## Marschner Shader Part II

In my last post I mentioned two functions that are needed to represent the hair model as depicted in Marschner’s paper.

*S* = *S _{R} * +

*S*+

_{TT}*S*

_{TRT,}*S _{p} * =

*M*(

_{p}*q*

_{ i },

*q*

_{r }) x

*N*(

_{p}*q*

_{ d },

*f*

_{ d }) for

*P*=

*R*,

*TT*,

*TRT*.

**M component**

This is actually just a probability density function and the best choice here is to use a Gaussian distribution (or normal distribution).

And the M components are as follows:

*M*(_{R}*q*) =_{ h }*g*(*Beta*,_{ R }*q*–_{ h }*Alpha*)._{ R}*M*(_{TT}*q*) =_{ h }*g*(*Beta*,_{ TT }*q*–_{ h }*Alpha*)._{ TT}*M*(_{TRT}*q*) =_{ h }*g*(*Beta*,_{ TRT }*q*–_{ h }*Alpha*)._{ TRT}

**N component**

The N component is actually a bit tricky to compute. Here are all the main steps:

- Convert to Miller-Bravais index.

This is done in order to change the index of refraction to 2D physics, so that the optics of a 3D cylindrical fiber may be reduced to the 2D analysis of the optics of its cross-section.

After looking into Snellâ€™s Law we define the indexes of refraction as:

- Solve a cubic equation

Remember this picture:

We need to find out who the incident angles are, and we can approximate the solution for this equation as:

- Solve Fresnel equation

Fresnel equation is used in order to simulate the reflection model from within the attenuation

- Find out the absorption factor

This is actually quite straightforward, just:

- The attenuation factor

This is obtain combining both the reflection and the absorption factor, hence the “Attenuation by absorption and reflection” model from Marschner’s paper.

where the first derivative is

- The N component (finally)

and the N are

*N*(_{R}*q*_{ d },*f*_{ d }) =_{ }*N*(0,_{P}*q*_{ d },*f*_{ d })._{ }*N*(_{TT}*q*_{ d },*f*_{ d }) =_{ }*N*(1,_{P}*q*_{ d },*f*_{ d })._{ }*N*(_{TRT}*q*_{ d },*f*_{ d )}=*N*(2,_{P}*q*_{ d },*f*_{ d })._{ }

For the last component Marschner proposes a more complex model in order to avoid singularities, but for my implementation I couldn’t tell any improvement so I stuck with the simpler version of *N _{TRT.
}*

**The whole model**

As a sum up this is the whole Marschner hair model in just an equation:

Hope I managed to keep everything simple and explicit alike.

## Marschner Shader Part I

I decided to write a trilogy (3 posts) explaining, as best as I can, what is discussed in Marschner’s paper “Light Scattering from Human Hair Fibers“.

First of all, I have to warn you that in order to understand this paper you must have some physics and math background, rather than knowing a lot about shaders, things such as Snell’s law or probability density functions being mentioned quite often.

The main advantage of the model proposed by Marschner is that it is based on the actual physical phenomenon that occurs when light passes through hair fibers. So by studying electron micrograph of hair fibers such as this one:

a model has been proposed, where each individual hair fiber is treated as a translucent cylinder, having the following components:

and the components that contribute to a distinct and visually significant aspect of hair reflectance are *R*, *TT *and *TRT*.

*R*– light that bounces off of the surface of the hair fiber toward the viewer.*TT*– light that refracts into the hair and refracts out again toward the viewer.*TRT*– light that refracts into the hair fiber, reflects off of the inside surface, and refracts out again toward the viewer.

The notation used throughout this paper is in tangent space, for the light and viewer position, reported to the current hair fiber.

These are all the variable inputs that are needed for Marschner hair shading model:

*u*– tangent to the hair, pointing in the direction from the root toward the tip.*w*– normal to the hair, pointing toward the viewer (the geometry faces the camera).*v*– binormal to the hair, pointing such that*v*and*w*complete a right-handed orthonormal basis, and are the*v*–*w*is the normal plane.*w*_{i }*w*– direction of camera (viewer)._{ r}*q*– inclinations with respect to the normal plane (measured so that 0 is perpendicular to the hair,_{i ,r }*PI*is*u*, and –*PI*is –*u*).*f*– azimuths around the hair (measured so that_{i ,r }*v*is 0 and*w*is +*PI*).

Several derived angles are used, as well:

*q*=_{d }*q*–_{r }*q*)/2_{i }*;*– the difference angle.*f*=*f*–_{r }*f*); – the relative azimuth_{i }*q*=_{ h }*q*+_{i }*q*)/2; – half angle_{r }*f*= (_{ h }*f*+_{i }*f*)/2; – half angle_{r }

Also, there are some constants parameters for hair fibers, surface and glints that you can find in Table 1 ( page 8 ) from Marschner’s paper.

Having all of this in mind we can approximate the hair model as:

*S* = *S _{R} * +

*S*+

_{TT}*S*

_{TRT,}*S _{p} * =

*M*(

_{p}*q*

_{ i },

*q*

_{r }) x

*N*(

_{p}*q*

_{ d },

*f*

_{ d }) for

*P*=

*R*,

*TT*,

*TRT*.

So it turns out the only thing we need is to find out who M and N are. My next post will do just that.

## Physics reloaded

A couple of days ago I managed to find some bullet ropes parameters that tend to make ropes act more like hair strands:

`body->m_cfg.kDP = 0.08f; // no elasticity`

body->m_cfg.piterations = 16; // no white zone

body->m_cfg.timescale = 2;

This is what each one of them does:

- kDP is the Damping coefficient [0, 1], zero means no damping and one full damping. This is a damping spring:

- piterations are the number of iterations for position solvers (if any). It goes from 1 to infinity.
- timescale is a factor of time step, that can be used to speed up, or slow down simulation, default=1.

There are a lot more settings that can be altered with from bullet’s soft bodies. More on bullet online documentation.

Next you can see how these settings look on Krystal, having 367 control points and 3670 hair strands. I guess I will have to further modify them, to get rid of some elasticity, but I find the overall simulation quite plausible:

## The plugin

Giving the fact that this project has two main parts, the physics simulation and the rendering, it would probably be a good designing idea if the plugin implementing it will also have two CS interfaces. This is what these interface should do:

**iFurMaterial**

- Attach fur to the base mesh

For this, a mesh (any type) would have to be specify along with some control points for the guide hairs.

- Generate geometry

Using the guide hairs and a density map (maybe a heightmap too) the rest of the hair strands will be generated using interpolation.

- Update position

This is done by synchronizing the position with iFurPhysicsControl. This interface can implement any type of physics not just ropes, and it can even be null, specifying that a particular instance of the iFurMaterial interface doesn’t have any physics simulation (might be used for static objects, or such).

- Implementing a shader

Or specify a shader/material to be used with this interface. This is especially important because although Marschner is a good model for hair rendering, it might be too complex for fur in general. Here Kajiya and Kay shading model could be used instead, because it has pretty good results too and it’s faster.

**iFurPhysicsControl**

- Initialize strand

Given a guide hair this function will create a physics object, a rope in my case, and make a connection with this strand by a unique id, or so.

- Animate strand

This function will update a strand’s position using the physics object especially created (via the initialize method) for this strand.

- Remove strand

Removing physics objects might be a good idea for a LOD scheme, because animating physics objects is quite computational expensive.

## Hair simulation types

There are various ways in which hair can be simulated using a physics engine.

Next I am going to present 3 of them. For these simulations I used CS for rendering, and for physics the Bullet plugin, that as of recently supports soft bodies, thanks to my mentor Christian Van Brussel.

**Solid geometry**

Perhaps the easiest way to simulate hair is as standard collision objects, such as spheres or cylinders. Although this representation has the best performance, it only covers some particular types of hair like the one below:

**Soft Body Dynamics**

Another approach is to use soft body dynamics, and represent the hair as a … cloth. A larger number of hair styles can be simulated using this method and it also looks more convincing. You can see in the next video both Krystal’s (that’s the model’s name BTW) hair and her skirt represented as soft bodies (drawn in green):

**Ropes**

This is probably the best and somehow the most intuitive way to simulate hair: as ropes. But, as you already know, there are way to much (i.e. millions) hair strands to be simulated individually as physics objects. So the trick here is to choose (either random or better yet using a density map) lets say a hundred hair strands to be guide hairs, represented as ropes. And for the rest of the hair strands just interpolate. You can see these guide hairs (hopefully) in black: