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3.1 Point operators 91

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domain S10 S9 S8 S7 S6 5 7
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(a) (b) (c) (d)
Figure 3.3 Visualizing image data: (a) original image; (b) cropped portion and scanline plot using an image in-
spection tool; (c) grid of numbers; (d) surface plot. For figures (c)–(d), the image was first converted to grayscale.

3.1.1 Pixel transforms

A general image processing operator is a function that takes one or more input images and
produces an output image. In the continuous domain, this can be denoted as

g(x)= h(f(x)) or g(x)= h(f 0 (x),...,f n (x)), (3.1)
where x is in the D-dimensional domain of the functions (usually D =2 for images) and the
functions f and g operate over some range, which can either be scalar or vector-valued, e.g.,
for color images or 2D motion. For discrete (sampled) images, the domain consists of a finite
number of pixel locations, x =(i, j), and we can write

g(i, j)= h(f(i, j)). (3.2)

Figure 3.3 shows how an image can be represented either by its color (appearance), as a grid
of numbers, or as a two-dimensional function (surface plot).
Two commonly used point processes are multiplication and addition with a constant,
g(x)= af(x)+ b. (3.3)

The parameters a> 0 and b are often called the gain and bias parameters; sometimes these
1
parameters are said to control contrast and brightness, respectively (Figures 3.2b–c). The
bias and gain parameters can also be spatially varying,
g(x)= a(x)f(x)+ b(x), (3.4)

e.g., when simulating the graded density filter used by photographers to selectively darken
the sky or when modeling vignetting in an optical system.
Multiplicative gain (both global and spatially varying) is a linear operation, since it obeys
the superposition principle,
h(f 0 + f 1 )= h(f 0 )+ h(f 1 ). (3.5)

(We will have more to say about linear shift invariant operators in Section 3.2.) Operators
such as image squaring (which is often used to get a local estimate of the energy in a band-
pass filtered signal, see Section 3.5) are not linear.

1 An image’s luminance characteristics can also be summarized by its key (average luminanance) and range
(Kopf, Uyttendaele, Deussen et al. 2007).