Spherical Hermite Maps

2026-02-23

AI Summary

Existing spherical function sampling methods struggle to balance accuracy, efficiency, and normal quality in real-time rendering: bilinear interpolation is fast but exhibits noticeable blockiness, while bicubic filtering achieves high accuracy at the cost of 16 samples per evaluation and relies on noisy finite differences for normals. This work proposes Spherical Hermite Mapping—a derivative-augmented cube map representation that stores both function values and scaled partial derivatives per texel. It enables bicubic Hermite reconstruction with only four texture samples and yields analytical gradients simultaneously. By introducing Hermite interpolation to spherical function representation for the first time, the method attains conventional bicubic accuracy at one-quarter the sampling cost (improving PSNR by 8–41 dB) and reduces normal angular error by 9–13%, significantly enhancing highlight stability. The approach has been successfully applied to spherical harmonic visualization, depth map LOD, and procedural planet rendering.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Computer Vision: Other Foundations of Computer Vision
• Machine Learning: Learning with Manifolds

Application Category

• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Representation, semantic annotation, enhancement, enrichments, access and/or integration of a variety of data on the Web

Abstract

Spherical functions appear throughout computer graphics, from spherical harmonic lighting and precomputed radiance transfer to neural radiance fields and procedural planet rendering. Efficient evaluation is critical for real-time applications, yet existing approaches face a quality-performance trade-off: bilinear LUT sampling is fast but produces faceting, while bicubic filtering requires 16 texture samples. Most implementations use finite differences for normals, requiring extra samples and introducing noise. This paper presents Spherical Hermite Maps, a derivative-augmented LUT representation that resolves this trade-off. By storing function values alongside scaled partial derivatives at each texel of a padded cubemap, bicubic-Hermite reconstruction is enabled from only four texture samples (a 2x2 footprint) while providing continuous gradients from the same samples. The key insight is that Hermite interpolation reconstructs smooth derivatives as a byproduct of value reconstruction, making surface normals effectively free. In controlled experiments, Spherical Hermite Maps improve PSNR by 8-41 dB over bilinear interpolation and match 16-tap bicubic quality at one-quarter the cost. Analytic normals reduce mean angular error by 9-13% on complex surfaces while yielding stable specular highlights. Three applications demonstrate versatility: spherical harmonic glyph visualization, radial depth-map impostors for mesh level-of-detail, and procedural planet/asteroid rendering with spherical heightfields.

Problem

• spherical functions
• quality-performance trade-off
• texture sampling
• surface normals
• real-time rendering

Innovation

• Spherical Hermite Maps
• cubemap interpolation
• analytic normals
• derivative-augmented LUT
• bicubic-Hermite reconstruction