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Last Time
Microfacet models
Diffuse
Oren-Nayar
Specular
Torrance-Sparrow
Blinn
Ashikhmin-Shirley
Ward
Schlick
Lafortune’s model
Glossy over Diffuse
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2. © 2005 University of Wisconsin
Today
Assignment 1, Part 2, h=4
More reflectance models
Materials
Textures
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3. © 2005 University of Wisconsin
Metallic Patinas
Dorsey and Hanrahan, SIGGRAPH 1996
Aim: capture the weathering of metallic surfaces
Actually, only did copper
Underlying ideas:
Layer model
Modulation by textures
Scripting for time control
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Layer Model
Surface consists of several layers
Each layer has constants:
Standard diffuse, specular, roughness parameters (roughness is Phong exponent)
Transmission and back-scatter parameters, K and S
Scripting controls the layers over time
Layers are not uniform
Consider a stack of layers at a single point
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Operating on Layers
coat material thickness thickness-map
Add a new layer of material
erode thickness thickness-map
Remove some material
fill height height-map
Fill in valleys to a certain minimum height
polish height height-map
Remove down to a given height
offset radius
Apply material in corners (places not reachable by a sphere of radius)
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6. Thickness/Height Maps
Use texture maps or triangulations to define the thickness of each layer
Growth models change the maps over time
Various methods
Steady thickening – quite uniform texture
Random deposition – drop points onto the surface
Ballistic deposition – point sticks when it first contacts something (gives overhangs)
Directed percolation depinning – points on surface are blocked or unblocked (with moisture) and surface grows into unblocked regions over time
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7. Kubelka-Munk (KM) Model
A model used in the paint, printing and textile industries to predict diffuse scattering in layers
Describes light distribution at a height in a medium with forward and backward flux density (energy per unit area)
K is absorption per unit length, S is backscattering per unit length B- dz B+ d
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Homogeneous Medium
R is total reflectance through a layer of thickness d
d for use in diffuse BRDF
T is total transmission
d for diffuse emission on the back side of the layer
But, hard to estimate S and K (they are colors)
Get ratio K/S from infinite thickness equation
Guess S
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Multiple Layers
Model can be extended for multiple layers of varying thickness
For two layers:
Subsurface compositing operators
To get more layers, just apply the operator again
Order doesn’t matter
Bottom layer has R but T=0
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BRDF
Diffuse term comes from previous equations: d=Rall
Gloss term is harder
Use physically viable Phong model at each interface, i, to compute a gloss color:
Scale by transmission of layers above:
Transmission composed for all layers above i
Sum terms from each layer:
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Copper Strips
Urban
Rural
Marine
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12. © 2005 University of Wisconsin
Others
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13. Subsurface Scattering
Kubelka-Munk is a gross approximation to real scattering
Assumes isotropic light distribution and glossy transmittance is all wrong
Subsurface scattering is very important to capturing the appearance of organic materials
More later
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14. Hemispherical-Directional Reflectance
Hemispheric-Directional Reflectance is the fraction of the incident irradiance in a given direction that is reflected by the surface
Useful for computation of some reflectance functions and for some algorithms
Note error in book on page 689 – see for errata
Derivation on board. Grooved surface with alternating black white.
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15. © 2005 University of Wisconsin
Computing hd (PBR )
For many reflectance functions, this value is not easy to compute, so use Monte Carlo integration to compute it
Evaluating this is easy in PBRT:
Samples are passed in (or you generate them somehow), as is 
Iterate over each sample
Ask for a sampled direction Sample_f. It returns both fr and p
Add up all samples and divide by N
In fact, if you inherit from BxDF then you already get it
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16. PBRT BSDF Structures (PBR Sect 10.1)
A BxDF is a component of a BSDF – one of the things we have been looking at so far
BSDF can be made up of multiple components
The PBRT BSDF structure has code for managing these components – read Chapter 10
When sampling, choose a BxDF
Uniform probability of choosing any one matching component
PDF of choosing some direction must account for all components
But it still assumes we’re talking about a single point
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17. PBRT Materials (PBR Sect 10.2)
A Material described how the BSDF varies over the entire surface of an object
All parameters to the components BxDFs are given in the form of textures
More on textures in a minute
The job of the Material is:
Manage local surface variations via textures and pass a single set of point properties (color, normal, etc) to BSDFs
Provide BSDFs with information about the neighborhood, particularly the geometric normal and the shading normal (bump mapped or otherwise modified)
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18. Bump Mapping (PBR Sect 10.3)
Bump mapping modifies the surface normal vector
In PBRT, a heightfield is given and normals are derived
Bump mapping changes the differential geometry
How fast the normal varies per unit length, etc
Book has nasty details for determining the modified differential geometry due to bump mapping
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Examples
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Textures (PBR Chap 11)
Read Chapter 11 for functionality (but not details)
Highlights:
Sampling and Aliasing – how to compute rate of change of surface u,v parameters with respect to image coordinates x,y
Needed for texture filtering
Generating texture coordinates
Various forms of textures
Texture maps (images)
Procedural textures of various forms
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Texture Coordinates
On the surface, we have u,v coordinates, defined as part of the shape definition
Sphere? Cylinder?
For texturing, we need s,t coordinates – not the same
There are various mappings that take (x,y,z) and convert to (s,t)
Actually, take entire differential geometry, such as normal vector and dp/du and dp/dv
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Mappings
Various formulas to convert point or surface parameterization to texture coords
All take a transformation to “orient” the texture
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Texture Types
Constant value, Scale: t1 * t2, Mix: (1-a)*t1+a*t2
Use these for texture blends
Bilinear: Bilinearly interpolate from four constant values
What type of shading interpolation does this perform?
Images
Lots of details about caching and mipmapping – ignore them
Procedural textures:
Program computes values – each texture must be a separate class
Perlin noise
Core functionality
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Texture Antialiasing
If you define a new procedural texture, you should correctly anti-alias it
Information about ds/dx, dt/dx, ds/dy, dt/dy is available
Tells you how fast texture coordinates change on screen, which tells you how many “texels” appear in one pixel
There are many examples in the book of how this might be done, but as a fallback you can always super-sample the texture function
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Next Time
Light Sources
Direct Lighting
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