02/2/05© 2005 University of Wisconsin Last Time Reflectance part 1 –Radiometry –Lambertian –Specular

02/2/05© 2005 University of Wisconsin Today Microfacet models –Diffuse Oren-Nayar –Specular Torrance-Sparrow –Blinn –Ashikhmin-Shirley –Ward –Schlick Lafortune’s model Glossy over Diffuse

02/2/05© 2005 University of Wisconsin Microfacet Models (PBR 9.4) Model fine detail as set of polygonal facets –Metals –Minerals –Things that solidified in crystal form, or were broken/cut by fracture/scraping Aim to capture the macroscopic effects of the many microscopic facets

02/2/05© 2005 University of Wisconsin Describing Microfacet Materials Surface normal distribution –How the surface normals of the facets are distributed about the macroscopic normal Facet BRDF –Are the facets diffuse or specular?

02/2/05© 2005 University of Wisconsin Microscopic Effects Masking – viewer can’t see a microfacet Shadowing – light can’t see a microfacet Interreflection – light off one facet hits another Aim is to capture these effects as efficiently as possible

02/2/05© 2005 University of Wisconsin Oren-Nayar (PBR 9.4.1) Model facet distribution as Gaussian with s.d.  (in radians) Facet BRDF is Lambertian Resulting model has no closed form solution, but a good approximation Sample using cosine-weighted sampling in hemisphere

02/2/05© 2005 University of Wisconsin Oren-Nayar Effects LambertianOren-Nayar

02/2/05© 2005 University of Wisconsin Torrance-Sparrow (PBR Sect 9.4.2) Specular BRDF for facets Arbitrary (in theory) distribution of facet normals Additional term for masking and shadowing ii n hh oo Half vector – facet orientation to get specular transfer

02/2/05© 2005 University of Wisconsin Torrance-Sparrow BRDF D(  h ) is the microfacet orientation distribution evaluated for the half angle –Changing this changes the surface appearance – but this equation doesn’t depend on the choice F r (  o ) is the Fresnel reflection coefficient

02/2/05© 2005 University of Wisconsin Geometry Term Masking: Shadowing: Together:

02/2/05© 2005 University of Wisconsin Blinn’s Microfacet Distribution Parameter e controls “roughness”

02/2/05© 2005 University of Wisconsin Sampling Blinn’s Microfacet (PBR ) Sampling from a Microfacet BRDF tries to account for all the terms: G, D, F, cos But D provides most variation, so sample according to D The sampled direction is completely determined by halfway vector,  h, so sample that –Then construct reflection ray based upon it So how do we sample such a direction …

02/2/05© 2005 University of Wisconsin More Blinn Sampling Need to sample spherical coords: ,  Book has details, and probably an error on page 684 Complication: We need to return the probability of choosing  i, but we have the probability of choosing  h –Simple conversion term We need to construct the reflection direction about an arbitrary vector …

02/2/05© 2005 University of Wisconsin Arbitrary Reflection Coordinate system is not nicely aligned, so use construction

02/2/05© 2005 University of Wisconsin Anisotropic Microfacet Distribution Parameters for x and y direction roughness, where x and y are the local BRDF coordinate system on the surface –Gives the reference frame for 

02/2/05© 2005 University of Wisconsin Sampling Anisotropic Microfacet Sampling is discussed in PBR Sect – similar to Blinn but with different distribution, and probably not quite right –Note that there are 4 symmetric quadrants in the tangent plane –Sample in a single quadrant, then map to one of 4 quadrants –Take care to maintain stratification 01 1st2nd3rd4th

02/2/05© 2005 University of Wisconsin Ward’s Isotropic Model “the simplest empirical formula that will do the job” Leaves out the geometry and Fresnel terms –Makes integration and sampling easier 3 terms, plus some angular values: –  d is the diffuse reflectance –  s is the specular reflectance –  is the standard deviation of the micro-surface slope

02/2/05© 2005 University of Wisconsin Ward’s Anisotropic Model For surfaces with oriented grooves 2 terms for anisotropy: –  x is the standard deviation of the surface slope in the x direction –  y is the standard deviation of the surface slope in the y direction

02/2/05© 2005 University of Wisconsin Sampling Ward’s Model Take  1 and  2 and transform to get  h and  h : Only samples one quadrant, use same trick as before to get all quadrants Not sure about correct normalization constant for solid angle measure

02/2/05© 2005 University of Wisconsin Schlick’s Model (Schlick94) Empirical model well suited to sampling Two parameters: – , a roughness factor (0 = Specular, 1 = Lambertian) – , an anisotropy term, (0 perfectly anisotropic, 1 = isotropic)

02/2/05© 2005 University of Wisconsin Schlick’s Model Facet Distribution: Geometry Terms:

02/2/05© 2005 University of Wisconsin Putting it Together Term to account for inter-reflection Not a Torrance-Sparrow model As before, sample a half vector: –Only samples in 1 quadrant Use trick from before –Normalization not given

02/2/05© 2005 University of Wisconsin More to it than that Both Ward and Schlick’s original papers define complete reflectance, including diffuse and pure specular components PBR calls these materials, because they are simply linear sums of individual components Schlick’s paper also includes a way to decide how to combine the diffuse, specular and glossy terms based on the roughness Both Ward and Schlick discuss sampling from the complete distribution

02/2/05© 2005 University of Wisconsin Phong Revisited The Phong Specularity model can be revised to make it physically viable – energy conserving and reciprocal In canonical BRDF coordinate system (z axis is normal)

02/2/05© 2005 University of Wisconsin Oriented Phong Define an orientation vector – the direction in which the Phong reflection is strongest For standard Phong, o=(-1,-1,1) To get “off specular” reflection, change o –Can get retro-reflection, more reflection at grazing, etc.

02/2/05© 2005 University of Wisconsin Lafortune’s Model (PBR 9.5) A diffuse component plus a sum of Phong lobes Allow all parameters to vary with wavelength Lots of parameters, 12 for each lobe, so suited for fitting to data –It’s reasonably easy to fit –Parameters for many surfaces are available

02/2/05© 2005 University of Wisconsin Lafortune’s Clay

02/2/05© 2005 University of Wisconsin Sampling From Lafortune First choose a lobe (or diffuse) –Could be proportional to lobe’s contribution to outgoing direction –But that might be expensive Then sample a direction according to that lobe’s distribution –Just like sampling from Blinn’s microfacet distribution, but sampling the direction directly

02/2/05© 2005 University of Wisconsin Two-Layer Models (PBR 9.6 and )) Captures the effects of a thin glossy layer over a diffuse substrate –Common in practice – polished painted surfaces, polished wood, … Glossy dominates at grazing angles, diffuse dominates at near-normal angles –Don’t need to trace rays through specular surface to hit diffuse

02/2/05© 2005 University of Wisconsin Fresnel Blend Model

02/2/05© 2005 University of Wisconsin Next Time Some specialized reflectance models Light sources Then we can start making pictures