Unbiased rendering

Mathematically speaking, the expected value (E) of an unbiased estimator is the population mean, regardless of the number of observations. The error found in a render produced by an unbiased rendering technique is due to random statistical variance, which manifests as high-frequency noise. Variance is reduced by ${\displaystyle n}$ (standard deviation by ${\displaystyle {\sqrt {n}}}$) for ${\displaystyle n}$ data, meaning that four times as many data are needed to halve the standard deviation of the error; this makes unbiased rendering techniques less attractive for realtime or interactive applications. This means that an image produced by an unbiased renderer that appears noiseless and smooth is probabilistically correct.

A biased rendering method is not necessarily wrong, and can still produce images close to those given by the rendering equation if the estimator is consistent. These methods, however, introduce a certain bias error (usually in the form of a blur) in efforts to reduce the variance (high-frequency noise). Often biased rendering is optimized to compute faster at the cost of accuracy.

It is important to note that an unbiased technique cannot consider all possible paths (because there is an infinite number of them), and may not select the ideal paths for a given render (because to select certain paths over others introduces bias). Path tracing, an unbiased approach at its core, cannot consistently handle caustics generated from a point light source, as it is highly unlikely to randomly generate the singular path that directly reflects into the point.

Progressive photon mapping (PPM), a biased rendering technique, can actually handle caustics quite well. Although biased, PPM is provably consistent, meaning that as the number of samples goes to infinity, the bias error goes to zero (like an unbiased technique), and the probability that the estimate is correct reaches one.

• Path tracing
• Bidirectional path tracing
• Metropolis light transport and the related "energy redistribution path tracing" (ESPT)[1]

• Arion
• Arnold[2]
• Cycles
• Kerkythea
• LuxRender
• Mantra
• Maxwell Render
• Octane Render
• Fstorm Render (external link)

• Global illumination (GI)
• Physically based rendering (PBR)
• Non-photorealistic rendering (NPR)