Digital Compositing - Mathematics

Mathematics

The basic operation used is known as 'alpha blending', where an opacity value, 'α' is used to control the proportions of two input pixel values that end up a single output pixel.

Consider three pixels;

• a foreground pixel, f
• a background pixel, b
• a composited pixel, c

and

• α, the opacity value of the foreground pixel. (α=1 for opaque foreground, α=0 for a completely transparent foreground). A monochrome raster image where the pixel values are to be interpreted as alpha values is known as a matte.

Then, considering all three colour channels, and assuming that the colour channels are expressed in a γ=1 colour space (that is to say, the measured values are proportional to light intensity), we have:

c_r = α f_r + (1 − α) b_r

c_g = α f_g + (1 − α) b_g

c_b = α f_b + (1 − α) b_b

Note that if the operations are performed in a colour space where γ is not equal to 1 then the operation will lead to non-linear effects which can potentially be seen as aliasing artifacts (or 'jaggies') along sharp edges in the matte. More generally, nonlinear compositing can have effects such as "halos" around composited objects, because the influence of the alpha channel is non-linear. It is possible for a compositing artist to compensate for the effects of compositing in non-linear space.

Performing alpha blending is an expensive operation if performed on an entire image or 3D scene. If this operation has to be done in real time video games there is an easy trick to boost performance.

c_out = α f_in + (1 − α) b_in

c_out = α f_in + b_in − α b_in

c_out = b_in + α (f_in − b_in)

By simply rewriting the mathematical expression one can save 50% of the multiplications required.