Performance
A path tracer continuously samples pixels of an image. The image starts to become recognisable after only a few samples per pixel, perhaps 100. However, for the image to "converge" and reduce noise to acceptable levels usually takes around 5000 samples for most images, and many more for pathological cases. Noise is particularly a problem for animations, giving them a normally-unwanted "film-grain" quality of random speckling.
The central performance bottleneck in Path Tracing is the complex geometrical calculation of casting a ray. Importance Sampling is a technique which is motivated to cast less rays through the scene while still converging correctly to outgoing luminance on the surface point. This is done by casting more rays in directions in which the luminance would have been greater anyway. If the density of rays cast in certain directions matches the strength of contributions in those directions, the result is identical, but far less rays were actually cast. Importance Sampling is used to match ray density to Lambert's Cosine law, and also used to match BRDFs.
Metropolis light transport can result in a lower-noise image with fewer samples. This algorithm was created in order to get faster convergence in scenes in which the light must pass through odd corridors or small holes in order to reach the part of the scene that the camera is viewing. It is also shown promise on correctly rendering pathological situations with caustics. Instead of generating random paths, new sampling paths are created as slight mutations of existing ones. In this sense, the algorithm "remembers" the successful paths from light sources to the camera.
Read more about this topic: Path Tracing
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