Zernike like functions on spherical cap: principle and applications in optical surface fitting and graphics rendering

Hemispherical harmonic illumination and reflectance angular spectrum

Light scattering plays an important role in physics, with wide applications in science and engineering. However, accurate and effective modeling of scattering remains a great challenge. In this study, we exploited the rendering equation using hemispherical harmonics to demonstrate an angular frequency representation that directly depicts scattering in a two-dimensional spectrum, free from any underlying assumptions. This representation offers a compact and intuitive characterization of mirror reflection, isotropic scattering, and anisotropic emission. The robust support of theoretical proofs and data-driven experimental results establishes the broad applicability of our computational model in conducting scattering analyses across diffuse, specular, and glossy materials. With the capability to characterize the scattering in angular frequency domain, we expect our proposed model to emerge as an essential tool in various domains, including surface feature recognition, reflectance data compression, and computer rendering.