The function a NeRF actually learns
Radiance fields, and how to render one
A radiance field is a function: give it a point in space and a direction to look from, and it returns a colour and a density. That is all a NeRF stores — no triangles, no surfaces. Turning it into a picture means marching a ray through it and accumulating, and that step is what the notes leave out.
This is a 2-D slice, so z and φ are pinned; a real field takes all five. Notice that σ does not depend on the direction — how solid something is cannot change with where you stand — but the colour does. That asymmetry is built into the architecture of a NeRF, and it is what stops the model explaining away a reflection by inventing geometry.
Every pixel of this strip is one ray, marched exactly as above. Rotate the camera and the picture changes — including the highlights, which slide across the surfaces because the colour returned by the field depends on direction. Training a NeRF means running this renderer backwards: adjust the field until the images it produces match the photographs.
The equation the notes stop just short of. March along the ray in steps of δ. At each sample the field returns a density σ and a colour c. The chance of the ray being stopped in that step is α = 1 − e−σδ, and the light still surviving from earlier is the transmittance T. Each sample contributes T · α · c, and then T is reduced by the factor (1 − α). Add it all up and that is the pixel. Nothing else happens — there is no surface, no intersection test, no shading model.
Why the weight curve matters more than the density curve. The blue curve peaks where the ray first meets something solid and then collapses, because everything behind is in shadow — transmittance has already fallen to nearly zero. That is how a volume without surfaces still produces hard occlusion, and where a rendering built purely on densities gets its notion of "the front of the object".
Drop the sample count to 8 and watch it fall apart. The integral is being estimated by a finite sum, and with too few samples thin structures are stepped over entirely and the image bands. Stratified sampling — jittering each sample within its own interval — converts that structured banding into unstructured noise, which is far less objectionable to the eye and averages away across frames. Real implementations go further and sample where the weights are already known to be large, which is why NeRF papers describe a coarse network whose job is to tell a fine network where to look.
And this is why NeRFs handle what photogrammetry cannot. There is no requirement anywhere above that the scene have surfaces, or that a point be visible from two cameras, or that anything be matched to anything. The field only has to be consistent with the photographs when rendered. Smoke, glass, fur and specular highlights all have perfectly good radiance fields; none of them has a well-defined surface to triangulate.