Spherical harmonic Gradients for Mid-Range Illumination

Spherical harmonics are often used for compact description of incident radiance in low-frequency but distant lighting environments. For interaction with nearby emitters, computing the incident radiance at the center of an object only is not sufficient. Previous techniques then require expensive sampling of the incident radiance field at many points distributed over the object. Our technique alleviates this costly requirement using a first-order Taylor expansion of the spherical-harmonic lighting coefficients around a point. We propose an interpolation scheme based on these gradients requiring far fewer samples (one is often sufficient). We show that the gradient of the incident-radiance spherical harmonics can be computed for little additional cost compared to the coefficients alone. We introduce a semi-analytical formula to calculate this gradient at run-time and describe how a simple vertex shader can interpolate the shading. The interpolated representation of the incident radiance can be used with any low-frequency light-transfer technique.