Editing Rendering (computer graphics) (section)

== Scientific and mathematical basis ==
{{Main|Unbiased rendering}}
{{Unreferenced section|date=December 2024}}

The implementation of a realistic renderer always has some basic element of physical simulation or emulation{{snd}} some computation which resembles or abstracts a real physical process.

The term "''[[physically based rendering|physically based]]''" indicates the use of physical models and approximations that are more general and widely accepted outside rendering. A particular set of related techniques have gradually become established in the rendering community.

The basic concepts are moderately straightforward, but intractable to calculate; and a single elegant algorithm or approach has been elusive for more general purpose renderers. In order to meet demands of robustness, accuracy and practicality, an implementation will be a complex combination of different techniques.

Rendering research is concerned with both the adaptation of scientific models and their efficient application.

Mathematics used in rendering includes: [[linear algebra]], [[calculus]], [[numerical analysis|numerical mathematics]], [[digital signal processing|signal processing]], and [[Monte Carlo methods]].

=== The rendering equation ===
{{Main|Rendering equation}}

This is the key academic/theoretical concept in rendering. It serves as the most abstract formal expression of the non-perceptual aspect of rendering. All more complete algorithms can be seen as solutions to particular formulations of this equation.

: <math>L_o(x, \omega) = L_e(x, \omega) + \int_\Omega L_i(x, \omega') f_r(x, \omega', \omega)  (\omega' \cdot n) \, \mathrm d \omega'</math>
Meaning: at a particular position and direction, the outgoing light (L<sub>o</sub>) is the sum of the emitted light (L<sub>e</sub>) and the reflected light. The reflected light being the sum of the incoming light (L<sub>i</sub>) from all directions, multiplied by the surface reflection and incoming angle. By connecting outward light to inward light, via an interaction point, this equation stands for the whole 'light transport'{{snd}} all the movement of light{{snd}} in a scene.

=== The bidirectional reflectance distribution function ===
The '''[[bidirectional reflectance distribution function]]''' (BRDF) expresses a simple model of light interaction with a surface as follows:

: <math>f_r(x, \omega', \omega) = \frac{\mathrm d L_r(x, \omega)}{L_i(x, \omega')(\omega' \cdot \vec n) \mathrm d \omega'}</math>

Light interaction is often approximated by the even simpler models: diffuse reflection and specular reflection, although both can ALSO be BRDFs.

=== Geometric optics ===
Rendering is practically exclusively concerned with the particle aspect of light physics{{snd}} known as [[geometrical optics]]. Treating light, at its basic level, as particles bouncing around is a simplification, but appropriate: the wave aspects of light are negligible in most scenes, and are significantly more difficult to simulate. Notable wave aspect phenomena include diffraction (as seen in the colours of [[Compact disc|CDs]] and [[DVD]]s) and polarisation (as seen in [[Liquid-crystal display|LCDs]]). Both types of effect, if needed, are made by appearance-oriented adjustment of the reflection model.

=== Visual perception ===
Though it receives less attention, an understanding of [[human visual perception]] is valuable to rendering. This is mainly because image displays and human perception have restricted ranges. A renderer can simulate a wide range of light brightness and color, but current displays{{snd}} movie screen, computer monitor, etc.{{snd}} cannot handle so much, and something must be discarded or compressed. Human perception also has limits, and so does not need to be given large-range images to create realism. This can help solve the problem of fitting images into displays, and, furthermore, suggest what short-cuts could be used in the rendering simulation, since certain subtleties will not be noticeable. This related subject is [[tone mapping]].

=== Sampling and filtering ===
One problem that any rendering system must deal with, no matter which approach it takes, is the '''sampling problem'''.  Essentially, the rendering process tries to depict a [[continuous function]] from image space to colors by using a finite number of pixels.  As a consequence of the [[Nyquist–Shannon sampling theorem]] (or Kotelnikov theorem), any spatial waveform that can be displayed must consist of at least two pixels, which is proportional to [[image resolution]].  In simpler terms, this expresses the idea that an image cannot display details, peaks or troughs in color or intensity, that are smaller than one pixel.

If a naive rendering algorithm is used without any filtering, high frequencies in the image function will cause ugly [[aliasing]] to be present in the final image.  Aliasing typically manifests itself as [[jaggies]], or jagged edges on objects where the pixel grid is visible.  In order to remove aliasing, all rendering algorithms (if they are to produce good-looking images) must use some kind of [[low-pass filter]] on the image function to remove high frequencies, a process called [[Spatial anti-aliasing|antialiasing]].