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.

翻译：全景影像的可靠水印技术面临任意三维旋转的根本性挑战。由于全景图定义在球面上，其变换遵循$SO(3)$群作用，这使得传统平面表示方法和基于数据增强的鲁棒性策略难以奏效，且缺乏理论保证。为解决该问题，我们将全景图建模为球面信号，并利用$SO(3)$表示理论推导出可证明的旋转不变描述符。尽管球谐系数在旋转下满足等变性，但天然的不变量构造通常局限于零阶统计量，这会消除方向信息并严重限制嵌入容量。本研究通过张量积耦合高阶$SO(3)$不可约表示并投影到平凡表示，首次提出一种原则性的三阶不变量构造方法，由此得到既保留相位信息又保持严格旋转不变性的球面不变双谱。利用该性质，我们将水印嵌入高阶球谐系数，并从不变双谱标量中恢复水印，从而在任意三维旋转下实现可靠提取。我们给出了该方法的$SO(3)$不变性理论证明，并通过实验证实其在连续旋转下近乎完美的鲁棒性，同时保持高视觉保真度。