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

arXiv: https://arxiv.org/abs/2605.26702 · HTML: https://mlsanity.com/paper/2605.26702/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 47 references that are in the map.
- First posted on arXiv 26 May 2026.

## Builds on (most cited first, 27 of 27)

- [Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds](https://mlsanity.com/paper/1802.08219/) (2018)
- [E(3)-Equivariant Graph Neural Networks for Data-Efficient and Accurate Interatomic Potentials](https://mlsanity.com/paper/2101.03164/) (2021)
- [Spherical CNNs](https://mlsanity.com/paper/1801.10130/) (2018)
- [HiDDeN: Hiding Data With Deep Networks](https://mlsanity.com/paper/1807.09937/) (2018)
- [StegaStamp: Invisible Hyperlinks in Physical Photographs](https://mlsanity.com/paper/1904.05343/) (2019)
- [Learning SO(3) Equivariant Representations with Spherical CNNs](https://mlsanity.com/paper/1711.06721/) (2017)
- [Learning Steerable Filters for Rotation Equivariant CNNs](https://mlsanity.com/paper/1711.07289/) (2017)
- [e3nn: Euclidean Neural Networks](https://mlsanity.com/paper/2207.09453/) (2022)
- [Clebsch-Gordan Nets: a Fully Fourier Space Spherical Convolutional Neural Network](https://mlsanity.com/paper/1806.09231/) (2018)
- [EditGuard: Versatile Image Watermarking for Tamper Localization and Copyright Protection](https://mlsanity.com/paper/2312.08883/) (2023)
- [Robust Watermarking Using Generative Priors Against Image Editing: From Benchmarking to Advances](https://mlsanity.com/paper/2410.18775/) (2024)
- [Distortion Agnostic Deep Watermarking](https://mlsanity.com/paper/2001.04580/) (2020)
- [DreamScene360: Unconstrained Text-to-3D Scene Generation with Panoramic Gaussian Splatting](https://mlsanity.com/paper/2404.06903/) (2024)
- [A Comprehensive Survey on World Models for Embodied AI](https://mlsanity.com/paper/2510.16732/) (2025)
- [Effective Rotation-invariant Point CNN with Spherical Harmonics kernels](https://mlsanity.com/paper/1906.11555/) (2019)
- [TrustMark: Universal Watermarking for Arbitrary Resolution Images](https://mlsanity.com/paper/2311.18297/) (2023)
- [SepMark: Deep Separable Watermarking for Unified Source Tracing and Deepfake Detection](https://mlsanity.com/paper/2305.06321/) (2023)
- [Scalable and Equivariant Spherical CNNs by Discrete-Continuous (DISCO) Convolutions](https://mlsanity.com/paper/2209.13603/) (2022)
- [Robust-Wide: Robust Watermarking against Instruction-driven Image Editing](https://mlsanity.com/paper/2402.12688/) (2024)
- [GenEx: Generating an Explorable World](https://mlsanity.com/paper/2412.09624/) (2024)
- [GuardSplat: Efficient and Robust Watermarking for 3D Gaussian Splatting](https://mlsanity.com/paper/2411.19895/) (2024)
- [PANORAMA: The Rise of Omnidirectional Vision in the Embodied AI Era](https://mlsanity.com/paper/2509.12989/) (2025)
- [QuaDreamer: Controllable Panoramic Video Generation for Quadruped Robots](https://mlsanity.com/paper/2508.02512/) (2025)
- [The Selective G-Bispectrum and its Inversion: Applications to G-Invariant Networks](https://mlsanity.com/paper/2407.07655/) (2024)
- [A General Framework for Robust G-Invariance in G-Equivariant Networks](https://mlsanity.com/paper/2310.18564/) (2023)
- [Deep Neural Watermarking for Robust Copyright Protection in 3D Point Clouds](https://mlsanity.com/paper/2510.27533/) (2025)
- [Rel-Zero: Harnessing Patch-Pair Invariance for Robust Zero-Watermarking Against AI Editing](https://mlsanity.com/paper/2603.17531/) (2026)

## Built on by (most cited first, 1 of 1)

- [Watermark Forensics for Generative Models: An Information-Theoretic Perspective](https://mlsanity.com/paper/2607.13003/) (2026)

## Abstract (arXiv)

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.