Lesson 15

Physically based shading

Microfacet BRDFs in practice: GGX, Smith and Schlick's Fresnel, the metalness and roughness workflow, energy conservation checked in a white furnace, and image-based lighting with the split-sum approximation. Spheres on your GPU, with the BRDF's lobe and energy computed beside them.

The last lesson added up lights with plain diffuse shading: albedo × light × cos θ. Real surfaces also reflect, and how sharply depends on how rough they are at a scale far below a pixel. Physically based shading is the set of conventions most engines settled on around 2012–2014 to model that: a microfacet BRDF with a handful of inputs that mean the same thing under any lighting. This lesson follows the model Brian Karis described for Unreal Engine 4 in Real Shading in Unreal Engine 4 (SIGGRAPH 2013), which built on Brent Burley’s Physically Based Shading at Disney (2012).

Loading…

The grid has roughness from 0 on the left to 1 on the right, a dielectric row above a metal row; One sphere uses the material’s own sliders. Light comes from a sun (the concept lessons’ light direction) and from an environment: a sky, a darker ground and two bright windows, which the smooth spheres reflect. Beside the frame, computed on the CPU with the same formulas as the shader, one plot at a time: the specular lobe for the current material, and a white furnace chart, explained below.

The microfacet BRDF

A BRDF, f(l, v), says how much of the light arriving from direction l leaves towards v. The light a pixel sees is the sum over all incoming light of f × incoming radiance × cos θ_l. Microfacet theory models a rough surface as many tiny perfect mirrors, and the specular BRDF as

f_spec = D(h) F(v, h) G(l, v) / (4 (n·l)(n·v))

where h is the half vector between l and v: only microfacets facing along h reflect l into v.

D, the normal distribution: how many microfacets face along h. UE4 uses GGX (Walter et al., 2007, the same function as Trowbridge and Reitz’s from 1975): D = α² / (π ((n·h)²(α² − 1) + 1)²). Its long tail gives highlights a soft glow that older Blinn-Phong lobes lack. Following Burley, α = roughness², which Burley found gives “a more perceptually linear change in the roughness”. D is normalised: ∫ D(h)(n·h) dω = 1, which the lesson’s test checks numerically for every roughness.
G, geometry or shadowing-masking: the share of those microfacets that aren’t hidden from l or v by their neighbours. UE4 uses Schlick’s approximation of Smith’s function, G₁(x) = n·x / (n·x (1 − k) + k), once for l and once for v (the separable form, G₁(l) G₁(v); later engines use Heitz’s height-correlated Smith, which couples the two), with k = α / 2 for the environment and k = (roughness + 1)² / 8 for lights: that’s Disney’s modification, which Karis adopted “to reduce ‘hotness’” of smooth surfaces, remapping roughness to (roughness + 1) / 2 before squaring.
F, Fresnel: how much light reflects rather than refracts, which rises towards 1 at grazing angles. Schlick’s approximation: F = F₀ + (1 − F₀)(1 − v·h)⁵, with F₀ the reflectance looking straight on. (Karis replaces the fifth power with a spherical Gaussian approximation; the widget uses the power.)

The diffuse part is Lambert: f_diff = albedo / π. Karis tried Burley’s diffuse and “saw only minor differences”.

View angle in the widget’s side panel moves the viewer; the lobe shows f_spec cos θ_l for light from every direction in the plane. Smooth surfaces give a narrow, tall lobe around the mirror direction; rough ones a wide, low one. Near grazing the lobe grows and tilts towards the horizon: that’s Fresnel, and the shadowing term pulling against it.

Metalness and roughness

The inputs are few on purpose: base colour, metalness, roughness, and for non-metals a reflectance.

A dielectric (plastic, wood, stone, skin) reflects a few per cent of light at its surface, the same for every colour, and scatters the rest under the surface, which comes back out tinted: that’s the diffuse. Naty Hoffman’s course notes put most dielectrics’ F₀ “between 2% and 5%”; water’s is 0.02, diamond’s 0.17. Base colour is the diffuse albedo, and F₀ defaults to 0.04 (UE4’s Specular of 0.5).
A metal has no diffuse: light that enters is absorbed. Its reflection is coloured and strong. In Hoffman’s table, linear F₀ is (1.00, 0.71, 0.29) for gold, (0.95, 0.64, 0.54) for copper and (0.56, 0.57, 0.58) for iron. Base colour becomes F₀.

So the shader computes F₀ = lerp(reflectance, base colour, metalness) and diffuse albedo = base colour × (1 − metalness). Metalness in between is for transitions in a texture (rust on metal, dust on paint). Asked whether metallic is binary, Karis found it “best to just say ‘Yes!’”. The Gold, Copper, Iron and Silver presets use Hoffman’s values; switch the view to Specular and Diffuse to see that metals have no diffuse at all.

Energy conservation and the white furnace

A surface lit equally from every direction by white light, inside a white furnace, should reflect at most what arrives: an albedo of 1 or less. Lambert’s 1 / π is what makes that work: ∫ (1 / π) cos θ dω = 1, so a white diffuse surface reflects exactly what it receives.

The furnace chart (choose White furnace beside the frame) integrates the specular BRDF the same way, on the CPU, with the 1,024 samples per entry Karis’s code uses. With F = 1 (blue line), a perfect mirror’s microfacets would return everything. GGX with the Smith term doesn’t: light that bounces off one microfacet into another is counted as lost, so rough surfaces come out too dark. In this model, looking straight on, the specular lobe reflects 100% at roughness 0, 90% at roughness 0.5 and only 31% at roughness 1. Kulla and Conty’s Revisiting Physically Based Shading at Imageworks (2017) add back the missing multiple scattering with a small extra term, read from a table like the one below.

The gold line is a white dielectric: Lambert diffuse plus the 0.04 specular, added together as UE4’s model does. It reflects 1.04 times the light that arrives at normal incidence, and up to 1.59 times at a grazing view (n·v = 0.1) when it’s smooth: the specular adds energy the diffuse never gave up. Some engines scale the diffuse down by (1 − F), or by one minus the specular’s albedo, to keep the total under 1. Turn the view angle up to see the gap open.

Image-based lighting and the split sum

A sun is one direction; the environment is light from all of them. Lighting with it means integrating the BRDF against the whole environment for every pixel, which is too slow to do by brute force. Karis’s split-sum approximation splits the integral into two factors that can each be precomputed:

The environment, prefiltered for each roughness: the light the lobe would gather, assuming the view looks straight along the normal (n = v = r). The widget stores it as an equirectangular map, 64 × 32 texels per level, for roughness 0, 0.25, 0.5, 0.75 and 1, and blends between levels. A real engine uses a cube map’s mip chain, one roughness per mip.
The BRDF integrated against white light, which depends only on roughness and n·v, so it fits in a 2D table. Fresnel’s F₀ factors out, leaving a scale and a bias: the integral is F₀ · A + B. The widget’s table is 32 × 32; at roughness 0.5 and n·v = 0.5, A is 0.727 and B is 0.019.

The shader multiplies the two. Karis notes it’s “exact for a constant L_i(l)”: in the white furnace, the table is the exact answer, and that’s how the furnace chart is computed. The error comes from the n = v = r assumption, which loses the stretched reflections you see on wet roads at grazing angles. Diffuse uses a third table: the environment convolved with a cosine, one more band of the same texture.

Untick Sun to see the environment’s contribution alone, and untick Environment to see how flat the spheres are with one light.

Takeaways

The specular BRDF is D F G / (4 (n·l)(n·v)): GGX for D, Smith (in Schlick’s form) for G, Schlick for F.
Metalness picks between a dielectric (diffuse = base colour, F₀ about 0.04) and a metal (no diffuse, F₀ = base colour). Roughness squared is GGX’s α.
Energy conservation is a test you can run: Lambert passes the white furnace exactly; single-scattering GGX loses energy at high roughness (31% left at roughness 1 here), and adding diffuse and specular without trading them off gains it.
Image-based lighting with the split sum: a prefiltered environment per roughness, times F₀ · A + B from a 2D table.