Concept: Radiometry
Why Radiometry Matters
- Path tracing computes physically correct light transport
- Without understanding radiometric units, you’ll get factor-of-π errors and wrong energy conservation
- Every quantity in the rendering equation has a specific unit — know them
Radiometric Quantities
- Radiant energy
Q — total energy of light (joules, J)
- Radiant flux
Φ — power, energy per unit time (watts, W)
- A 100W light bulb emits 100W of radiant flux (most as heat, ~5% as visible light)
- Irradiance
E — flux arriving per unit area (W/m²)
E = dΦ / dA- Depends on angle of incidence:
E = L * cos(θ) for a single beam
- This is what a solar panel measures
- Radiance
L — flux per unit area per unit solid angle (W/m²/sr)
L = d²Φ / (dA * dω * cos(θ))- The
cos(θ) accounts for projected area
- Radiance is what cameras measure — it’s the “brightness” of a ray
- Key property: radiance is constant along a ray in vacuum (no participating media)
Radiance is What Path Tracing Computes
- Each ray in a path tracer returns a radiance value
L
- The rendering equation computes outgoing radiance
L_o
- The camera integrates radiance over the pixel area and lens aperture
- Pixel value =
∫∫ L(ray(x,y)) * W(x,y) dx dy where W is the pixel filter
Relationship Between Quantities
- Irradiance from radiance:
E(x) = ∫_Ω L(x, ω) cos(θ) dω
- Integrate radiance over the hemisphere, weighted by cosine
- This is the integral in the rendering equation
- Radiance from irradiance:
L = dE / (dω * cos(θ))
- For a Lambertian emitter:
L_e = M / π where M is exitance (W/m²)
- Exitance
M = total power emitted per unit area
- The
1/π comes from integrating over the hemisphere
Spectral Quantities
- All quantities above have spectral versions (per wavelength)
- Spectral radiance:
L_λ(λ) in W/m²/sr/nm
- RGB rendering: approximate by sampling at 3 wavelengths (R≈700nm, G≈546nm, B≈436nm)
- Spectral rendering: sample many wavelengths, reconstruct color at end
Common Mistakes
- Confusing radiance and irradiance
- Radiance: per solid angle (directional)
- Irradiance: integrated over hemisphere (omnidirectional)
- Missing the
cos(θ) factor
E = L * cos(θ) — a surface tilted away from a light receives less irradiance- This is Lambert’s cosine law — it’s a geometric fact, not a material property
- Wrong units for emissive materials
- If you want a light to emit
X watts total: L_e = X / (area * π) for Lambertian emitter
- If you set
L_e = X directly: the total emitted power scales with area