Efficient Irradiance Normal Mapping

Ralf Habel, Michael Wimmer

Institute of Computer Graphics and Algorithms

Vienna University of Technology

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Motivation

Combining Light Mapping and Normal MappingAlso know as:

Radiosity Normal Mapping

Directional Light Mapping

Spherical Harmonics Light Mapping

Popular in GamesHalf-Life 2, Halo 3 …

Cheap and good looking:Normal maps can be reused

Per vertex/per texel light map pipeline

Fast and trivial evaluation

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Motivation

Light mapped

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Motivation

Irradiance normal mapped

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Motivation

Irradiance normal mapped no albedo

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Introduction

Goal: Represent irradiance on all surfaces for all possible directions (S x Ω)

Allows illumination to be stored sparsely similar to light mapping

Local variation is transported by normal maps

Representation:

Environment maps (piecewise linear)

Basis function sets (Spherical Harmonics)

Evaluation: Look up/calculate irradiance value in normal direction

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Irradiance Environment Maps

Ramamoorthi et al. 2001: Spherical Harmonics up to the quadratic band (RGB: 27 coefficients) is enough for an accurate representation (avg. error < 3%).

9 RGB textures containing SH coefficients

Irradiance over all directions is a low frequency signal

Can we do better?Only hemispherical signal (Ω+) needed on opaque surfaces

Other basis functions than Spherical Harmonics?

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Hemispherical bases

Set of functions defined over the hemisphere (Ω+)

Desired attributes for irradiance:No discontinuities for smooth interpolation

Orthonormality: simplifies projections and other calculations (just like in Euclidian space)

Band structure for LOD/increasing accuracy (like Spherical Harmonics)

Not important:Locality

High-frequency behavior

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Half-Life 2 Basis

Consists of 3 orthonormal cosine lobes (linear SRBFs)

Orthonormal over Ω+

Equivalent toDirectional occlusion (one general cosine lobe)

Linear Spherical Harmonics band normed on Ω+

All require 3 coefficients and arelinear

No quadratic terms

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Hemispherical bases

General orthonormal hemispherical bases:Hemispherical Harmonics [Gautron et al. 04]

Makhotkin Basis [Makhotkin 96] All basis functions are 0 or constant on border

of Ω+ due to generation through shifting Non-polynomial

Zernike Basis [Koenderink 96] Different band structure:

1,2,3..instead of 1,3,5.. Non-polynomial

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Creating Directional Irradiance

We need irradiance on all surface points in all Ω+ directions:

Convolution with diffuse kernel far too expensive in Cartesian coordinates

Tens of millions of convolutions

Instead: Spherical Harmonics as an intermediate basis [Ramamoorthi 01, Basri and Jacobs 00]

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Creating Directional Irradiance

Create radiance estimate in precomputationFrom photon mapping, path tracing, shadow mapping…

In tangent space (for tangent space normal maps)

Expand radiance into Spherical Harmonics by integrating against SH basis functions:

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Creating Directional Irradiance

Perform diffuse convolution directly in SH to getUsing Funk-Hecke Theorem, diffuse convolution is carried out by scaling SH coefficients in each band l with al :

a0 = 1, a1 = 2/3, a2 = ¼, a3 = 0, a4 = -1/24

There is never a cubic contribution in an SH irradiance signal

All l >=4 are very small This is why SH up to the quadratic band is so

efficient for irradiance!

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H-Basis

We would like something similar to SH on Ω+

Polynomial As fast as SH to evaluate

Same interpolation behavior

Orthonormal on Ω+

Targeted for irradiance representation

Take a close look at SH functions and polynomial Hilbert space to derive basis functions

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H-Basis

SH functions that are symmetric to the z-axis are orthogonal on the hemisphere as well:

Y00,Y1

-1,Y11,Y2

-2,Y22

Renormed to Ω+

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H-Basis

SH functions that are symmetric to the z-axis are orthogonal on the hemisphere as well:

Y00,Y1

-1,Y11,Y2

-2,Y22

Renormed to Ω+

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H-Basis

Apply shifting to Y10 (cos θ = 2 cos θ -1)

Similar to Hemispherical Harmonics/ Makhotkin basis

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H-Basis

Results in Ω+ orthonormal polynomial basis with 1 constant, 3 linear and 2 quadratic basis functions

There is a mathematical rigorous derivation!

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H-Basis

Band structure allows to use only the constant+linear functions (H4) or all six (H6) similar to SH

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SH to H-Basis

Directional irradiance signals are calculated in SH

Project SH coefficient vector into H-Basis with matrix multiplication:

Sparse due to closeness to SH

Both bases are polynomialNo loss due to change in used function space

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Bases Comparison

Visual/perceptual comparison of all basesReplace H-Basis with any other

In “very bad case” lighting situation All basis functions are contributing

SH comparison is least-square hemispherically projected [Sloan 03]

Makes optimal use of SH on Ω+

Shown with increasing number of coefficientsOnly few are shown

See paper for all of them

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Bases Comparison: 3 Coefficients

Ground truth

Half-Life 2 (not how the game evaluates)

Zernike 2 bands

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Bases Comparison: 4 Coefficients

Ground truth

H4

SH 2 bands (Ω+ projected)Makhotkin 2 bands(Artefacts at border)

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Bases Comparison: 6/9 Coefficients

Ground truth

H6

(6 coefficients)SH 3 bands(9 coefficients)

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Bases Comparison

Integrated Mean Square Error

averaged over 10 000 random irradiance signals

6 coefficients is enough for a numerically accurate representation

What about difference between H4 and H6?

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H-Basis Comparison

H4 - 4 coefficients

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H-Basis Comparison

H6 - 6 coefficients

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Conclusion

H-Basis is very efficient and very simple solution for hemispherical irradiance signals

4 coeffs. for perceptually accurate representationProbably sufficient for almost all practical cases

6 coeffs. for numerically accurate representationSome lighting situations may benefit from 6 coeffs.

Orthonormality :Shader LOD (functions are delocalized)

Easy expansion of other low frequency signals

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Future Work

There is a general mathematical description and derivation similar to Spherical Harmonics

H-Basis is a special caseEfficient generating procedures

Clarify correlations to SH

Other hemispherical signalsVisibility?

BRDFs?

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Thanks foryour attention