Sampling Visible GGX Normals with Spherical Caps

Sampling Visible GGX Normals
with Spherical Caps

Jonathan Dupuy	Anis Benyoub
Intel Corporation	Intel Corporation

GGX VNDF

spherical caps

linked to

🔗

Contribution

[Heitz18]

Ours

✓ simpler

✓ faster

This paper is about changing a simple code snippet due to Eric Heitz shown on the left here into another one, ours, which is even simpler and faster. I want to emphasize that this change is not the result of instruction-level optimization over the original code. It actually builds upon a new simple insight that relates the GGX VNDF to spherical caps and leads to a completely different implementation. This insight is that I want to talk about in this presentation and so I will spend most of my time talking about it. To start things off, I will start by recalling a few things about what GGX is and what it is useful for in the context of physically based rendering. The term GGX refers to a distribution that controls the appearance of most surfaces that are rendered in video games and production. This is done by influencing the direction a ray takes after intersecting a surface. Microfacet theory.

Ours

✓ simpler

✓ faster

GGX VNDF

spherical caps

linked to

🔗

Contribution

[Heitz18]

Background

[Walter et al. 07]

GGX = "Ground Glass (X)unknown"

(microsurface)

(equivalent ellipsoid)

Background

[Walter et al. 07]

GGX = "Ground Glass (X)unknown"

(microsurface)

(equivalent ellipsoid)

Background

[Walter et al. 07]

GGX = "Ground Glass (X)unknown"

(microsurface)

(equivalent ellipsoid)

Background

[Walter et al. 07]

GGX = "Ground Glass (X)unknown"

(microsurface)

(equivalent ellipsoid)

Background

[Walter et al. 07]

GGX = "Ground Glass (X)unknown"

(microsurface)

(equivalent ellipsoid)

Background

[Walter et al. 07]

GGX = "Ground Glass (X)unknown"

(microsurface)

(equivalent ellipsoid)

Background

[Walter et al. 07]

GGX = "Ground Glass (X)unknown"

(microsurface)

(equivalent ellipsoid)

Background

[Walter et al. 07]

GGX = "Ground Glass (X)unknown"

(microsurface)

(equivalent ellipsoid)

Background

[Walter et al. 07]

GGX = "Ground Glass (X)unknown"

NDF

(microsurface)

(equivalent ellipsoid)

Background

[Heitz18]

GGX VNDF sampler

VNDF

NDF

incident direction

visible / occluded surface

(microsurface)

(equivalent ellipsoid)

visible / occluded surface

(1) linear transformation

(2) sample

(3) inverse transformation

(equivalent ellipsoid)

visible / occluded surface

GGX sampling code

Background

[Heitz18]

GGX VNDF sampler

(1) linear transformation

(2) sample

(3) inverse transformation

(equivalent ellipsoid)

visible / occluded surface

Background

GGX sampling code

[Heitz18]

GGX VNDF sampler

(1) linear transformation

(2) sample

(3) inverse transformation

(equivalent ellipsoid)

visible / occluded surface

hemisphere sampling code

overview

GGX sampling code

Background

[Heitz18]

GGX VNDF sampler

Ours

✓ simpler

✓ faster

GGX VNDF

spherical caps

linked to

🔗

[Heitz18]

Contribution

Ours

✓ simpler

✓ faster

GGX VNDF

spherical caps

linked to

🔗

[Heitz18]

Contribution

Key Insight

GGX VNDF	🔗	spherical caps

incident direction

$\boldsymbol{\omega}_i = (x_i, \, y_i, z_i)$

Key Insight

GGX VNDF	🔗	spherical caps

incident direction

$\boldsymbol{\omega}_i = (x_i, \, y_i, z_i)$

Key Insight

GGX VNDF	🔗	spherical caps

incident direction

$\boldsymbol{\omega}_i = (x_i, \, y_i, z_i)$

Key Insight

GGX VNDF	🔗	spherical caps

incident direction

$\boldsymbol{\omega}_i = (x_i, \, y_i, z_i)$

Key Insight

GGX VNDF	🔗	spherical caps

incident direction

$\boldsymbol{\omega}_i = (x_i, \, y_i, z_i)$

Key Insight

GGX VNDF	🔗	spherical caps

incident direction

$\boldsymbol{\omega}_i = (x_i, \, y_i, z_i)$

Key Insight

GGX VNDF	🔗	spherical caps

incident direction

$\boldsymbol{\omega}_i = (x_i, \, y_i, z_i)$

Key Insight

GGX VNDF	🔗	spherical caps

incident direction

$\boldsymbol{\omega}_i = (x_i, \, y_i, z_i)$

Key Insight

GGX VNDF	🔗	spherical caps

incident direction

$\boldsymbol{\omega}_i = (x_i, \, y_i, z_i)$

Key Insight

GGX VNDF	🔗	spherical caps

incident direction

$\boldsymbol{\omega}_i = (x_i, \, y_i, z_i)$

Key Insight

GGX VNDF	🔗	spherical caps

incident direction

$\boldsymbol{\omega}_i = (x_i, \, y_i, z_i)$

cutoff plane

A hemispherical mirror reflects parallel rays uniformly and within a spherical cap

(see paper for mathematical proof)

Key Insight

GGX VNDF	🔗	spherical caps

incident direction

$\boldsymbol{\omega}_i = (x_i, \, y_i, z_i)$

cutoff plane

$z = -$$z_i$

A hemispherical mirror reflects parallel rays uniformly and within a spherical cap

(see paper for mathematical proof)

Key Insight

GGX VNDF	🔗	spherical caps

incident direction

$\boldsymbol{\omega}_i = (x_i, \, y_i, z_i)$

cutoff plane

$z = -$$z_i$

A hemispherical mirror reflects parallel rays uniformly and within a spherical cap

(see paper for mathematical proof)

Key Insight

GGX VNDF	🔗	spherical caps

incident direction

$\boldsymbol{\omega}_i = (x_i, \, y_i, z_i)$

cutoff plane

$z = -$$z_i$

A hemispherical mirror reflects parallel rays uniformly and within a spherical cap

(see paper for mathematical proof)

Key Insight

GGX VNDF	🔗	spherical caps

incident direction

$\boldsymbol{\omega}_i = (x_i, \, y_i, z_i)$

cutoff plane

$z = -$$z_i$

A hemispherical mirror reflects parallel rays uniformly and within a spherical cap

(see paper for mathematical proof)

Key Insight

GGX VNDF	🔗	spherical caps

incident direction

$\boldsymbol{\omega}_i = (x_i, \, y_i, z_i)$

cutoff plane

$z = -$$z_i$

A hemispherical mirror reflects parallel rays uniformly and within a spherical cap

(see paper for mathematical proof)

Key Insight

GGX VNDF	🔗	spherical caps

incident direction

$\boldsymbol{\omega}_i = (x_i, \, y_i, z_i)$

cutoff plane

$z = -$$z_i$

A hemispherical mirror reflects parallel rays uniformly and within a spherical cap

(see paper for mathematical proof)

Key Insight

GGX VNDF	🔗	spherical caps

incident direction

$\boldsymbol{\omega}_i = (x_i, \, y_i, z_i)$

cutoff plane

$z = -$$z_i$

A hemispherical mirror reflects parallel rays uniformly and within a spherical cap

(see paper for mathematical proof)

Key Insight

GGX VNDF	🔗	spherical caps

incident direction

$\boldsymbol{\omega}_i = (x_i, \, y_i, z_i)$

cutoff plane

$z = -$$z_i$

A hemispherical mirror reflects parallel rays uniformly and within a spherical cap

(see paper for mathematical proof)

Intuitions

environment:

what you see:

Intuitions

environment:

what you see:

Intuitions

environment:

what you see:

Intuitions

Key Insight

GGX VNDF	🔗	spherical caps

incident direction

$\boldsymbol{\omega}_i = (x_i, \, y_i, z_i)$

cutoff plane

$z = -$$z_i$

A hemispherical mirror reflects parallel rays uniformly and within a spherical cap

Our Sampling Algorithm

GGX VNDF	🔗	spherical caps

incident direction

$\boldsymbol{\omega}_i = (x_i, \, y_i, z_i)$

cutoff plane

$z = -$$z_i$

Our Sampling Algorithm

GGX VNDF	🔗	spherical caps

incident direction

$\boldsymbol{\omega}_i = (x_i, \, y_i, z_i)$

cutoff plane

$z = -$$z_i$

$\boldsymbol{\omega}_o$

1) sample the spherical cap

Our Sampling Algorithm

GGX VNDF	🔗	spherical caps

incident direction

$\boldsymbol{\omega}_i = (x_i, \, y_i, z_i)$

cutoff plane

$z = -$$z_i$

$\boldsymbol{\omega}_o$

1) sample the spherical cap
2) compute half vector

Our Sampling Algorithm

GGX VNDF	🔗	spherical caps

incident direction

$\boldsymbol{\omega}_i = (x_i, \, y_i, z_i)$

cutoff plane

$z = -$$z_i$

$\boldsymbol{\omega}_o$

$\boldsymbol{\omega}_i$

$\boldsymbol{\omega}_o$ $+$ $\boldsymbol{\omega}_i$ $\propto$ $\boldsymbol{\omega}_m$

1) sample the spherical cap
2) compute half vector

Our Sampling Algorithm

GGX VNDF	🔗	spherical caps

incident direction

$\boldsymbol{\omega}_i = (x_i, \, y_i, z_i)$

cutoff plane

$z = -$$z_i$

$\boldsymbol{\omega}_o$

$\boldsymbol{\omega}_i$

$\boldsymbol{\omega}_o$ $+$ $\boldsymbol{\omega}_i$ $\propto$ $\boldsymbol{\omega}_m$

$\boldsymbol{\omega}_m$

1) sample the spherical cap
2) compute half vector

Our Sampling Algorithm

GGX VNDF	🔗	spherical caps

incident direction

$\boldsymbol{\omega}_i = (x_i, \, y_i, z_i)$

cutoff plane

$z = -$$z_i$

$\boldsymbol{\omega}_o$

$\boldsymbol{\omega}_i$

$\boldsymbol{\omega}_o$ $+$ $\boldsymbol{\omega}_i$ $\propto$ $\boldsymbol{\omega}_m$

$\boldsymbol{\omega}_m$

1) sample the spherical cap
2) compute half vector

Our Sampling Algorithm

GGX VNDF	🔗	spherical caps

incident direction

$\boldsymbol{\omega}_i = (x_i, \, y_i, z_i)$

cutoff plane

$z = -$$z_i$

$\boldsymbol{\omega}_o$ $+$ $\boldsymbol{\omega}_i$ $\propto$ $\boldsymbol{\omega}_m$

$\boldsymbol{\omega}_o$

$\boldsymbol{\omega}_m$

1) sample the spherical cap
2) compute half vector
3) return half vector

Ours

✓ simpler

✓ faster

sample spherical cap

compute half vector

GGX VNDF

spherical caps

linked to

🔗

[Heitz18]

Contribution

Ours

✓ simpler

✓ faster

sample spherical cap

compute half vector

GGX VNDF

spherical caps

linked to

🔗

[Heitz18]

build basis

sample projected area

reproject

Contribution

Ours

✓ simpler

✓ faster

sample spherical cap

compute half vector

GGX VNDF

spherical caps

linked to

🔗

[Heitz18]

build basis

sample projected area

reproject

Contribution

Performances

Ours

CPU (Intel i7-13700K): 36.67% faster

GPU (Intel Arc A770): 39.25% faster

[Heitz18]

[Heitz18]	ours	[Heitz18]	ours

	7.59% faster		3.28% faster

Ours

✓ simpler

✓ faster

GGX VNDF

spherical caps

linked to

🔗

[Heitz18]

Contribution

End

Ours

✓ simpler

✓ faster

GGX VNDF

spherical caps

linked to

🔗

Contribution

[Heitz18]

GGX VNDF

spherical caps

linked to

🔗

Contribution

[Heitz18]

Ours

✓ simpler

✓ faster

Spherical Caps

Ours

✓ simpler

✓ faster

GGX VNDF

spherical caps

linked to

🔗

Contribution

[Heitz18]

Ours

✓ simpler

✓ faster

GGX VNDF

spherical caps

linked to

🔗

[Heitz18]

Contribution

Ours

✓ simpler

✓ faster

GGX VNDF

spherical caps

linked to

🔗

[Heitz18]

Contribution

Ours

✓ simpler

✓ faster

GGX VNDF

spherical caps

linked to

🔗

[Heitz18]

Contribution

Background

[Heitz18]

GGX VNDF sampler

VNDF

NDF

(microsurface)

(equivalent ellipsoid)

Background

[Heitz18]

GGX VNDF sampler

VNDF

NDF

incident direction

visible / occluded surface

(microsurface)

(equivalent ellipsoid)

visible / occluded surface