Code Notes made by Vaibhav Rathod

PathTracer Learning - Concept - Environment Map

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Written by Vaibhav Rathod

Concept: Environment Map (IBL)

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 
    // 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:
    • 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 
    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

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