Eurographics Symposium on Rendering (2004)H. W. Jensen, A. Keller (Editors)Spherical Harmonic Gradients for Mid-Range IlluminationThomas Annen1Jan Kautz2Frédo Durand2Hans-Peter Seidel11MPI Informatik2Massachusetts Institute of Technology – CSAILSaarbrücken, Germany Cambridge, USAFigure 1: From left to right. A hand model shaded using only one sample at the center. The middle image shows the same model shadedusing one sample and the analytical gradient. The last image shows a reference rendering where the incident radiance field is sampledper-vertex.AbstractSpherical harmonics are often used for compact description of incident radiance in low-frequency but distantlighting environments. For interaction with nearby emitters, computing the incident radiance at the center of anobject only is not sufficient. Previous techniques then require expensive sampling of the incident radiance field atmany points distributed over the object. Our technique alleviates this costly requirement using a first-order Taylorexpansion of the spherical-harmonic lighting coefficients around a point. We propose an interpolation schemebased on these gradients requiring far fewer samples (one is often sufficient). We show that the gradient of theincident-radiance spherical harmonics can be computed for little additional cost compared to the coefficientsalone. We introduce a semi-analytical formula to calculate this gradient at run-time and describe how a simplevertex shader can interpolate the shading. The interpolated representation of the incident radiance can be usedwith any low-frequency light-transfer technique.Categories and Subject Descriptors (according to ACM CCS): I.3.3 [Computer Graphics]: Color, Shading, Shadow-ing and Texture1. IntroductionIn recent years, several methods have been proposed thatpermit the usage of global incident lighting in real-time ren-dering [SKS02, NRH03, SHHS03]. These approaches rep-resent the incident radiance in spherical harmonics (SH).They, however, assume distant lighting. As a result, they areincapable of rendering scenes with mid-range lighting ef-fects without visual error (Figure 1).To alleviate this problem, Sloan et al. [SKS02] sample theincident light field at multiple points over the object. Whilethis was shown to be a possible solution, the high compu-tational cost of multiple sampling leaves room for improve-ment.In this paper, we propose to compute a first-order Tay-lor expansion of the spherical harmonic coefficients arounda sampling point. We show how the gradient of the inci-dent radiance (represented in SH) can be computed for lit-tle additional cost compared to the coefficients alone. Asemi-analytical formula is introduced to calculate this gra-dient at run-time. The incident radiance can now be extrap-olated to different positions around the original sample lo-c The Eurographics Association 2004.T. Annen, J. Kautz, F. Durand, and H.-P. Seidel / Spherical Harmonic Gradientscation. In case of multiple samples, the interpolation qualityis greatly improved, thus requiring less samples. Extrapola-tion/interpolation can be easily performed in a vertex shaderon the GPU. The extrapolated/interpolated incident radiancecan then used together with any radiance transfer technique,e.g. [RH01, SKS02].2. Previous WorkOur work uses the same framework as the recent pre-computed radiance transfer technique [SKS02]. This ap-proach permits the illumination of objects with low-frequency incident lighting represented in spherical har-monics [Edm60]. The object can either be diffuse [SKS02]or glossy [KSS02, SHHS03, LK03]. Rendering can be per-formed in real-time, but requires precomputing the transferfor self-shadowing and other global illumination effects.Precomputed radiance transfer is limited to distant illumi-nation, unless multiple incident radiance samples are takenand interpolated [SKS02]. We improve on this by computingthe gradient of the spherical harmonics coefficients around asample point. This enables extrapolation of the incident ra-diance to other points in space, which in turn can be used toimprove interpolation of multiple samples.Our technique is similar in spirit to the irradiance gradi-ents for ray-tracing by Ward and Heckbert [WH92]. Theypropose to compute gradients of the view-independent irra-diance at various sample points in order to improve interpo-lation. This involves a translational gradient (for the changeof position) as well as rotational gradient (for the changeof relative surface orientation). In contrast to their work, wechoose to compute a gradient for the whole sphere of in-cident radiance (given in SH), independent of any incidentsurface orientation. Therefore we only need a translationalgradient, but not the rotational gradient. However, our gradi-ent is higher-dimensional, as we encode the directional radi-ance information through the vector of lighting coefficients.Essentially, we trade dependencies on the receiver orienta-tion for a more comprehensive directional treatment of inci-dent radiance. Furthermore, we focus on real-time rendering,whereas their application area was offline rendering.Irradiance gradients were generalized by Arvo [Arv94],who additionally accounted for occlusions. As Ward andHeckbert, we have to decided to neglect occlusion changes.This is motivated by implementation robustness and simplic-ity goals, and is justified by the use of low-frequency inci-dent illumination, which is not very susceptible to visibilitychanges.3. Review: Shading with Spherical HarmonicsComputing exit radiance at a diffuse surface is usually com-puted by the following integral:Lp=ZΩIp(ω) ·V∗p(ω) dω, (1)n(x)n(x)dA(x)dA(x)xpxsps’pdFigure 2: At a point p, we see point x in direction s := x −p.The point x has a differential solid area dA(x). When wemove from p to p0along d, the direction s changes, as wellas the angle between s and n.V∗p(ω) = Vp(ω)(np· ω), (2)where Lpis the exitant radiance at point p, Ipis the inci-dent radiance at p, Vpis the visibility function at p, V∗pis thecosine-weighted visbility, and npis the normal at p. Integra-tion is performed over all directions ω.This is an expensive integral, and it needs to be com-puted at every point p on an object. Recently, techniques[SKS02, SHHS03] were introduced to speed up the com-putation of this integral under the following assumptions:Lighting is assumed to be low-frequency, the object is static,and the illumination is distant (i.e., Ip(ω) = I(ω) is