Spherical Hermite Maps

TL;DR

Spherical Hermite Maps is presented, a derivative-augmented LUT representation that resolves the quality-performance trade-off between bilinear LUT sampling and bicubic filtering, and reconstructs smooth derivatives as a byproduct of value reconstruction, making surface normals effectively free.

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.

Figures (8)

• Figure 1: Interpolation at LUT face resolution $N=10$. From left to right: ground truth (direct SH evaluation), nearest-neighbor lookup, bilinear reconstruction with finite-difference normals, 16-tap bicubic reconstruction with finite-difference normals, and Spherical Hermite Maps with analytic normals. Insets zoom a challenging silhouette/highlight region: bilinear exhibits faceting, finite differences introduce gradient noise, and bicubic retains residual error; Hermite interpolation remains faithful to the ground truth at $4\times$ lower sampling cost than 16-tap bicubic.
• Figure 2: Image-based reconstruction accuracy versus LUT resolution for a representative $L=8$ spherical-harmonic glyph. Plot shows PSNR (higher is better) of LUT-based rendering compared to direct SH evaluation. Hermite consistently improves over bilinear across resolutions and exceeds 16-tap bicubic quality while using only a $2\times2$ (four-sample) footprint.
• Figure 3: Procedural planet terrain comparison at $N=48$. Shaded surface. From left to right: ground truth, bilinear (4 taps), bicubic (16 taps), and Hermite (4 taps). Hermite achieves the lowest normal error (7.28°) while outperforming bicubic's sample efficiency advantage over the 16-tap baseline.
• Figure 4: Multi-glyph shading comparison at $N=10$. From left to right: ground truth, bilinear + finite-difference normals, bicubic + finite-difference normals, and Hermite with analytic normals. Insets highlight visible artifacts caused by discontinuous or noisy gradients (silhouette waviness and unstable specular structure); Hermite produces smooth silhouettes and stable highlights from the same four-sample reconstruction.
• Figure 5: Equal-storage comparison (shaded SH glyph). At matched scalar storage per face (32$^2\times$1ch vs 16$^2\times$4ch), Hermite achieves high quality results using only a 2$\times$2 footprint and generates a slightly higher image PSNR than 16-tap bicubic (+1.2 dB), while also providing analytic derivatives for shading. Fast bicubic shows visible banding under specular lighting due to finite-difference normals, showing that 4 fetches alone are insufficient without analytic gradients.