Rendering Optics
Solving the physics of light transport—hardware rasterization pipelines, logarithmic BVH ray traversal, James Kajiya's Rendering Equation, Multiple Importance Sampling (MIS), and real-time ReSTIR global illumination.
Indexed Rendering Paradigms (8)
Rasterization vs. Stochastic Path TracingRasterization Pipeline & The Z-Buffer
The workhorse of interactive 3D graphics for 40+ years. Projects 3D triangles onto a 2D screen grid and resolves visibility via a per-pixel depth buffer in sub-millisecond time.
\mathbf{p}_{\text{screen}} = \mathbf{M}_{\text{proj}} \cdot \mathbf{M}_{\text{view}} \cdot \mathbf{M}_{\text{world}} \cdot \mathbf{p}_{\text{local}}, \quad \text{Depth Test: } z_{\text{new}} < z_{\text{buffer}}(x, y)Bounding Volume Hierarchies (BVH & SAH)
The tree structure that makes ray tracing fast. Organizes millions of triangles into nested bounding boxes, allowing rays to bypass 99.9% of scene geometry with logarithmic efficiency.
C_{\text{split}} = C_t + \frac{\text{SA}(L)}{\text{SA}(P)} N_L C_i + \frac{\text{SA}(R)}{\text{SA}(P)} N_R C_i \quad \text{(Surface Area Heuristic)}Whitted Classical Recursive Ray Tracing
Turner Whitted's 1980 breakthrough. Casts primary rays from camera, shadow rays toward lights, and recursive reflection/refraction rays to render flawless mirrors and glass.
I = I_{\text{local}} + k_r I_{\text{reflected}} + k_t I_{\text{refracted}} \quad \text{(evaluated recursively along specular directions)}The Rendering Equation (Kajiya 1986)
The grand unified equation of computer graphics. James Kajiya's 1986 formulation states that outgoing light is the sum of emitted light and the integral of all incoming light reflected across the hemisphere.
L_o(\mathbf{x}, \vec{\omega}_o) = L_e(\mathbf{x}, \vec{\omega}_o) + \int_{\Omega} f_r(\mathbf{x}, \vec{\omega}_i, \vec{\omega}_o) L_i(\mathbf{x}, \vec{\omega}_i) (\mathbf{n} \cdot \vec{\omega}_i) d\vec{\omega}_iMonte Carlo Path Tracing & Russian Roulette
Solves Kajiya's integral by tracing random stochastic photon paths through virtual scenes. Russian roulette statistically terminates low-energy paths without introducing bias.
\langle L_o \rangle = \frac{1}{N} \sum_{k=1}^N \frac{f_r(\vec{\omega}_k) L_i(\vec{\omega}_k) (\mathbf{n} \cdot \vec{\omega}_k)}{p(\vec{\omega}_k)}, \quad \text{Survival Prob: } q = \min(1, \max(\text{throughput}))Multiple Importance Sampling (MIS)
Erich Veach & Leonidas Guibas' 1995 masterpiece. Combines light source sampling with BSDF surface sampling, eliminating firefly artifacts and drastically reducing render noise.
w_i(\mathbf{x}) = \frac{p_i(\mathbf{x})^\beta}{\sum_j p_j(\mathbf{x})^\beta} \quad \text{(Power Heuristic, } \beta=2\text{)}Next Event Estimation (NEE / Direct Lighting)
Explicitly casts shadow rays to light sources at every bounce rather than waiting for random rays to hit a tiny bulb by chance, accelerating convergence by orders of magnitude.
L_o = L_e + \sum_{\text{lights}} \frac{f_r(\vec{\omega}_l) L_{\text{light}} V(\mathbf{x} \leftrightarrow \mathbf{y}) (\mathbf{n} \cdot \vec{\omega}_l) (\mathbf{n}_l \cdot -\vec{\omega}_l)}{p_l(\mathbf{y}) \|\mathbf{x}-\mathbf{y}\|^2} + \text{Indirect Bounce}ReSTIR & Real-Time Global Illumination
The modern holy grail of gaming graphics. ReSTIR shares and re-weights light candidate samples across neighboring pixels and past frames, bringing cinematic path tracing to real-time 60 FPS gameplay.
W = \frac{1}{M} \sum_{i=1}^M w_i, \quad \text{Spatial-Temporal Resampling: } \mathcal{R}_{\text{new}} = \text{Merge}(\mathcal{R}_{t-1}, \mathcal{R}_{\text{spatial}})