Euler-Mamba: Learning Resolution-Invariant State Evolution on Polar Manifolds for Asymmetric Pansharpening

The goal of asymmetric pansharpening is to synthesize a high-resolution multispectral (HR-MS) image by integrating panchromatic (PAN) and low-resolution multispectral (LR-MS) images. While State Space Models (SSMs), particularly Mamba, offer a promising paradigm for global dependency modeling, their standard selective scan mechanisms operate in discrete Cartesian grids, which are ill-suited for distinguishing between structural and spectral features in complex-valued frequency representations. To resolve this, we propose Euler-Mamba, a physics-inspired framework that reformulates the selective scan as a continuous state evolution on polar manifolds. We develop an Eulerian Selective Scan (ESS) mechanism that treats the fusion task as a trajectory of frequency growth. By performing radial scans within the polar domain, ESS governs complex-valued state transitions through the decoupled architecture: the Explicit Phase Rotation (EPR) block facilitates explicit phase rotations for geometric alignment, while the Implicit Magnitude Mapping (IMM) block performs magnitude scaling for spectral calibration. This polar-decoupled evolution allows the model to capture the continuous transition from global spectral backgrounds to fine structural details with linear complexity. Consequently, by modeling the fusion as a continuous functional evolution flow, Euler-Mamba achieves resolution-invariant performance and zero-shot generalization across varying spatial scales, bypassing the computational complexity inherent in traditional discrete-integration frameworks. Extensive experiments demonstrate that Euler-Mamba establishes a new Pareto frontier for asymmetric pansharpening, achieving a superior balance between physical interpretability, computational efficiency, and reconstruction quality.