Stable Geometric Capacity Reallocation: A Unified Framework for Foveated Immersive Video Compression via Bounded Diffeomorphic Warping

Retinal-resolution Extended Reality (XR) video demands raw bitrates approaching 157 Gbps, far exceeding the 100 kbps–2 Mbps per-user budgets available in satellite and high-latency networks. Prior work on Geometric Rate–Distortion Alignment (GRDA) introduced Hyperbolic Foveated Warping (HFW) as a signal-geometry layer to redistribute spatial information density according to human retinal acuity. However, empirical and analytical evaluation reveals a structural failure mode, termed Magnification Debt: the coupling of foveal scaling and peripheral compression through a single parameter induces an irreducible geometric error at the gaze center, leading to a degenerate rate–distortion regime in which foveal quality becomes invariant to bitrate. This work introduces Stable Geometric Capacity Reallocation (SGCR), a unified theoretical framework that resolves this limitation through five formal constraints: bounded discrete capacity, foveal identity preservation, digital diffeomorphism, C² smoothness, and adaptive anti-aliasing. SGCR is instantiated via the Piecewise Smoothstep Radial Transform (PSRT), a three-zone bounded diffeomorphic mapping composed of an identity core, a Hermite spline transition, and a hyperbolic periphery aligned with the Watson retinal ganglion cell density model. We prove that PSRT eliminates Magnification Debt by construction, guaranteeing exact foveal signal preservation (|J(0)| = 1), globally bounded Jacobians, discrete invertibility, and O(1) GPU LUT-based inverse decoding within real-time constraints. Experimental results confirm that, unlike HFW (which systematically produces C/P < 1 and degenerate R–D behavior), PSRT achieves stable foveated compression with central-to-peripheral quality ratios consistently greater than one and near-lossless foveal reconstruction. Phase 1 synthetic experiments demonstrate −68% to −83% foveal BD-rate reductions relative to ERP-HEVC at matched foveal quality. Phase 2 validation on real 4K video confirms consistent bitrate savings in the range of 15–27%, with negligible foveal quality deviation and statistically preserved perceptual fidelity. Additional causality and attribution experiments show that foveal identity (C2) is the necessary condition for perceptual correctness, while adaptive anti-aliasing (C5) is responsible for the transition from bitrate overhead to net compression gain. SGCR establishes a principled shift from bitrate optimization to geometric capacity reallocation, defining the signal-geometry layer as a fundamental and necessary component for perceptually aligned immersive video compression systems, particularly under high-latency and bandwidth-constrained conditions.