A Spherical Cap Preserving Parameterization for Spherical Distributions

Doing Stuff with Spherical Caps

Jonathan Dupuy	Eric Heitz	Laurent Belcour
Unity Technologies	Unity Technologies	Unity Technologies

A Spherical Cap Preserving Parameterization for Spherical Distributions

"Doing Stuff with Spherical Caps"

Jonathan Dupuy	Eric Heitz	Laurent Belcour
Unity Technologies	Unity Technologies	Unity Technologies

Overview

Theoretical contribution

Spherical distributions
Analytic cap integration
Analytic sampling

Applied to sphere lighting

Real-time shading
Specular occlusion
Variance reduction

Overview

Theoretical contribution

Spherical distributions
Analytic cap integration
Analytic sampling

Applied to sphere lighting

Real-time shading
Specular occlusion
Variance reduction

Overview

Theoretical contribution

Spherical distributions
Analytic cap integration
Analytic sampling

Applied to sphere lighting

Real-time shading
Specular occlusion
Variance reduction

Overview

Theoretical contribution

Spherical distributions
Analytic cap integration
Analytic sampling

Applied to sphere lighting

Real-time shading
Specular occlusion
Variance reduction

Overview

Theoretical contribution

Spherical distributions
Analytic cap integration
Analytic sampling

Applied to sphere lighting

Real-time shading
Specular occlusion
Variance reduction

Overview

Theoretical contribution

Spherical distributions
Analytic cap integration
Analytic sampling

Applied to sphere lighting

Real-time shading
Specular occlusion
Variance reduction

Overview

Theoretical contribution

Spherical distributions
Analytic cap integration
Analytic sampling

Applied to sphere lighting

Real-time shading
Specular occlusion
Variance reduction

MIS@64spp: Previous	MIS@64spp: Ours

Motivation

Problem

$$\mathscr{C}$$

$$ \boxed{\; I = \int_\mathscr{C} D(\boldsymbol{\omega}) \, d\boldsymbol{\omega} \;} $$

How to compute efficiently ?

[Note: this slide is progressive. Press the left key on your keyboard each time you read [click] to keep the text in sync with the contents of the slide]
When reproducing such images virtually, [click] each pixel is associated to a material, which is typically represented by [click] a BRDF. The BRDF is a spherical distribution (illustrated as a color-mapped sphere here) that tells us how much light the material is expected to emit towards a given direction for any possible incident direction. In the case of a sphere light [click], we actually have to deal with [click] the set of directions that intersect it. This set [click] defines a spherical cap in the domain of the BRDF, i.e., [click] the unit sphere. In practice, this means that we need to compute [click] the integral of the BRDF over a spherical cap. If we denote by $\mathscr{C}$ [click] the spherical cap, [click] this is typically the integral that we need to compute [click] as fast as possible. In order to tackle this problem, we studied the very nature of this integration problem: what can we tell about the geometry of this problem ? And what kind of properties can we infer about it ?

Related Work

Uniform	Clipped Uniform	Cosine

vMF (Spherical Gaussian)

Phong Lobes

Low frequency

No closed-form

O(exponent)

Contribution

Uniform	Clipped Uniform	Cosine

vMF (Spherical Gaussian)

Phong Lobes

Low frequency

No closed-form

O(exponent)

NEW Pivot Distributions

All frequency + closed-form + O(1)

Pivot Distributions

Overview

Pivot transformation
Jacobian as PDF

Properties

Analytic sampling
Cap-preserving
Analytic cap integrals

Pivot Distributions

Overview

Pivot transformation
Jacobian as PDF

Properties

Analytic sampling
Cap-preserving
Analytic cap integrals

\begin{align} {\color{red}\boldsymbol{\omega}} \end{align}

Pivot Distributions

Overview

Pivot transformation
Jacobian as PDF

Properties

Analytic sampling
Cap-preserving
Analytic cap integrals

\begin{align} {\color{red}\boldsymbol{\omega}} \mapsto {\color{green}\boldsymbol{\omega}} \end{align}

Pivot Distributions

Overview

Pivot transformation
Jacobian as PDF

Properties

Analytic sampling
Cap-preserving
Analytic cap integrals

\begin{align} {\color{red}\boldsymbol{\omega}} \mapsto {\color{green}\boldsymbol{\omega}} &= g({\color{red}\boldsymbol{\omega}}; {\color{RoyalBlue}\mathbf{r}_p}) \end{align}

\begin{align} \Rightarrow {\color{red}\boldsymbol{\omega}} &= g({\color{green}\boldsymbol{\omega}}; {\color{RoyalBlue}\mathbf{r}_p}) \end{align}

Pivot Distributions

Overview

Pivot transformation
Jacobian as PDF

Properties

Analytic sampling
Cap-preserving
Analytic cap integrals

\begin{align} {\color{red}\boldsymbol{\omega}} \mapsto {\color{green}\boldsymbol{\omega}} &= g({\color{red}\boldsymbol{\omega}}; {\color{RoyalBlue}\mathbf{r}_p}) \end{align}

\begin{align} \Rightarrow {\color{red}\boldsymbol{\omega}} &= g({\color{green}\boldsymbol{\omega}}; {\color{RoyalBlue}\mathbf{r}_p}) \end{align}

$$D_\textrm{std}({\color{red}\boldsymbol{\omega}})$$

\begin{align} D({\color{red}\boldsymbol{\omega}}; {\color{RoyalBlue}\mathbf{r}_p}) = D_\textrm{std}({\color{green}\boldsymbol{\omega}}) \frac{ d {\color{green}\boldsymbol{\omega}} }{ d {\color{red}\boldsymbol{\omega}} } \end{align}

Pivot Distributions

Overview

Pivot transformation
Jacobian as PDF

Properties

Analytic sampling
Cap-preserving
Analytic cap integrals

\begin{align} {\color{red}\boldsymbol{\omega}} \mapsto {\color{green}\boldsymbol{\omega}} &= g({\color{red}\boldsymbol{\omega}}; {\color{RoyalBlue}\mathbf{r}_p}) \end{align}

\begin{align} \Rightarrow {\color{red}\boldsymbol{\omega}} &= g({\color{green}\boldsymbol{\omega}}; {\color{RoyalBlue}\mathbf{r}_p}) \end{align}

$$D_\textrm{std}({\color{red}\boldsymbol{\omega}})$$

Pivot Distributions

Overview

Pivot transformation
Jacobian as PDF

Properties

Analytic sampling
Cap-preserving
Analytic cap integrals

\begin{align} {\color{red}\boldsymbol{\omega}} \mapsto {\color{green}\boldsymbol{\omega}} &= g({\color{red}\boldsymbol{\omega}}; {\color{RoyalBlue}\mathbf{r}_p}) \end{align}

\begin{align} \Rightarrow {\color{red}\boldsymbol{\omega}} &= g({\color{green}\boldsymbol{\omega}}; {\color{RoyalBlue}\mathbf{r}_p}) \end{align}

$$D_\textrm{std}({\color{red}\boldsymbol{\omega}})$$

Pivot Distributions

Overview

Pivot transformation
Jacobian as PDF

Properties

Analytic sampling
Cap-preserving
Analytic cap integrals

\begin{align} {\color{red}\boldsymbol{\omega}} \mapsto {\color{green}\boldsymbol{\omega}} &= g({\color{red}\boldsymbol{\omega}}; {\color{RoyalBlue}\mathbf{r}_p}) \end{align}

\begin{align} \Rightarrow {\color{red}\boldsymbol{\omega}} &= g({\color{green}\boldsymbol{\omega}}; {\color{RoyalBlue}\mathbf{r}_p}) \end{align}

$$\mathbf{u} \mapsto {\color{red}\boldsymbol{\omega}}$$

$$\mathbf{u} \mapsto {\color{green}\boldsymbol{\omega}} = g({\color{red}\boldsymbol{\omega}}; {\color{RoyalBlue}\mathbf{r}_p})$$

Pivot Distributions

Overview

Pivot transformation
Jacobian as PDF

Properties

Analytic sampling
Cap-preserving
Analytic cap integrals

\begin{align} {\color{red}\mathscr{C}} \end{align}

Pivot Distributions

Overview

Pivot transformation
Jacobian as PDF

Properties

Analytic sampling
Cap-preserving
Analytic cap integrals

\begin{align} {\color{red}\mathscr{C}} \mapsto {\color{green}\mathscr{C}} \end{align}

Pivot Distributions

Overview

Pivot transformation
Jacobian as PDF

Properties

Analytic sampling
Cap-preserving
Analytic cap integrals

\begin{align} {\color{red}\mathscr{C}} \mapsto {\color{green}\mathscr{C}} &= g({\color{red}\mathscr{C}}; {\color{RoyalBlue}\mathbf{r}_p}) \end{align}

\begin{align} \Rightarrow {\color{red}\mathscr{C}} &= g({\color{green}\mathscr{C}}; {\color{RoyalBlue}\mathbf{r}_p}) \end{align}

Pivot Distributions

Overview

Pivot transformation
Jacobian as PDF

Properties

Analytic sampling
Cap-preserving
Analytic cap integrals

\begin{align} {\color{red}\mathscr{C}} \mapsto {\color{green}\mathscr{C}} &= g({\color{red}\mathscr{C}}; {\color{RoyalBlue}\mathbf{r}_p}) \end{align}

\begin{align} \Rightarrow {\color{red}\mathscr{C}} &= g({\color{green}\mathscr{C}}; {\color{RoyalBlue}\mathbf{r}_p}) \end{align}

$$\mathbf{u} \mapsto {\color{red}\boldsymbol{\omega}} \in {\color{red}\mathscr{C}}$$

$$\mathbf{u} \mapsto {\color{green}\boldsymbol{\omega}} = g({\color{red}\boldsymbol{\omega}}; {\color{RoyalBlue}\mathbf{r}_p}) \in {\color{green}\mathscr{C}}$$

Pivot Distributions

Overview

Pivot transformation
Jacobian as PDF

Properties

Analytic sampling
Cap-preserving
Analytic cap integrals

\begin{align} {\color{red}\mathscr{C}} \mapsto {\color{green}\mathscr{C}} &= g({\color{red}\mathscr{C}}; {\color{RoyalBlue}\mathbf{r}_p}) \end{align}

\begin{align} \Rightarrow {\color{red}\mathscr{C}} &= g({\color{green}\mathscr{C}}; {\color{RoyalBlue}\mathbf{r}_p}) \end{align}

$\int_{\color{red}\mathscr{C}} D_\textrm{std}({\color{red}\boldsymbol{\omega}}) \, d {\color{red}\boldsymbol{\omega}}$

$\int_{\color{green}\mathscr{C}} D ({\color{green}\boldsymbol{\omega}}; {\color{RoyalBlue}\mathbf{r}_p}) \, d {\color{green}\boldsymbol{\omega}}$

Pivot Distributions

Overview

Pivot transformation
Jacobian as PDF

Properties

Analytic sampling
Cap-preserving
Analytic cap integrals

\begin{align} {\color{red}\mathscr{C}} \mapsto {\color{green}\mathscr{C}} &= g({\color{red}\mathscr{C}}; {\color{RoyalBlue}\mathbf{r}_p}) \end{align}

\begin{align} \Rightarrow {\color{red}\mathscr{C}} &= g({\color{green}\mathscr{C}}; {\color{RoyalBlue}\mathbf{r}_p}) \end{align}

$\int_{\color{red}\mathscr{C}_1 \bigcap \mathscr{C}_2} D_\textrm{std}({\color{red}\boldsymbol{\omega}}) \, d {\color{red}\boldsymbol{\omega}}$

$\int_{\color{green}\mathscr{C}_1 \bigcap \mathscr{C}_2} D ({\color{green}\boldsymbol{\omega}}; {\color{RoyalBlue}\mathbf{r}_p}) \, d {\color{green}\boldsymbol{\omega}}$

Applications

Realtime Sphere Light Shading

Realtime Shading

BRDF approximation
Specular occlusion
Validation

Realtime Shading

BRDF approximation
Specular occlusion
Validation

≈

$$\int_{\color{green}\mathscr{C}} f_r \; \cos \theta_i \; d\boldsymbol{\omega}_i$$

$$\int_{\color{green}\mathscr{C}} D(\boldsymbol{\omega}_i; {\color{RoyalBlue}\mathbf{r}_p}) \; d\boldsymbol{\omega}_i$$

$$\int_{\color{red}\mathscr{C}} D_\textrm{std}(\boldsymbol{\omega}) \; d\boldsymbol{\omega}$$

Realtime Shading

BRDF approximation
Specular occlusion
Validation

≈

$$\int_{\color{green}\mathscr{C}} f_r \; \cos \theta_i \; d\boldsymbol{\omega}_i$$

$$\int_{\color{green}\mathscr{C}} D(\boldsymbol{\omega}_i; {\color{RoyalBlue}\mathbf{r}_p}) \; d\boldsymbol{\omega}_i$$

$$\int_{\color{red}\mathscr{C}} D_\textrm{std}(\boldsymbol{\omega}) \; d\boldsymbol{\omega}$$

Realtime Shading

BRDF approximation
Specular occlusion
Validation

≈

$$\int_{\color{green}\mathscr{C}} f_r \; \cos \theta_i \; d\boldsymbol{\omega}_i$$

$$\int_{\color{green}\mathscr{C}} D(\boldsymbol{\omega}_i; {\color{RoyalBlue}\mathbf{r}_p}) \; d\boldsymbol{\omega}_i$$

$$\int_{\color{red}\mathscr{C}} D_\textrm{std}(\boldsymbol{\omega}) \; d\boldsymbol{\omega}$$

Realtime Shading

BRDF approximation
Specular occlusion
Validation

≈

$$\int_{\color{green}\mathscr{C}} f_r \; \cos \theta_i \; d\boldsymbol{\omega}_i$$

$$\int_{\color{green}\mathscr{C}} D(\boldsymbol{\omega}_i; {\color{RoyalBlue}\mathbf{r}_p}) \; d\boldsymbol{\omega}_i$$

$$\int_{\color{red}\mathscr{C}} D_\textrm{std}(\boldsymbol{\omega}) \; d\boldsymbol{\omega}$$

Realtime Shading

BRDF approximation
Specular occlusion
Validation

≈

$$\int_{\color{green}\mathscr{C} \bigcap \mathscr{C}_v} f_r \; \cos \theta_i \; d\boldsymbol{\omega}_i$$

$$\int_{\color{green}\mathscr{C} \bigcap \mathscr{C}_v} D(\boldsymbol{\omega}_i; {\color{RoyalBlue}\mathbf{r}_p}) \; d\boldsymbol{\omega}_i$$

$$\int_{\color{red}\mathscr{C} \bigcap \mathscr{C}_v} D_\textrm{std}(\boldsymbol{\omega}) \; d\boldsymbol{\omega}$$

Specular Occlusion: OFF

Specular Occlusion: ON

Realtime Shading

BRDF approximation
Specular occlusion
Validation

	≈
$f_r \cdot \cos \theta_i$		$D$

Ours	Ref (raytraced)

$\alpha = 0.001$

Ours	Ref (raytraced)

$\alpha = 0.10$

Ours	Ref (raytraced)

$\alpha = 0.25$

Ours	Ref (raytraced)

$\alpha = 0.50$

Ours	Ref (raytraced)

$\alpha = 1.00$

Ours	Ref (raytraced)

$\alpha = \textrm{map}$

Variance Reduction

BRDF MIS
MIS demo
Pivot phase function

MIS (previous)

MIS (ours)

BRDF	Light

BRDF	Pivot-Light

BRDF

Pivot

Variance Reduction

BRDF MIS
MIS demo
Pivot phase function

MIS (previous)

MIS (ours)

BRDF	Light

BRDF	Pivot-Light

MIS (previous)

MIS (ours)

$\alpha = 0.01$

MIS (previous)

MIS (ours)

$\alpha = 0.25$

MIS (previous)

MIS (ours)

$\alpha = 0.75$

Variance Reduction

BRDF MIS
MIS demo
Pivot phase function

Variance Reduction

BRDF MIS
MIS demo
Pivot phase function

Henyey-Greenstein	Pivot

$$f_s(\mu; g) = \frac{1}{4\pi}\frac{1 - g^2}{(1 + g^2 - 2 \,g\, \mu)^\frac{3}{2}} $$	$$f_s(\mu; g) = \frac{1}{4\pi}\left(\frac{1 - g^2}{1 + g^2 - 2 \,g\, \mu}\right)^2 $$

Phase Function Fitting


Henyey-Greenstein ($g = 0$)	Our fit

Phase Function Fitting


Henyey-Greenstein ($g = -0.8$)	Our fit

Phase Function Fitting


Henyey-Greenstein ($g = +0.8$)	Our fit

Variance Reduction

BRDF MIS
MIS demo
Pivot phase function

MIS (previous)

Perfect (ours)

Phase function	Light

Phase function-Light

Variance Reduction

BRDF MIS
MIS demo
Pivot phase function

Conclusion

New spherical distributions

Pivot parameterization
Analytic properties
Useful for sphere lights

Future Work

Unify SPTDs with LTSDs ?
Generalize further ?

\begin{align} {\color{red}\boldsymbol{\omega}} \mapsto {\color{green}\boldsymbol{\omega}} &= g({\color{red}\boldsymbol{\omega}}; {\color{RoyalBlue}\mathbf{r}_p}) \end{align}

$$D_\textrm{std}({\color{red}\boldsymbol{\omega}})$$

Acknowledgements

Morgan Villedieu
Hamza Cheggour
Laurent Harduin

Link to shadertoy: www.shadertoy.com

Acknowledgements

Morgan Villedieu
Hamza Cheggour
Laurent Harduin

Free download: www.emirage.org

Acknowledgements

Morgan Villedieu
Hamza Cheggour
Laurent Harduin

Lighting by Laurent Harduin

Acknowledgements

Morgan Villedieu
Hamza Cheggour
Laurent Harduin

Lighting by Jonathan Dupuy

Questions

\begin{align} {\color{red}\boldsymbol{\omega}} \mapsto {\color{green}\boldsymbol{\omega}} &= g({\color{red}\boldsymbol{\omega}}; {\color{RoyalBlue}\mathbf{r}_p}) \end{align}

$$D_\textrm{std}({\color{red}\boldsymbol{\omega}})$$

Get our source code online: https://github.com/jdupuy/pivot

Related Work

$$ \boxed{\quad I = \int_\mathscr{C} D(\boldsymbol{\omega}) \, d\boldsymbol{\omega} \quad} $$

Polygon lighting
Approximations

[Heitz et al. 2016]

$\mathscr{C} \approx \mathscr{P}_1 + \mathscr{P}_2 + \cdots + \mathscr{P}_N$

Related Work

$$ \boxed{\quad I = \int_\mathscr{C} D(\boldsymbol{\omega}) \, d\boldsymbol{\omega} \quad} $$

Polygon lighting
Approximations

[Lecocq et al. 2016]	[Drobot2014]

$\mathscr{C} \approx \mathscr{P}$	$\mathscr{C} \approx \boldsymbol{\omega}_i$

Rendering: Point Lights

Rendering: Ours

Closed-form solutions

Uniform ✓

Clipped Uniform ✓

Clamped Cosine ✓

Closed-form solutions

Uniform ✓

Clipped Uniform ✓

Clamped Cosine ✓

Closed-form solutions

Uniform ✓

Clipped Uniform ✓

Clamped Cosine ✓

Parametric BRDF Shapes

BRDF:

(varying roughness)

Rendering:

Closed-form solutions

Uniform ✓

Clipped Uniform ✓

Clamped Cosine ✓

	vMF ✘
k = 2
k = 30
k = 200

Cheap Analytic Approximation ?

Closed-form solutions

Uniform ✓

Clipped Uniform ✓

Clamped Cosine ✓

	Phong ✓
k = 2
k = 30
k = 200

Cheap Analytic Approximation ?

Closed-form solutions

Uniform ✓

Clipped Uniform ✓

Clamped Cosine ✓

O(1)

	Phong ✓
k = 2		✓
k = 30		✘
k = 200		✘✘
	O(k)

Cheap Analytic Approximation ?

Closed-form solutions

Uniform ✓

Clipped Uniform ✓

Clamped Cosine ✓

O(1)

	Phong ✓
k = 2		✓
k = 30		✘
k = 200		✘✘
	O(k)

Closed-form solutions

Uniform ✓

Clipped Uniform ✓

Clamped Cosine ✓

O(1)

	Phong ✓
k = 2		✓
k = 30		✘
k = 200		✘✘
	O(k)

Can we extend the set of exact solutions ?

Closed-form solutions

Uniform ✓

Clipped Uniform ✓

Clamped Cosine ✓

O(1)

	Phong ✓
k = 2		✓
k = 30		✘
k = 200		✘✘
	O(k)

Our Distributions ✓

O(1)

Pivot Distribution

Video

Pivot Distribution

So now I am going to focus on the intuitions that led us to this. We often view distributions as mathematical functions. But there is also another way of seeing them, namely as a set of points. The values of the distribution then tell us how concentrated these points are on the surface of the sphere. There are two useful insights we can get from this new interpretation. First, our spherical integration problem turns into knowing what fraction of the point set is located inside the cap. Second, the problem is invariant to transformations that preserve the unit sphere. In order to illustrate this second point, I am going to show a transformation we introduce in the paper. It is complicated, but what I want you to see is that the point set located inside the spherical cap remains inside during the entire transformation. Hence, this means that our integral is constant. And since the concentration of points has changed, it means that we can solve this problem for various distributions ! So the key problem is to find transformations that preserve spheres and circles.