Concept: Environment Map (IBL)

• Parent: PathTracer Learning - Phase 2 - CPU Ray Tracing

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
  • 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

  // Compute luminance for each texel
  for (int y = 0; y < height; y++) {
      float sin_theta = sin(π * (y + 0.5) / height);  // solid angle weight
      for (int x = 0; x < width; x++) {
          float lum = luminance(hdr[y * width + x]) * sin_theta;
          pdf_2d[y][x] = lum;
      }
  }
  // Build marginal CDF (over rows) and conditional CDFs (per row)

IBL (Image-Based Lighting) in PBR

• Split-sum approximation (Karis 2013, used in Unreal Engine)
  • Precompute two terms separately:
  • 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

  layout(location=0) rayPayloadInEXT vec3 payload;
  void main() {
      vec2 uv = directionToEquirectangular(gl_WorldRayDirectionEXT);
      payload = texture(envMap, uv).rgb;
  }

• 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

Related

• PathTracer Learning - Concept - Importance Sampling
• PathTracer Learning - Concept - MIS
• PathTracer Learning - Concept - Tone Mapping