SLIDE 1
02941 Physically Based Rendering
Path Tracing
Jeppe Revall Frisvad June 2020
SLIDE 2 Heckbert’s Light Transport Notation
◮ Path tracing includes all light paths: L(S|D)∗E .
References:
- Heckbert, P. S. Adaptive radiosity textures for bidirectional ray tracing. Computer Graphics (Proceedings of ACM SIGGRAPH 90) 24(4),
- pp. 145–154, August 1990.
SLIDE 3 Progressive unidirectional path tracing
- 1. Generate rays from the camera through pixel positions.
- 2. Trace the rays and evaluate the rendering equation for each ray.
- 3. Randomize the position within the pixel area to Monte Carlo integrate
(measure) the radiance arriving in a pixel.
p0 p1 p2 p3
http://www.pbr-book.org/
◮ Noise is reduced by progressive updates
◮ Update the rendering result in a pixel Lj after rendering a new frame with result Lnew using Lj+1 = Lnew + jLj j + 1 . ◮ Progressive (stop and go) rendering is convenient for several reasons:
◮ No need to start over. ◮ Result can be stored and refined later if need be. ◮ Convergence can be inspected during progressive updates.
SLIDE 4 Monte Carlo integration
◮ The rendering equation: Lo(x, ω) = Le(x, ω) +
fr(x, ω′, ω)Li(x, ω′) cos θ dω′ . ◮ The Monte Carlo estimator: LN(x, ω) = Le(x, ω) + 1 N
N
fr(x, ω′
j,
ω)Li(x, ω′
j) cos θ
pdf( ω′
j)
with cos θ = ω′
j ·
n, where n is the surface normal at x.
SLIDE 5
Splitting the evaluation
◮ Distinguishing between:
◮ Direct illumination Ldirect.
◮ Light reaching a surface directly from the source.
◮ Indirect illumination Lindirect.
◮ Light reaching a surface after at least one bounce.
◮ The rendering equation is then L = Le + Ldirect + Lindirect .
◮ Le is emission. ◮ Ldirect is sampling of lights. ◮ Lindirect is sampling of the BRDF excluding lights.
SLIDE 6 Path tracing diffuse objects
◮ The diffuse BRDF: fr = ρd/π . ◮ Computing direct illumination:
◮ Sample positions uniformly on light sources. ◮ Estimator for Ldirect is Ldirect,N = ρd(x) π 1 N
N
Le(xℓ,j, − ω′
j)V (x, xℓ,j)
ω′
j)
xℓ,j − x2 Aℓ ( ω′
j ·
n ) .
◮ Computing indirect illumination:
◮ Set Le = 0 . ◮ Sample directions using cosine-weighted hemisphere: pdf( ω′
j) = cos θ/π .
◮ Estimator for Lindirect is Lindirect,N = ρd(x) 1 N
N
V ( ω′
j)Li(x,
ω′
j) .
SLIDE 7
Example
◮ 100 samples per pixel. ◮ Time: c. 3 minutes and 39 seconds.
SLIDE 8
Example
◮ 100 samples per pixel. ◮ Split into 5 samples at first diffuse surface. ◮ Time: c. 8 minutes and 41 seconds.
SLIDE 9
Example
◮ 100 samples per pixel. ◮ Split into 5 samples at first diffuse surface. ◮ Russian roulette for remaining bounces. ◮ Time: c. 43 minutes.
SLIDE 10 Splitting vs. Russian roulette for diffuse objects
◮ Splitting is very expensive. Russian roulette is very noisy. ◮ A compromise: Final gathering. ◮ Final gathering: Trace from the eye. Split at the first diffuse surface encountered. Russian roulette afterwards. ◮ Russian roulette: Either diffuse reflection or absorption. ◮ What is the probability? The simplest idea is the average: probability of diffuse reflection = ρd,R + ρd,G + ρd,B 3 . ◮ Could we importance sample this probability? Do some colours have more importance than
◮ The eye is more sensitive to some colours compared to others. ◮ This is estimated by luminance (photometric radiance). ◮ The luminance of an RGB colour is a weighted average of R, G, and B. The weights depend on the RGB colour space used. ◮ Luminance according to the NTSC (1953) colour space: Y = 0.2989R + 0.5866G + 0.1145B .
SLIDE 11
Path tracing specular objects
◮ See slides on Reflection and Transmission.
SLIDE 12
Example
◮ 288 samples per pixel. ◮ Split into 5 samples at first diffuse surface. ◮ Split in 2 at first 2 specular surfaces. ◮ Russian roulette for remaining bounces. ◮ Time: c. 2 hours.
SLIDE 13
Exercises
◮ Implement path tracing for diffuse objects. ◮ Render the Cornell box with blocks. ◮ Render the Cornell box with specular spheres. ◮ Explain light paths leading to secondary caustics. ◮ Extended light transport notation:
L − Light E − Eye D − Diffuse surface S − Specular surface (Sr – reflection, St – transmission) ∗ − 0 or more interactions + − 1 or more interactions ? − 0 or 1 interaction | − either the path on the left or the right side All possible paths: L(D|S)∗E Caustics: LS+DS∗E Primary caustics: L(Sr|StSt)DE | LStD(St?)E
