes increasingly important to track where these images come from and whether they have been reused without permission. Digital watermarks can help, but panoramic images create a special challenge: when a viewer turns their head or changes the viewing direction, the image is effectively rotated on a sphere, and many existing watermarking methods can fail. We address this problem by designing a watermarking method that respects the spherical geometry of panoramic images. Instead of treating a panorama as a flat picture, our method represents it as a signal on a sphere and hides the watermark in patterns that can still be recovered after arbitrary 3D rotations. The key idea is to use mathematical relationships among spherical frequency components that remain unchanged when the panorama is rotated. This allows the watermark to be extracted reliably without needing to undo the rotation or train on many rotated examples. Our experiments show that the method maintains high image quality while achieving strong robustness to 3D rotations and common image distortions. This can support more trustworthy provenance tracking for immersive media, virtual reality, and AI-generated panoramic content.