Vaibhav Rathod's Notes

PathTracer Learning - Path Tracing Algorithm

4 min read

  • Path Tracing Algorithm

  • The Rendering Equation

    • L_o(x, ω_o) = L_e(x, ω_o) + ∫_Ω f_r(x, ω_i, ω_o) L_i(x, ω_i) (N · ω_i) dω_i
    • This is an integral equation — L_i depends on L_o at other points
    • Path tracing solves it by Monte Carlo integration
      • Sample a random direction ω_i from the hemisphere
      • Trace a ray in that direction to find L_i
      • Recursively evaluate the rendering equation at the hit point
      • Average many such estimates to converge to the true integral

  • Path Tracing vs Ray Tracing vs Whitted RT

    • Whitted ray tracing (1980)
      • Only traces specular reflections and refractions
      • Direct illumination only (shadow rays to lights)
      • Fast but physically incorrect for diffuse interreflection
    • Distributed ray tracing (Cook 1984)
      • Stochastic sampling for soft shadows, glossy reflections, DOF
      • Still limited — no full global illumination
    • Path tracing (Kajiya 1986)
      • Traces complete light paths from camera to light
      • Handles all light transport: diffuse, specular, caustics, subsurface
      • Unbiased — converges to ground truth with enough samples

  • The Algorithm Step by Step

    • Step 1: Generate camera ray
      • For pixel (i, j), compute ray direction through the pixel center (or jittered for AA)
      • ray.origin = camera.position
      • ray.direction = normalize(pixel_world_pos - camera.position)
      • See PathTracer Learning - Concept - Camera Model
    • Step 2: Find closest intersection
      • Traverse BVH to find nearest triangle hit
      • If no hit → return sky/environment radiance (sample env map)
      • If hit emissive surface → return emission (with MIS weight if applicable)
    • Step 3: Russian roulette (path termination)
      • Compute survival probability q = max(throughput.r, throughput.g, throughput.b)
      • Clamp: q = clamp(q, 0.05, 0.95)
      • If random() > q → return 0 (path terminated)
      • Otherwise divide contribution by q to maintain unbiasedness
    • Step 4: Sample new direction
      • Sample ω_i from the BRDF’s importance sampling distribution
      • Compute the PDF p(ω_i) for the sampled direction
      • Transform from tangent space to world space using TBN matrix
    • Step 5: Evaluate BRDF
      • f_r = brdf.eval(ω_o, ω_i, N, material)
      • Compute cos(θ) = max(0, dot(N, ω_i))
    • Step 6: NEE — Direct lighting
    • Step 7: Recurse
      • L_i = trace(new_ray, depth + 1)
      • contribution = f_r * L_i * cos(θ) / (p(ω_i) * q)
      • Apply MIS weight for BRDF sample: w_brdf = p_brdf² / (p_brdf² + p_light²)

  • Throughput Tracking

    • Instead of recursion, track a running “throughput” multiplier
    • More efficient — avoids deep call stacks

  • Variance and Convergence

    • Each pixel estimate has variance σ²
    • After N samples: variance = σ²/N, std dev = σ/√N
    • To halve noise: need 4× more samples
    • Variance reduction techniques
      • Importance sampling — reduce σ² by sampling proportional to integrand
      • Next event estimation — directly sample lights (huge variance reduction)
      • MIS — combine multiple strategies optimally
      • PathTracer Learning - ReSTIR — reuse samples across pixels and frames
      • Path guiding — learn the light distribution

  • Bias vs Variance

    • Unbiased estimator: expected value = true value (no systematic error)
    • Biased estimator: expected value ≠ true value (systematic error)
    • Path tracing is unbiased (with Russian roulette)
    • Clamping radiance introduces bias (but reduces fireflies)
    • Denoising introduces bias (but dramatically reduces variance)
    • In practice: biased but low-variance is often preferred for real-time

  • Fireflies

    • Extremely bright pixels from rare high-energy paths
    • Cause: f(x) / p(x) is very large for some samples
    • Mitigation strategies
      • Clamp radiance: L = min(L, max_radiance) — introduces bias
      • Better importance sampling: reduce variance at the source
      • MIS: prevents double-counting which amplifies fireflies
      • Firefly rejection: discard samples that deviate too far from neighbors

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