31:T516,It indicates how radiance towards the viewer/eye/direction of interest from a point on a surface ${tex`L_o\left(\mathrm p, \omega_o\right)`} is the sum of the radiance emitted from the surface at the point ${tex`L_e\left(\mathrm p, \omega_o\right)`} and the reflected light ${tex`L_r\left(\mathrm p, \omega_o\right)`} from elsewhere in the scene. From our discussion of BRDFs, we know the reflected light is composed of the projected incoming radiance ${tex`L_i\left(\mathrm p, \omega_i\left(\mathrm{\mathbf{n} \cdot \omega_i}\right)\right)`} modulated and scattered towards ${tex`\omega_o`} via the BRDF ${tex`f_r(\mathrm{p}, \omega_o, \omega_i)`}; the radiance is gathered from the hemisphere above the point ${tex`\Omega^+`} (for translucent objects, we will need to gather luminance from both hemispheres). This equation has two unknowns: the outgoing radiance are we solving for, and the incoming radiance from the scene. We can express the incoming radiance as a function of outgoing radiance by noting that, so long as we do not lose energy during transport, the incoming radiance from ${tex`\omega_i`} is the same as the outgoing radiance from another surface going the opposite direction ${tex`-\omega_i`}; this is a good assumption if we do not have volumetric scattering such as from smoke:32:T77e,The ray tracing function ${tex`t\left(\mathrm p, \omega\right)`} handles both the visibility test for lights and gathering energy from the specular (towards ${tex`\omega_r`}) and transmissive (towards ${tex`\omega_t`}) bounce. If the traced ray for a light hits an object before it hits the light, then no light energy is returned. The last two terms trace rays into the scene for more diffuse/specular/transmissive interactions; the direct lighting and specular/transmissive bounces are calculated at the new location based upon even further bounces—ray tracing is naturally recursive. The specular/transmissive bounces are ideal interactions (there are ray tracing formulations which use glossy/rough bounces instead); in addition, in the classical approach, ray tracing does not add radiance from direct specular illumination. Care must be taken in a few cases for ray tracing: - If the number of bounces becomes to long, the ray tracer can blow the stack. As each bounce adds less and less energy, one can terminate after a set number of bounces, when the accumulated attenuation becomes larger than some set factor, or probabilistically via a Russian roulette method. - Classical ray tracing is designed for scenes with only distant environmental or directional lighting. If local sources are used (such as point, spot, or area lights), a direct specular component should be added at each surface. In this case, if a recursive ray hits a light directly, its contribution should not be added (as it has already been accounted for by direct lighting). - Rays that exit the scene without hitting an object should either add energy from an environment map or surrounding direct light, if any. Since this and the methods below gather radiance from both lights *and* objects, they are called global illumination methods whereas the direct illumination methods are only local illumination.33:T4b6,Path tracing, introduced by Kajiya in his Rendering Equation paper, generates path samples and combines them together to compose an image. Because the the possible variance in the scene, **many** such paths per pixel are required, though methods such as importance sampled Monte Carlo approaches can speed the calculation and improve visual fidelity. To generate a path, one first shoots a ray from the pixel. At the first object, determine its material properties from the BRDF. The BRDF should indicate whether it is diffuse, specular, or transmissive; it possible to be all three. Randomly select which of the three cases this path follows (using the Fresnel factor as the probability of a specular/transmissive interaction is common); a random location above the hemisphere is chosen for the bounce and the process repeats. Note, for pure specular/transmissive bounces, the "random" location has to be the only angle that reflects. Finally, if the ray hits a light, the emission is propagated back up the ray. For rays that exit the scene or bounce too much, a random light is selected and (so long as it is visible), it is treated as the last light hit. Another ray is chosen and the process repeats.34:[] 2f:{"title":"Lighting & Shading IV: Light Transport","cells":[{"id":0,"value":"# Lighting & Shading IV: Light Transport\n\nNow that we have models for materials and lights, we can put these together with our understanding of radiometry to render images.","pinned":false,"hidden":false,"mode":"md"},{"id":9,"value":"toc({\n headers: \"h2,h3,h4\",\n title: null\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":12,"value":"## The Rendering Equation\n\nReturning to our energy balance equation for light, we have:","pinned":false,"hidden":false,"mode":"md"},{"id":14,"value":"tex.block`\\begin{aligned}L_o\\left(\\mathrm{p}, \\omega_o\\right) &= L_e\\left(\\mathrm{p}, \\omega_o\\right) + L_r\\left(\\mathrm{p}, \\omega_o\\right)\\\\&= L_e\\left(\\mathrm{p}, \\omega_o\\right) \n + \\int_{\\Omega^+} f_r(\\mathrm{p}, \\omega_o, \\omega_i) L_i\\left(\\mathrm p, \\omega_i\\right)\\left(\\mathrm{\\mathbf{n} \\cdot \\omega_i}\\right) d\\omega_i \\end{aligned}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":22,"value":"$31","pinned":false,"hidden":false,"mode":"md"},{"id":224,"value":"tex.block`L_i\\left(\\mathrm p, \\omega_i\\right) = L_o\\left(\\mathrm p', -\\omega_i\\right)`","pinned":false,"hidden":false,"mode":"ojs"},{"id":228,"value":"The last piece of the puzzle is finding where on the other surface ${tex`\\mathrm p'`} the incoming light is coming from. For this, we can employ ray tracing: Starting at ${tex`\\mathrm p`}, we cast a ray towards ${tex`-\\omega_i`}; the closest object hit is the one we gather the radiance from. If a light is hit first, we gather the luminance directly; if the ray does not hit anything, we gather luminance from any environment map or directional lights. We will indicate the ray tracing function as ${tex`t\\left(\\mathrm p, \\omega\\right)`}.\n\nWe can finally express the rendering equation that allows us to calculate the lighting based upon all other known objects in the scene:","pinned":false,"hidden":false,"mode":"md"},{"id":238,"value":"tex.block`\\begin{aligned}L_o\\left(\\mathrm{p}, \\omega_o\\right) = L_e\\left(\\mathrm{p}, \\omega_o\\right) \n + \\int_{\\Omega^+} f_r(\\mathrm{p}, \\omega_o, \\omega_i) L_o\\left(t\\left(\\mathrm p, \\omega_i\\right), -\\omega_i\\right)\\left(\\mathrm{\\mathbf{n} \\cdot \\omega_i}\\right) d\\omega_i \\end{aligned}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":218,"value":"In one of the original Rendering Equation papers, Kajiya outlined different ways to solve this integral. As the term we are solving for occurs on both sides (one in the integral), there is generally no closed form analytical solution. All rendering algorithms are approximations to the true solution.","pinned":false,"hidden":false,"mode":"md"},{"id":38,"value":"### Local Illumination (Direct Illumination)","pinned":false,"hidden":false,"mode":"md"},{"id":57,"value":"local_fig = {\n const canvas = html`