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3.1 Point operators 93

(a) (b) (c) (d)

Figure 3.4 Image matting and compositing (Chuang, Curless, Salesin et al. 2001) c 2001 IEEE: (a) source
image; (b) extracted foreground object F; (c) alpha matte α shown in grayscale; (d) new composite C.

× (1− ) + =

B α αF C
(a) (b) (c) (d)
Figure 3.5 Compositing equation C =(1 − α)B + αF. The images are taken from a close-up of the region of
the hair in the upper right part of the lion in Figure 3.4.

1996), while the process of inserting it into another image (without visible artifacts) is called
compositing (Porter and Duff 1984; Blinn 1994a).
The intermediate representation used for the foreground object between these two stages
is called an alpha-matted color image (Figure 3.4b–c). In addition to the three color RGB
channels, an alpha-matted image contains a fourth alpha channel α (or A) that describes the
relative amount of opacity or fractional coverage at each pixel (Figures 3.4c and 3.5b). The
opacity is the opposite of the transparency. Pixels within the object are fully opaque (α =1),
while pixels fully outside the object are transparent (α =0). Pixels on the boundary of the
object vary smoothly between these two extremes, which hides the perceptual visible jaggies
that occur if only binary opacities are used.
To composite a new (or foreground) image on top of an old (background) image, the over
operator, first proposed by Porter and Duff (1984) and then studied extensively by Blinn
(1994a; 1994b), is used,
C =(1 − α)B + αF. (3.8)

This operator attenuates the influence of the background image B by a factor (1 − α) and
then adds in the color (and opacity) values corresponding to the foreground layer F, as shown
in Figure 3.5.
In many situations, it is convenient to represent the foreground colors in pre-multiplied
form, i.e., to store (and manipulate) the αF values directly. As Blinn (1994b) shows, the
pre-multiplied RGBA representation is preferred for several reasons, including the ability
to blur or resample (e.g., rotate) alpha-matted images without any additional complications
(just treating each RGBA band independently). However, when matting using local color
consistency (Ruzon and Tomasi 2000; Chuang, Curless, Salesin et al. 2001), the pure un-