Vaibhav Rathod's Notes

PathTracer Learning - Concept - Environment Map

3 min read

  • What Is an Environment Map?

    • A texture that represents the radiance arriving from all directions in the scene
    • Typically an HDR (high dynamic range) image — captures real-world lighting
    • Used as the “sky” and distant lighting in a path tracer
    • When a ray misses all geometry: sample the environment map

  • Formats

    • Equirectangular (lat-long)
      • Maps the sphere to a 2:1 rectangle
      • u = (atan2(dir.z, dir.x) / (2π)) + 0.5
      • v = acos(dir.y) / π
      • Simple but has pole distortion (oversampled at poles)
    • Cubemap
      • 6 square faces (±X, ±Y, ±Z)
      • No pole distortion, hardware-accelerated sampling
      • Harder to importance sample
    • Octahedral
      • Maps sphere to a square via octahedral projection
      • No distortion, easy to importance sample
      • Used in some modern engines

  • Sampling the Environment Map

    • Naive: sample random direction, look up environment map
      • PDF: p(ω) = 1 / (4π) (uniform sphere)
      • High variance — most directions contribute little
    • Importance sampling: sample proportional to luminance
      • Build a 2D CDF from the luminance of each texel
      • Sample row (θ) then column (φ) using inverse CDF
      • PDF: p(ω) = L(ω) / ∫L(ω)dω — proportional to luminance
      • Dramatically reduces variance for scenes with bright sun/sky
    • Building the 2D CDF

  • IBL (Image-Based Lighting) in PBR

    • Split-sum approximation (Karis 2013, used in Unreal Engine)
      • Precompute two terms separately:
      • L_IBL = ∫ f_r(ω_i, ω_o) L_i(ω_i) cos(θ_i) dω_i
      • ≈ (∫ L_i(ω_i) D(ω_i, roughness) dω_i) * (∫ f_r(ω_i, ω_o) cos(θ_i) dω_i)
      • Term 1: pre-filtered environment map (one mip per roughness level)
      • Term 2: BRDF integration LUT (precomputed 2D texture)
    • Pre-filtered environment map
      • Convolve environment map with GGX NDF at each roughness level
      • Store in mip chain: mip 0 = sharp (roughness=0), mip N = blurry (roughness=1)
      • Sample: textureLod(envMap, reflect(-V, N), roughness * MAX_MIP)
    • BRDF integration LUT
      • 2D texture indexed by (NdotV, roughness)
      • Stores (scale, bias) for Schlick Fresnel: F_approx = F0 * scale + bias
      • Precomputed once, reused for all materials
    • This is an approximation — path tracing with environment map sampling is exact

  • In a Path Tracer

    • Miss shader: sample environment map at ray direction
    • NEE with environment map: sample a direction from the env map CDF
      • Compute PDF in solid angle: p(ω) = luminance(L(ω)) / total_luminance * (W*H) / (2π²*sin(θ))
      • Trace shadow ray in that direction
      • Apply MIS weight: w = p_env² / (p_env² + p_brdf²)

  • HDR File Formats

    • .hdr (Radiance RGBE format) — most common, 8-bit mantissa + 8-bit exponent
    • .exr (OpenEXR) — 16-bit or 32-bit float, lossless, industry standard
    • .dds with BC6H compression — GPU-compressed HDR, used in games
    • Loading in C++: stb_image.h for .hdr, tinyexr for .exr

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