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Importance Sampling Microfacet-Based BSDFs using the Distribution of Visible Normals Eric Heitz 1 & Eugene dEon 2 1 INRIA ; CNRS ; Univ. Grenoble Alpes 2 The Jig Lab Introduction Dielectric material (e.g. water, plastic) 1 Introduction

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Importance Sampling Microfacet-Based BSDFs using the Distribution of Visible Normals

Eric Heitz1 & Eugene d’Eon2

1INRIA ; CNRS ; Univ. Grenoble Alpes 2The Jig Lab

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Introduction Dielectric material (e.g. water, plastic)

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Introduction Rough dielectric material

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Introduction Conductor material (e.g. metal)

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Introduction Rough conductor material

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Introduction Microfacet theory:

statistical model of the rough interface and its interaction with light.

→

microfacet normals

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Introduction State of the art in microfacet theory

Walter et al., Microfacet Models for Refraction through Rough Surfaces, EGSR 2007

Physical model → the equations.

BRDF

fr(ωi,ωo)= F(ωi,ωhr )G2(ωi,ωo,ωhr )D(ωhr )

4|ωi·ωg||ωo·ωg|

BTDF

ft(ωi,ωo)=

|ωi·ωht ||ωo·ωht | |ωi·ωg||ωo·ωg| n2

0 (1−F(ωi,ωht ))G2(ωi,ωo,ωht )D(ωht )

(ni(ωi·ωht )+no(ωo·ωht ))

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Importance sampling → how to solve the equations.

Where does the light path continue? Choose a direction at random.

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Introduction Available in Mitsuba (Physically Based Renderer)

Mitsuba’s documentation

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Introduction Rendering with Mitsuba (path tracing+MIS)

32 spp 128 spp A dielectric glass plate (n = 1.5) with anisotropic Beckmann roughness (αx = 0.05, αy = 0.4).

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Introduction Rendering with Mitsuba (path tracing+MIS)

512 spp 4096 spp A dielectric glass plate (n = 1.5) with anisotropic Beckmann roughness (αx = 0.05, αy = 0.4).

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Introduction What if we had perfect BSDF importance sampling?

Precomputed data (inverse CDF)

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Introduction What if we had perfect BSDF importance sampling?

Perfect importance sampling Previous with precomputed data (512 spp) (512 spp)

This is NOT what we propose to do.

But this shows that there is clearly room for improvement.

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Introduction OK, so why not just use perfect BSDF importance sampling?

Precomputed data (inverse CDF) Precomputed data (inverse CDF) Precomputed data (inverse CDF)

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Introduction Per-BSDF precomputed data?

OK for “toy-scenes” with a low number of BSDFs.

  • nly 1 BSDF in this scene

Not affordable for complex scenes with a huge number of

different BSDFs.

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Introduction Example: Textured Assets

Parametric BRDF models (roughness parameters αx,αy) Per-texel parameters GigaBytes of textures

Varying isotropic GGX Roughness map (α)

∞ number

  • f BSDFs

in this scene Courtesy of Christian Bense

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Introduction Example: Textured Assets

Parametric BRDF models (roughness parameters αx,αy) Per-texel parameters GigaBytes of textures

Varying anisotropic Beckmann Roughness map (αx,αy)

∞2 number

  • f BSDFs

in this scene Courtesy of WETA Digital

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Introduction We propose

A better BSDF importance sampling scheme with lower variance, not perfect, but that can be used in production rendering.

Fulfilled requirements

Parametric BSDF models used in production (typically Beckmann

and GGX)

Varying anisotropic roughness parameters αx,αy No assumption on the kind of integrator (bidir, MLT, etc.) Plug & Play (trivial update of an existing implementation of the

previous method)

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Introduction Our approach

We investigate the physical “meaning” of the BSDF model and use it to our advantage.

fr(ωi,ωo) = F(ωi,ωhr)G2(ωi,ωo,ωhr)D(ωhr) 4|ωi ·ωg||ωo ·ωg| ft(ωi,ωo) = |ωi ·ωht||ωo ·ωht| |ωi ·ωg||ωo ·ωg| n2

0 (1−F(ωi,ωht))G2(ωi,ωo,ωht)D(ωht)

  • ni(ωi ·ωht)+no(ωo ·ωht)

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Looks complicated. But the model behind the equations is actually very simple and intuitive.

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The Problem with the Previous Method

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The Problem with the Previous Method

How to sample outgoing directions?

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The Problem with the Previous Method

Available information: the distribution of normals.

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The Problem with the Previous Method

Use it to generate a random normal sample.

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The Problem with the Previous Method

Apply a transport operator on the normal (“reflect” or ”transmit”).

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The Problem with the Previous Method

Importance sampling is based on the distribution of normals.

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The Problem with the Previous Method

But in the physical model, the incident ray can only intersect normals that are visible.

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The Problem with the Previous Method

In the physical model, the transport operators are applied only on visible normals.

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The Problem with the Previous Method

This difference is what makes importance sampling inefficient.

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The Problem with the Previous Method

Samples are generated with the distribution of normals, but are weighted by the distribution of visible normals.

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The Problem with the Previous Method

Samples are generated with the distribution of normals, but are weighted by the distribution of visible normals.

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The Problem with the Previous Method

Samples are generated with the distribution of normals, but are weighted by the distribution of visible normals.

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The Problem with the Previous Method

Samples are generated with the distribution of normals, but are weighted by the distribution of visible normals.

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The Problem with the Previous Method

Samples are generated with the distribution of normals, but are weighted by the distribution of visible normals.

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The Problem with the Previous Method

Samples are generated with the distribution of normals, but are weighted by the distribution of visible normals.

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The Problem with the Previous Method Problem 1: sampling space wasting.

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The Problem with the Previous Method

Samples are generated with the distribution of normals, but are weighted by the distribution of visible normals.

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The Problem with the Previous Method

Samples are generated with the distribution of normals, but are weighted by the distribution of visible normals.

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The Problem with the Previous Method Problem 2: high sampling weights.

firefly artifacts →

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Our Method: Main Idea

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Our Method: Main Idea Importance sampling with the distribution of normals distribution of visible normals.

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Our Method: Main Idea

Our importance sampling method does exactly what is represented in this picture.

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Our Method: Main Idea

Only one component of the model is missing: most of the rays leave the surface...

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Our Method: Main Idea

...but other are occluded (microsurface shadowing).

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Our Method: Main Idea

Previous weights Our weights masking*shadowing*projected area shadowing

[0,∞[ [0,1] 28

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Our Method: Main Idea

Previous 512 spp (88.9s) Our 408 spp (87.1s) A dielectric glass plate (n = 1.5) with anisotropic Beckmann roughness (αx = 0.05, αy = 0.4). same rendering time

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Our Method: Main Idea

Previous 256 spp (194.5s) Our 192 spp (198.9s) A rough dielectric with isotropic GGX roughness (α = 0.05). same rendering time

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Our Method: Main Idea

Previous 32 spp (2.9s) Our 28 spp (2.8s) A rough conductor with isotropic GGX roughness (α = 0.10). same rendering time

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Our Method: Main Idea

Previous 8192 spp (31.1min) Our 1024 spp (5.2m) A rough conductor with isotropic GGX roughness (α = 0.15). 8× less samples per pixel 6× less rendering time

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Our Method: in Practice

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Our Method: in Practice The distribution of visible normals

Dωi(ωm) = G1(ωi,ωm) cosθi 〈ωi,ωm〉D(ωm,αx,αy)

D(ωm,αx,αy) distribution of normals (αx,αy) roughness parameters G1(ωi,ωm) masking function 〈ωi,ωm〉 cosine projection factor

Sampling this PDF is not simple! 34

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Our Method: in Practice

microfacet normals ⇔ microsurface slope

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Our Method: in Practice

scaling the roughness ⇔ stretching the microsurface

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Our Method: in Practice

The distribution of visible normals is stretch-invariant.

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Our Method: in Practice

If a sampling scheme is available for let’s say αx = 1 and αy = 1... ...it can be used for any other αx and αy.

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Our Method: in Practice

Example: we want to sample Dωi for α = 1

2.

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Our Method: in Practice

First, we stretch into a configuration where α = 1.

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Our Method: in Practice

Now, since α = 1 we know how to sample Dωi.

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Our Method: in Practice

Finally, we go back to the initial configuration.

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Our Method: in Practice

Implementing the sampling scheme for α = 1

↓

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Our Method: in Practice

Spherical coordinates

ωm ⇔ (θm,φm)

The PDF of θm depends on θi and φm The PDF of φm depends on θi and θm 2D PDFs difficult

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Our Method: in Practice

Spherical coordinates

ωm ⇔ (θm,φm)

The PDF of θm depends on θi and φm The PDF of φm depends on θi and θm 2D PDFs difficult Slope coordinates

ωm ⇔ (x ˜

m,y ˜ m)

The PDF of x ˜

m depends on θi

The PDF of y ˜

m depends on x ˜ m

1D PDFs simple

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Our Method: in Practice

  • 1. Main idea: sample the distribution of visible normals.
  • 2. In practice: sampling Dωi(ωm,αx,αy) is not simple

2.1 Stretch invariance: solve only for αx = 1 and αy = 1. 2.2 Slope space: sample only two 1D PDFs.

Now it’s simple

  • 3. Implementation (C++ code in our supplemental material)

3.1 Beckmann and GGX distributions: analytic sampling. 3.2 Other distributions: generic sampling with precomputed data.

  • 4. Validation: passes χ2 test in Mitsuba.

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Our Method: in Practice

  • 1. Main idea: sample the distribution of visible normals.
  • 2. In practice: sampling Dωi(ωm,αx,αy) is not simple

2.1 Stretch invariance: solve only for αx = 1 and αy = 1. 2.2 Slope space: sample only two 1D PDFs.

Now it’s simple

  • 3. Implementation (C++ code in our supplemental material)

3.1 Beckmann and GGX distributions: analytic sampling. 3.2 Other distributions: generic sampling with precomputed data.

  • 4. Validation: passes χ2 test in Mitsuba.

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Our Method: in Practice

  • 1. Main idea: sample the distribution of visible normals.
  • 2. In practice: sampling Dωi(ωm,αx,αy) is not simple

2.1 Stretch invariance: solve only for αx = 1 and αy = 1. 2.2 Slope space: sample only two 1D PDFs.

Now it’s simple

  • 3. Implementation (C++ code in our supplemental material)

3.1 Beckmann and GGX distributions: analytic sampling. 3.2 Other distributions: generic sampling with precomputed data.

  • 4. Validation: passes χ2 test in Mitsuba.

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Our Method: in Practice

  • 1. Main idea: sample the distribution of visible normals.
  • 2. In practice: sampling Dωi(ωm,αx,αy) is not simple

2.1 Stretch invariance: solve only for αx = 1 and αy = 1. 2.2 Slope space: sample only two 1D PDFs.

Now it’s simple

  • 3. Implementation (C++ code in our supplemental material)

3.1 Beckmann and GGX distributions: analytic sampling. 3.2 Other distributions: generic sampling with precomputed data.

  • 4. Validation: passes χ2 test in Mitsuba.

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Conclusion

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Conclusion Contribution

Efficient BSDF Importance sampling Easy to implement Analytic for Beckmann and GGX distributions

Lessons Learned

Physical models have often a “meaning” Spend time working out the intuitions behind the math

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Conclusion Contribution

Efficient BSDF Importance sampling Easy to implement Analytic for Beckmann and GGX distributions

Lessons Learned

Physical models have often a “meaning” Spend time working out the intuitions behind the math

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Questions?

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A Few Words on Microfacet Theory

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A Few Words on Microfacet Theory

The distribution of visible normals

Dωi(ωm) = G1(ωi,ωm) cosθi 〈ωi,ωm〉D(ωm,αx,αy)

D(ωm,αx,αy) distribution of normals G1(ωi,ωm) masking function

Is it actualy a PDF? I.e.

  • Ω

G1(ωi,ωm) cosθi 〈ωi,ωm〉D(ωm,αx,αy)dωm

?

= 1 46

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A Few Words on Microfacet Theory

The distribution of visible normals

Dωi(ωm) = G1(ωi,ωm) cosθi 〈ωi,ωm〉D(ωm,αx,αy)

D(ωm,αx,αy) distribution of normals G1(ωi,ωm) masking function

Is it actualy a PDF? I.e.

  • Ω

G1(ωi,ωm) cosθi 〈ωi,ωm〉D(ωm,αx,αy)dωm

?

= 1 46

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A Few Words on Microfacet Theory

Masking functions G1 in the CG literature Smith Cook & Torrance V-cavities Implicit Kelemen Schlick-Smith

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A Few Words on Microfacet Theory

Masking functions G1 in the CG literature Smith Cook & Torrance V-cavities Implicit Kelemen Schlick-Smith “approximate” “unrealistic” “accurate”

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A Few Words on Microfacet Theory

Masking functions G1 in the CG literature Smith Cook & Torrance V-cavities Implicit Kelemen Schlick-Smith “approximate” “unrealistic” “accurate” Does it work?

  • Ω

G1(ωi,ωm) cosθi 〈ωi,ωm〉D(ωm,αx,αy)dωm

?

= 1 47

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A Few Words on Microfacet Theory What does it mean for a masking function to be “correct” or “exact”?

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A Few Words on Microfacet Theory

The masking function is “correct” or “exact”.

⇔

The distribution of visible normals is a PDF, i.e.

  • Ω

G1(ωi,ωm) cosθi 〈ωi,ωm〉D(ωm,αx,αy)dωm = 1 49

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A Few Words on Microfacet Theory

The distribution of visible normals is NOT a PDF, i.e.

  • Ω

G1(ωi,ωm) cosθi 〈ωi,ωm〉D(ωm,αx,αy)dωm = 1 ⇔

The masking function is NOT “correct” or “exact” and should NOT be called “physically based”.

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A Few Words on Microfacet Theory

Masking functions G1 in the CG literature Name

Dωi is a PDF?

“physically based”? Smith Cook & Torrance V-cavities Implicit Kelemen Schlick-Smith

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A Few Words on Microfacet Theory

Masking functions G1 in the CG literature Name

Dωi is a PDF?

“physically based”? Smith ✓ ✓ Cook & Torrance V-cavities Implicit Kelemen Schlick-Smith

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A Few Words on Microfacet Theory

Masking functions G1 in the CG literature Name

Dωi is a PDF?

“physically based”? Smith ✓ ✓ Cook & Torrance V-cavities ✓ ✓ Implicit Kelemen Schlick-Smith

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A Few Words on Microfacet Theory

Masking functions G1 in the CG literature Name

Dωi is a PDF?

“physically based”? Smith ✓ ✓ Cook & Torrance V-cavities ✓ ✓ Implicit ✗ ✗ Kelemen ✗ ✗ Schlick-Smith ✗ ✗

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A Few Words on Microfacet Theory

Masking functions G1 in the CG literature Name

Dωi is a PDF?

“physically based”? Smith ✓ ✓ Cook & Torrance V-cavities ✓ ✓ Implicit ✗ ✗ Kelemen ✗ ✗ Schlick-Smith ✗ ✗ More details in “Understanding the Masking-Shadowing Function in Microfacet-Based BRDFs”, Eric Heitz, JCGT 2014, to appear.

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