volume rendering

Volume rendering produces an image of a scene described not by hard surfaces but by a participating medium that fills space — think of fog, smoke, clouds, a glowing nebula, or medical CT data where each point in a 3D grid has some density. Instead of finding where a ray hits a surface, you let the ray travel through the volume, accumulating color and getting progressively blocked by density as it goes. The final pixel is the blended result of everything the ray passed through. This is the rendering model that NeRF borrowed to turn a neural density field into images.

The physics is a single integral along each ray. As the ray marches from the camera into the scene, two things compete. Density at each point contributes some emitted color, weighted by how visible that point still is from the camera. Visibility is captured by transmittance: the fraction of light that survives from the camera to that point without being absorbed, which starts at one and decays as the ray passes through dense regions. The rendered color is the integral, over distance along the ray, of transmittance times local density times local color. In words: each point's color counts only to the extent that it is dense (it actually emits) and unoccluded (light from it reaches the camera).

Because no computer integrates exactly, the integral is discretized into samples along the ray and evaluated by alpha compositing — the same front-to-back blending used throughout computer graphics. Each sample is converted to an opacity (alpha) from its density and the step length, and samples are composited in order: nearer, more opaque samples dominate; farther samples show through only where nearer ones are transparent. The accumulated transmittance is just the running product of one-minus-alpha over the samples already passed. This discrete form is exactly the equation NeRF computes for every pixel during training and rendering.

The crucial property for learning is that this whole pipeline is differentiable: the rendered pixel is a smooth function of every sample's color and density. That means you can compare the rendered image to a real photo and backpropagate the error all the way into whatever produced the densities — a neural network (NeRF), a voxel grid, or a set of Gaussians. Volume rendering is therefore the bridge that lets 2D photographs supervise a 3D representation, and it is the integral that NeRF-style methods are, in effect, optimizing.