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

So now we are going to look at some of the intuitive properies of our new pivot distributions. First note that pivot distributions are parametric: they are controled by a 3D point located inside the sphere. This point is shown on the left and directly impacts the shape of the distribution, as well as its importance sampling both on the full sphere (center figure) and arbitrary spherical caps (rightmost figure). Notice how the motion of the samples is quite complex; that's because they are warped by a 4D quaternionic Mobius transformation, which our pivot distributions build upon. The 4D quaternionic Mobius transformations is defined over a full 4D space and, as such, is very difficult to visualize. Fortunately, the transformation has a very simple interpretation when restricted to the unit sphere, which we are going to look at now.

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}

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

[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]
Here is some source code to give more perspective to what we have seen so far.
[click] First, here is our GLSL code for the pivot transformation. Notice that it is fairly cheap to compute (no fancy special function involved, not even an sqrt()).
[click] Next is the code for the Jacobian of the pivot transformation. Again, very simple and computationally cheap code.
[click] And finally, the code to evaluate the PDF of a pivot-transformed distribution. Given that you have already implemented the pdf() function of your standard distribution, this routine is also extremely simple.
That's all there is to evaluating the PDF of pivot-transformed distributions! Using these three code snippets, you can turn your favorite spherical distribution into a pivot distribution. Note that it also follows that if your standard distribution has an analytic PDF, then so does its pivot-transformed variants (by construction).

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}

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

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

[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]
Importance sampling is also extremely straightforward: If you know how to [click] importance sample your input distribution, then you can importance sample its pivot-transformed variant by simply [click] transforming the samples with the pivot transformation. Again, here is some source code.
[click] Here, the same pivot transformation code we saw in the previous slide.
[click] And here, the importance sampling code for the pivot distribution.
That's all there is to importance sampling pivot-transformed distributions! Note that it follows that if your standard distribution has analytic importance sampling, then so does its pivot-transformed variants (by construction).

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

≈

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

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

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

[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]
[click] Here, we need to [click] integrate the BRDF against the spherical cap produced by the spherical light. Most realtime rendering engines use a GGX microfacet BRDF for their materials so what we did was that we precomputed a small table that fits a pivot distribution to any isotropic GGX microfacet configuration. At runtime, we simply lookup this table to get [click] a good approximation of the problem with a pivot transformed distribution. We then simply return [click] the analytic integral of the pivot distribution. By leveraging the analytic integration properties of pivot distributions, we get a fast technique for spherical area lights.

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

[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]
Just as in the real time rendering case, we start by looking at sphere lighting. This time though, we want to converge to the exact solution. Traditionally, exact solutions are computed using [click] multiple importance sampling (MIS). State-of-the-art MIS techniques combine the importance sampling of [click] the BRDF and [click] the spherical light cap. In our paper [click], we still rely on [click] BRDF importance sampling, but improve over the light cap strategy by [click] relying on a pivot distribution whose density is close to that of the BRDF. That way, we can concentrate more samples towards the lobe of the BRDF, effectively resuling in faster convergence.

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