Cone tracing[1] andbeam tracing are a derivative of theray tracingalgorithm that replaces rays, which have no thickness, with thick rays.
Inray tracing, rays are often modeled as geometric ray with no thickness to perform efficient geometric queries such as a ray-triangle intersection. From aphysics of light transport point of view, however, this is an inaccurate model provided the pixel on the sensor plane has non-zero area.
In the simplifiedpinhole camera optics model, the energy reaching the pixel comes from the integral of radiance from thesolid angle by which the sensor pixel sees the scene through the pinhole at the focal plane. This yields the key notion ofpixel footprint on surfaces or in thetexture space, which is the back projection of the pixel on to the scene. Note that this approach can also represent a lens-based camera and thusdepth of field effects, using a cone whose cross-section decreases from the lens size to zero at thefocal plane, and then increases.
Real optical system do not focus on exact points because ofdiffraction and imperfections. This can be modeled with apoint spread function (PSF) weighted within a solid angle larger than the pixel.
From a signal processing point of view, ignoring thepoint spread function and approximating the integral of radiance with a single, central sample (through a ray with no thickness) can lead to strongaliasing because the "projected geometric signal" has very high frequencies exceeding theNyquist-Shannon maximal frequency that can be represented using the uniform pixel sampling rate.
The physically based image formation model can be approximated by the convolution with thepoint spread function assuming the function is shift-invariant and linear. In practice, techniques such asmultisample anti-aliasing estimate this cone-based model by oversampling the signal and then performing a convolution (the reconstruction filter). The backprojected cone footprint onto the scene can also be used to directly pre-filter the geometry and textures of the scene.
Note that contrary to intuition, the reconstruction filter should not be the pixel footprint (as the pinhole camera model would suggest), since abox filter has poor spectral properties.[2] Conversely, the idealsinc function is not practical, having infinite support with possibly negative values which often createsringing artifacts due to theGibbs phenomenon. AGaussian or aLanczos filter are considered good compromises.[3]
Cone and Beam early papers rely on different simplifications: the first considers a circular section and treats the intersection with various possible shapes. The second treats an accurate pyramidal beam through the pixel and along a complex path, but it only works forpolyhedrical shapes.
Cone tracing solves certain problems related tosampling and aliasing, which can plague conventional ray tracing. However, cone tracing creates a host of problems of its own. For example, just intersecting a cone with scene geometry leads to an enormous variety of possible results. For this reason, cone tracing has remained mostly unpopular. In recent years, increases in computer speed have madeMonte Carlo algorithms likedistributed ray tracing - i.e. stochastic explicit integration of the pixel - much more used than cone tracing because the results are exact provided enough samples are used. But the convergence is so slow that even in the context of off-line rendering a huge amount of time can be required to avoidnoise.
Differential cone-tracing, considering a differential angular neighborhood around a ray, avoids the complexity of exact geometry intersection but requires a LOD representation of the geometry and appearance of the objects.MIPmapping is an approximation of it limited to the integration of the surface texture within a cone footprint. Differential ray-tracing[4] extends it to textured surfaces viewed through complex paths of cones reflected or refracted by curved surfaces.
Raymarching methods oversigned distance fields (SDFs) naturally allow easy use of cone-like tracing, at zero additional cost to the tracing, and both speeds up tracing and improves quality.