Geometric Rate–Distortion Alignment: A Theoretical Framework for Hyperbolic Foveated Immersive Video Transmission

Retinal-resolution Extended Reality (XR) at 60 pixels-per-degree (PPD) requires raw bitrates of approximately 157 Gbps, far exceeding satellite in-flight entertainment (IFE) per-user bandwidth budgets of 100 kbps–2 Mbps. Existing standards such as MPEG OMAF rely on viewport-adaptive streaming, which fails under high-latency conditions (>200 ms), typical of satellite communication. This work introduces Geometric Rate–Distortion Alignment (GRDA), a foundational principle stating that rate–distortion efficiency is maximized when spatial information density is geometrically aligned with the human visual system’s perceptual weighting function. GRDA is instantiated via Hyperbolic Foveated Warping (HFW), a smooth, time-invariant transform whose Jacobian matches retinal ganglion cell density. We prove two key theorems: (i) Temporal Consistency, demonstrating preservation of motion vectors under fixed warping, and (ii) LUT Decode, showing that inverse warping can be computed in O(1) time using a precomputed lookup table. Preliminary results indicate up to 2.36× foveal compression-perception gains, with projected 35–45% BD-rate savings. GRDA establishes a new signal-geometry layer in the video compression stack, enabling efficient immersive streaming in high-latency environments where existing approaches fail.