Dense Local Operators on the Sphere

Spherical representations are ideal for wide-field-of-view (FoV) perception because they describe local visual geometry independently of the viewing direction. A spherical local operator should respect this geometric consistency by producing consistent responses to the same upright local pattern wherever it is relocated on the sphere—a property we call *translation consistency*. Achieving this property on a 2D grid is challenging, because no global 2D parameterization can preserve the spherical metric everywhere. On an equirectangular projection (ERP) grid, a common representation for spherical data, the same physical neighborhood has a latitude-dependent grid footprint. Prior spherical local operators recover this neighborhood through location-dependent sampling, but the resulting sparse memory access patterns limit GPU utilization. We propose **D**ense **L**ocal Operator on the **S**phere (DLS), a spherical local attention mechanism realizing *fully dense*, *approximately translation-consistent* operation on ERP grids. DLS combines distance-corrected longitude positional encoding (PE) with linear attention using separate row and column key–value (KV) aggregation. This separation cancels the cross-term involving global longitude, allowing the global PE to encode local spherical geometry therefore enabling KV-first aggregation without explicitly constructing local KV fields through sparse operations. The formulation yields approximately consistent responses to the same upright spherical patch at different locations, using only dense operations. On the panoramic segmentation task, the DLS-based network achieves a 4.9× speedup while improving mIoU by 2.9 points over the prior sparse-sampling-based network.