## Sunday, December 15, 2013

### Tone mapping

When working with HDR values, two troublesome situations often arise.

The first happens when one tries to encode an HDR color using an encoding that has a limited range, for instance RGBM. Values outside the range still need to be handled gracefully, ie not clipped.

The second happens when an HDR signal is under sampled. One very bright sample can completely dominate the result. In path tracing these are commonly called fireflies.

In both cases the obvious solution is to reduce the range. This sounds exactly like tonemapping so break out those tone mapping operators, right? Well yes and no. Common tone mapping operators work on color channels individually. This has the downside of desaturating the colors which can look really bad if later operations attenuate the values, for instance reflections, glare, or DOF.

Instead I use a function that modifies only the luminance of the color. The simplest of which is this:
$$T(color) = \frac{color}{ 1 + \frac{luma}{range} }$$
Where $T$ is the tone mapping function, $color$ is the color to be tone mapped, $luma$ is the luminance of $color$, and $range$ is the range that I wish to tone map into. If the encoding must fit RGB individually in range then $luma$ is the max RGB component.

Inverting this operation is just as easy. $$T_{inverse}(color) = \frac{color}{ 1 - \frac{luma}{range} }$$
This operation, when used to reduce fireflies, can also be thought of as a weighting function for each sample: $$weight = \frac{1}{ 1 + luma }$$
For a weighted average, sum all samples and divide by the summed weights. The result will be the same as if the samples were tone mapped using $T$ with $range$ of 1, averaged, then inverse tone mapped using $T_{inverse}$.

If a more expensive function is acceptable then keeping more of the color range linear is best. To do this use the functions below where 0 to $a$ is linear and $a$ to $b$ is tone mapped. $$T(color) = \left\{ \begin{array}{l l} color & \quad \text{if luma \leq a}\\ \frac{color}{luma} \left( \frac{ a^2 - b*luma }{ 2a - b - luma } \right) & \quad \text{if luma \gt a} \end{array} \right.$$ $$T_{inverse}(color) = \left\{ \begin{array}{l l} color & \quad \text{if luma \leq a}\\ \frac{color}{luma} \left( \frac{ a^2 - ( 2a - b )luma }{ b - luma } \right) & \quad \text{if luma \gt a} \end{array} \right.$$
These are same as the first two functions if $a=0$ and $b=range$.

I have used these methods for lightmap encoding, environment map encoding, fixed point bloom, screen space reflections, path tracing, and more.

## Saturday, August 3, 2013

### Specular BRDF Reference


First let me define alpha that will be used for all following equations using UE4's roughness: $$\alpha = roughness^2$$

## Normal Distribution Function (NDF)

The NDF, also known as the specular distribution, describes the distribution of microfacets for the surface. It is normalized [12] such that: $$\int_\Omega D(\mv) (\ndotm) d\omega_i = 1$$ It is interesting to notice all models have $\frac{1}{\pi \alpha^2}$ for the normalization factor in the isotropic case.

Blinn-Phong [2]: $$D_{Blinn}(\mv) = \frac{1}{ \pi \alpha^2 } (\ndotm)^{ \left( \frac{2}{ \alpha^2 } - 2 \right) }$$ This is not the common form but follows when $power = \frac{2}{ \alpha^2 } - 2$.

Beckmann [3]: $$D_{Beckmann}(\mv) = \frac{1}{ \pi \alpha^2 (\ndotm)^4 } \exp{ \left( \frac{(\ndotm)^2 - 1}{\alpha^2 (\ndotm)^2} \right) }$$

GGX (Trowbridge-Reitz) [4]: $$D_{GGX}(\mv) = \frac{\alpha^2}{\pi((\ndotm)^2 (\alpha^2 - 1) + 1)^2}$$

GGX Anisotropic [5]: $$D_{GGXaniso}(\mv) = \frac{1}{\pi \alpha_x \alpha_y} \frac{1}{ \left( \frac{(\mathbf{x} \cdot \mv)^2}{\alpha_x^2} + \frac{(\mathbf{y} \cdot \mv)^2}{\alpha_y^2} + (\ndotm)^2 \right)^2 }$$

The geometric shadowing term describes the shadowing from the microfacets. This means ideally it should depend on roughness and the microfacet distribution.

Implicit [1]: $$G_{Implicit}(\lv,\vv,\hv) = (\ndotl)(\ndotv)$$

Neumann [6]: $$G_{Neumann}(\lv,\vv,\hv) = \frac{ (\ndotl)(\ndotv) }{ \mathrm{max}( \ndotl, \ndotv ) }$$

Cook-Torrance [11]: $$G_{Cook-Torrance}(\lv,\vv,\hv) = \mathrm{min}\left( 1, \frac{ 2(\ndoth)(\ndotv) }{\vdoth}, \frac{ 2(\ndoth)(\ndotl) }{\vdoth} \right)$$

Kelemen [7]: $$G_{Kelemen}(\lv,\vv,\hv) = \frac{ (\ndotl)(\ndotv) }{ (\vdoth)^2 }$$

### Smith

The following geometric shadowing models use Smith's method[8] for their respective NDF. Smith breaks $G$ into two components: light and view, and uses the same equation for both: $$G(\lv, \vv, \hv) = G_{1}(\lv) G_{1}(\vv)$$ I will define $G_1$ below for each model and skip duplicating the above equation.

Beckmann [4]: $$c = \frac{\ndotv}{ \alpha \sqrt{1 - (\ndotv)^2} }$$ $$G_{Beckmann}(\vv) = \left\{ \begin{array}{l l} \frac{ 3.535 c + 2.181 c^2 }{ 1 + 2.276 c + 2.577 c^2 } & \quad \text{if c < 1.6}\\ 1 & \quad \text{if c \geq 1.6} \end{array} \right.$$

Blinn-Phong:
The Smith integral has no closed form solution for Blinn-Phong. Walter [4] suggests using the same equation as Beckmann.

GGX [4]: $$G_{GGX}(\vv) = \frac{ 2 (\ndotv) }{ (\ndotv) + \sqrt{ \alpha^2 + (1 - \alpha^2)(\ndotv)^2 } }$$ This is not the common form but is a simple refactor by multiplying by $\frac{\ndotv}{\ndotv}$.

Schlick-Beckmann:
Schlick [9] approximated the Smith equation for Beckmann. Naty [1] warns that Schlick approximated the wrong version of Smith, so be sure to compare to the Smith version before using. $$k = \alpha \sqrt{ \frac{2}{\pi} }$$ $$G_{Schlick}(\vv) = \frac{\ndotv}{(\ndotv)(1 - k) + k }$$

Schlick-GGX:
For UE4, I used the Schlick approximation and matched it to the GGX Smith formulation by remapping $k$ [10]: $$k = \frac{\alpha}{2}$$

## Fresnel

The Fresnel function describes the amount of light that reflects from a mirror surface given its index of refraction. Instead of using IOR we instead use the parameter or $F_0$ which is the reflectance at normal incidence.

None: $$F_{None}(\mathbf{v}, \mathbf{h}) = F_0$$

Schlick [9]: $$F_{Schlick}(\mathbf{v}, \mathbf{h}) = F_0 + (1 - F_0) ( 1 - (\vdoth) )^5$$

Cook-Torrance [11]: $$\eta = \frac{ 1 + \sqrt{F_0} }{ 1 - \sqrt{F_0} }$$ $$c = \vdoth$$ $$g = \sqrt{ \eta^2 + c^2 - 1 }$$ $$F_{Cook-Torrance}(\mathbf{v}, \mathbf{h}) = \frac{1}{2} \left( \frac{g - c}{g + c} \right)^2 \left( 1 + \left( \frac{ (g + c)c - 1 }{ (g - c)c+ 1 } \right)^2 \right)$$

## Optimize

Be sure to optimize the BRDF shader code as a whole. I choose these forms of the equations to either match the literature or to demonstrate some property. They are not in the optimal form to compute in a pixel shader. For example, grouping Smith GGX with the BRDF denominator we have this: $$\frac{ G_{GGX}(\lv) G_{GGX}(\vv) }{4(\ndotl)(\ndotv)}$$ In optimized HLSL it looks like this:

float a2 = a*a;
float G_V = NoV + sqrt( (NoV - NoV * a2) * NoV + a2 );
float G_L = NoL + sqrt( (NoL - NoL * a2) * NoL + a2 );
return rcp( G_V * G_L );

If you are using this on an older non-scalar GPU you could vectorize it as well.

### References

[1] Hoffman 2013, "Background: Physics and Math of Shading"
[2] Blinn 1977, "Models of light reflection for computer synthesized pictures"
[3] Beckmann 1963, "The scattering of electromagnetic waves from rough surfaces"
[4] Walter et al. 2007, "Microfacet models for refraction through rough surfaces"
[5] Burley 2012, "Physically-Based Shading at Disney"
[6] Neumann et al. 1999, "Compact metallic reflectance models"
[7] Kelemen 2001, "A microfacet based coupled specular-matte brdf model with importance sampling"
[8] Smith 1967, "Geometrical shadowing of a random rough surface"
[9] Schlick 1994, "An Inexpensive BRDF Model for Physically-Based Rendering"
[10] Karis 2013, "Real Shading in Unreal Engine 4"
[11] Cook and Torrance 1982, "A Reflectance Model for Computer Graphics"
[12] Reed 2013, "How Is the NDF Really Defined?"

## Sunday, July 28, 2013

### Epic, SIGGRAPH, etc

I'm resurrecting this blog from the dead. I'm sorry it's been neglected for a year but I've been busy. If you follow me on twitter (@BrianKaris) then this probably isn't news, but for those that don't here's an update:

A year ago I left Human Head and accepted a position on the rendering team at Epic Games. Since then we made the UE4 Infiltrator demo. I've worked on temporal AA, reflections, shading, materials, and other misc cool stuff for UE4 and games being developed here at Epic. I'm surrounded by a bunch of really smart, talented people, with whom it has been a pleasure to work.

Just this last week I presented in the SIGGRAPH 2013 course: Physically Based Shading in Theory and Practice. If you saw my talk and are interested in the subject but haven't looked at the course notes I highly suggest you follow that link and check them out as well as the other presenter's materials. Like previous years, the talks are only a taste of the content that the course notes cover in detail.

Now with that out of the way, hopefully I can start making some good posts again.