# Dielectric Microfacet BSDF Model ## Overview This project is an implementation of a **Dielectric Microfacet Bidirectional Scattering Distribution Function (BSDF)** using the **GGX normal distribution function** for physically-based rendering. The BSDF accounts for both **ideal reflection** and **refraction** using **Fresnel equations** and supports varying roughness parameters for realistic rendering of dielectric materials such as glass.

Roughness 0.005 Roughness 0.5

Figure 1: Rendered images with Roughness values - Left: 0.005 (smooth surface), Right: 0.5 (rough surface).

## Features - **Ideal Dielectric Model**: Implements perfect reflection and transmission based on Fresnel equations. - **GGX Microfacet Distribution**: Samples microfacet normals to simulate rough surfaces. - **Roughness Parameterization**: Adjusts the model to simulate surfaces ranging from smooth to rough. - **Physically Accurate Fresnel Effect**: Computes Fresnel reflectance and transmittance for varying incident angles. - **Efficient Sampling**: Includes optimized sampling for reflection and refraction directions. ## Implementation Details ### Scene Setup - **Modified Cornell Box**: Includes a glass bunny with: - Green back wall and red floor to distinguish reflection and transmission effects. - **Index of Refraction**: Fixed at **1.5**, representative of glass. ### Key Components #### 1. Ideal Reflection and Transmission - **Reflection**: Light reflects at the same angle as incidence. - **Transmission**: Handles refraction with Snell’s law and accounts for orientation changes. - **Sampling**: Fresnel probabilities determine reflection vs transmission. ```cpp tuple sample(Vec3f wo_world, Vec3f n, Vec3f u) const { Vec3f wo = normalize(to_local(wo_world, n)); Vec3f wi = mirror_reflect(-wo, m); Vec3f wi_world = from_local(normalize(wi), n); return {wi_world, 1.0f}; } ```

Reflection Transmission

Figure 2: Ideal Reflection (left) and Ideal Transmission (right) results using the BSDF model.

#### 2. GGX Microfacet Distribution - Models the microfacet distribution using a roughness parameter `g`. - Implements **Shadow-Masking** functions to ensure energy conservation. ```cpp float D(Vec3f m) const { if (m.z < 0.0f) return 0.0f; float cos_thetaM2 = m.z * m.z; float tan_thetaM2 = (1.0f - cos_thetaM2) / cos_thetaM2; float alpha2 = roughness * roughness; float sqrt_denom = alpha2 + tan_thetaM2; return alpha2 * INV_PI / (cos_thetaM2 * cos_thetaM2 * sqrt_denom * sqrt_denom); } ``` The GGX Distribution Function determines the probability of microfacet orientations based on the roughness parameter, producing smoother or rougher surfaces depending on its value. #### 3. Fresnel Equations - Computes reflectance based on the angle of incidence and relative indices of refraction. - Determines probability of reflection vs transmission. ```cpp float Fresnel(float cos_thetaI) const { cos_thetaI = clamp(cos_thetaI, -1.0f, 1.0f); bool entering = cos_thetaI > 0.0f; float ei = eta, et = 1.0f; if (!entering) swap(ei, et); float sin_thetaT = ei / et * sqrt(max(0.0f, 1.0f - cos_thetaI * cos_thetaI)); if (sin_thetaT >= 1.0f) return 1.0f; float cos_thetaT = sqrt(max(0.0f, 1.0f - sin_thetaT * sin_thetaT)); float Rs = (et * cos_thetaI - ei * cos_thetaT) / (et * cos_thetaI + ei * cos_thetaT); float Rp = (ei * cos_thetaI - et * cos_thetaT) / (ei * cos_thetaI + et * cos_thetaT); return 0.5f * (Rs * Rs + Rp * Rp); } ``` The Fresnel equations calculate the reflectance based on the angle of incidence, producing realistic behavior where reflection increases at grazing angles and transmission dominates at normal incidence. #### 4. Shadow-Masking Functions - Ensures that microfacets are not shadowed or masked, preserving energy conservation in the model. ```cpp float G(Vec3f wo, Vec3f wi, Vec3f m) const { return G1(wo, m) * G1(wi, m); } float G1(Vec3f w, Vec3f m) const { if (dot(w, m) * w.z <= 0.0f) return 0.0f; float tan_thetaW2 = (1.0f - w.z * w.z) / (w.z * w.z); if (tan_thetaW2 <= 0.0f) return 1.0f; float tan_thetaW = sqrt(tan_thetaW2); float root = roughness * tan_thetaW; return 2.0f / (1.0f + sqrt(1.0f + root * root)); } ``` These functions account for geometric occlusion, which occurs when facets block light from reaching others. #### 5. Sampling the BSDF - The sampling function generates microfacet normals using the GGX distribution and evaluates reflection or transmission based on Fresnel probabilities. ```cpp tuple sample(Vec3f wo_world, Vec3f n, Vec3f u) const { Vec3f wo = normalize(to_local(wo_world, n)); const auto [m, pdf_m] = sample_normal(Vec2f{u[1], u[2]}); float cos_thetaO = dot(wo, m); float F = Fresnel(cos_thetaO); if (u[0] < F) { Vec3f wi = mirror_reflect(-wo, m); Vec3f wi_world = from_local(normalize(wi), n); return {wi_world, F * pdf_m}; } bool entering = wo.z > 0.0f; float ei = eta, et = 1.0f; if (!entering) swap(ei, et); float sin_thetaO2 = 1.0f - cos_thetaO * cos_thetaO; float etaEff = ei / et; float sin_thetaT2 = etaEff * etaEff * sin_thetaO2; if (sin_thetaT2 >= 1.0f) return {Vec3f{}, 0.0f}; float cos_thetaT = sqrt(1.0f - sin_thetaT2); if (entering) cos_thetaT *= -1; Vec3f wt = {etaEff * -wo.x, etaEff * -wo.y, cos_thetaT}; Vec3f wt_world = from_local(normalize(wt), n); return {wt_world, (1.0f - F) * pdf_m}; } ``` This function calculates probabilities for both reflection and transmission and determines which behavior occurs based on random sampling and Fresnel equations. ## References - **PBRT Textbook** - Concepts for BSDF and Fresnel equations. - **Course Material** - Provided project guidelines and sample implementations. ## Acknowledgments - **Professor's Guidance** - Inspired by the CS190I: Offline Rendering Final Project requirements. - **PBRT Textbook** - Key references for BSDF theory and implementations. ## License This project is licensed under the **MIT License**.