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**Fast, Arbitrary BRDF Shading for Low-Frequency Lighting Using Spherical Harmonics**

Jan Kautz, MPI Informatik Peter-Pike Sloan, Microsoft Research John Snyder, Microsoft Research

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**Motivation – BRDF vs. Light Complexity**

Lighting ? area lights point lights Initially HW accelerated rendering was only able to render objects with point light sources and the Phong model. In recent years, research tried on the one hand to allow for arbitrary BRDFs with point lights and on the other hand limited BRDFs (Phong again) but with area lights (often given as an environment map). Our work tries to fill in the last gap, i.e. arbitrary BRDFs and area lights BRDF Complexity Phong + diffuse arbitrary aniso. BRDFs

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**Motivation – What we want**

Illuminate objects with environment maps Use arbitrary BRDFs Change lighting on-the-fly Possibly include self-shadowing and interreflections At real-time rates Phong Anisotropic

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**Related Work – Interactive Techniques**

Lighting DiffSH RefSpace high-frequency area lighting PhoDiff FreqSpace ApproxEnv HomEnv low-frequency area lighting PRT Our Technique ArbBRDF point lights BRDF Complexity diffuse Phong isotropic anisotropic Our Technique Phong/Diffuse Prefiltered Environment Maps [Miller84] [Greene86] [Heidrich99] Precomputed Radiance Transfer [Sloan02] Frequency Space Environment Mapping [Ramamoorthi02] BRDF Approximation for Environment Maps [Kautz99] Reflection Space Rendering [Cabral99] Diffuse Environment Maps using Spherical Harmonics [Ramamoorthi01] Homomorphic Factorization of Environment Maps [Latta02] Arbitrary BRDFs with Point Lights [Kautz99] [McCool01]

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**Related Work Previous use of Spherical Harmonics**

[Cabral87] Bidirectional Reflection Functions from Surface Bump Maps [Westin92] Predicting Reflectance Functions from Complex Surfaces

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**Background – Spherical Harmonics**

Orthonormal basis over the sphere Analogous to Fourier transform over 1D circle Important properties: Rotational invariance no aliasing artifacts Projection: Integration: Rotation: linear xform on coefficients

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**Background – Spherical Harmonics**

Basis functions (examples) i = 1 i = 2 i = 3 i = 4 i = 8 i = 12 i = 15 i = 19

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**Background – Spherical Harmonics**

Example: projection of environment n=4 n=9 n=25 n=262 original

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**Environment Mapping + Spherical Harmonics**

Rendering Equation (no shadows): Rewrite with Project Lighting and BRDF Mention cosine explicitly as important Rewrite: mention tabulate light function: into SH BRDF:

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**Evaluating the Integral**

The integral becomes But BRDF defined in local frame Rotate lighting (or BRDF) to match:

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**Preprocessing – BRDF Texture**

Project BRDF into SH: Put coefficients in texture map Use parabolic parameterization for … Project the BRDF into Spherical Harmonics for every (local) view direction v individually and tabulate the results in a texture map. One texture map here shows the coefficients for some basis function i, but for all view direction v. i=1 i=3 i=4 i=5 i=6 i=7

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**Rendering Lookup (local ) Project lighting per object * =**

… Lookup (local ) Project lighting per object * = Rotate lighting (to local) = Compute integral per pixel/vertex

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**Anisotropic brushed in X Anisotropic brushed in Y**

Examples Phong Anisotropic brushed in X Anisotropic brushed in Y

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**Rendering – Fixed Light**

Lookup (local ) Project lighting … ONCE Rotate lighting (local) * = Compute integral =

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**Rendering – Fixed View Lookup (local ) Project lighting**

… Rotate BRDF (to global) * = ONCE Compute integral =

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**Example Bird model 48K vert. Measured Vinyl FPS: 6.04 free light/view**

28.4 fixed light 128 fixed view

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**Precomputed Radiance Transfer**

[SIG02] Tell cosine again Without PRT PRT: Shadows+Interrefl. SIG02: Phong only

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**Precomputed Radiance Transfer – Transfer Matrix**

Precompute how global incident lighting local incident * p1 p1 lighting p2 p2 * transfer matrices transferred radiance

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**Arbitrary BRDF with PRT**

Lookup (local ) Project lighting … per object Transfer & rotate light per pixel/vertex * * = Compute integral =

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**Example Stanford buddha 50K vert. Ashikhmin- BRDF FPS: 4.05 no xfer**

15.6 fixed light 127 fixed view

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**Example 2: PRT with different BRDFs**

Phong [SIG02] Measured Vinyl

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**Results – Different BRDFs**

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**Results – Brushed Metal-Patch**

Anisotropic AS brushed radially Anisotropic AS brushed tangentally

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**Results – Spatially Varying BRDF**

Varying Exponent Varying Anisotropy

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**Comparison of SH order vs. Glossiness**

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**Conclusions Pros: Fast, arbitrary dynamic lighting**

Works for arbitrary BRDFs Combined with PRT: includes shadows and interreflections Cons: Works only for low-frequency lighting Not real-time (yet)

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Thank you! Questions? Please visit us at

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**Glossy Transfer – Rendering**

transfer matrix lighting coefficients transferred radiance exiting radiance * BRDF kernel evaluate at R convolution transfer computation lookup

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**Precomputation – Transfer Matrix**

Glossy Transfer More difficult, but works similarly Have to compute matrix instead of vector Update matrices for interreflections Neighborhood Transfer Same as for glossy, just for points not on surface

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**Precomputation – Diffuse Transfer**

Visually: . Basis 16 Basis 17 Basis 18 illuminate result

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**Rendering Project lighting into SH Per-vertex:**

Project into local tangent frame Lookup : Rotate lighting: Compute dot-product: … * = =

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**Results Glossy object, 50K mesh**

No Shadows/Inter Shadows Shadows+Inter Glossy object, 50K mesh Runs at 3.6/16/125fps on 2.2Ghz P4, ATI Radeon 8500

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**Dynamic Lighting Sample incident lighting on-the-fly Results**

Precompute textures for SH basis functions in cube map parameterization Render 6 cube map faces around p Read them back Projection: simple dot-product between cube maps Results Low-resolution cube maps sufficient: 6x16x16 Average error: 0.2%, worst-case: 0.5% Takes 1.16 ms on P3-933Mhz, ATI 8500

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**Introduction – Light Integration**

Integrate over all incoming light Emitter 1 Emitter 2 Diffuse: * cos<n, s> Glossy: * f(v, s) * cos<n, s> Receiver

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**Background – Spherical Harmonics**

Projection: Reconstruction: Integration: Convolution/Rotation: Simple and efficient formulas

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**Overview Previous Work Background Fast Environment Mapping with SH**

Spherical Harmonics Fast Environment Mapping with SH Theory Rendering Combine with Precomputed Radiance Transfer Results Conclusions

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**Comparison – Size Light vs. SH Order**

0° 20° 40° n= n= n= n= n= n= n= RT linear quadratic cubic quartic quintic windowed

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**Introduction – Filtered Environment Maps**

Environment map over sphere Source Target BRDF maps to: Shift-variant & radially symmetric kernel 2D filtered environment map But: General anisotropic BRDF 5D filtered environment map apply filter Filter kernel

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**Precomputed Radiance Transfer**

Precompute how global incident radiance is transferred to local incident radiance at points : For self-shadowing and interreflections Transfer is represented as a Transfer Matrix global local transfer

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**Introduction – Environment Maps**

Approximate incident light field with a single sample at object’s center Assumptions: Environment at infinity No self-shadowing or interreflections (concave object)

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**Motivation – BRDF Complexity**

HW rendering: Increased BRDF complexity But only for point light sources! [Heidrich98] [Kautz00]

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**Results – Head in Various Environments**

Max Planck head 50K vertices Ashikhmin- BRDF FPS: 5.24 no xfer 4.18 xfer 25.3 fixed light 130 fixed view

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**Results – Different BRDFs**

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**Motivation – Environment Maps**

Store light incident at single position Prefilter to get glossy reflections: Only limited Phong-like BRDFs Complex BRDFs require up to 5D table Dynamic lighting difficult, no self-shadowing [Miller & Hoffmann84] filter

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