k
k k
k
2.2 Multidimensional Signals 7
because traditionally images have been scanned from top to bottom, most image file
formats store the image line-by-line, with the top line first, and line numbers increasing
from top to bottom of the image. is makes the orientation shown in Figure 2.2(b) more
convenient, with the origin in the upper left corner of the image and the y-axis pointing
downward. For this reason, we will generally use the orientation of Figure 2.2(b).
2.2 Multidimensional Signals
A multiD signal can be considered to be a function from the domain, here ℝ
D
,tothe
range. In this and the next few chapters, we consider only real and complex valued
signals, which we call scalar signals. In later chapters, where we consider color signals,
we will take the range to be a suitable vector space. In addition to naturally occurring
continuous space-time images, many analytically defined D-dimensional functions are
useful in image processing theory. A few of these are introduced here.
2.2.1 Zero–One Functions
Let be a region in the D-dimensional space, ⊂ ℝ
D
. We define the zero–one function
p
(x) as illustrated in Figure 2.3(a) by
p
(x)=
{
1, if x ∈ ;
0, otherwise.
(2.1)
Sometimes, p
(x) is called the indicator function of the region . Different functions are
obtained with different choices of the region . Such functions arise frequently in mod-
eling sensor elements or display elements (sub-pixels). e most commonly used ones
in image processing are the rect and the circ functions in two dimensions. Specifically,
for a unit-square region we obtain (Figure 2.3(b))
rect(x, y)=
{
1, if |x|≤0.5 and |y|≤0.5;
0, otherwise.
(2.2)
For a circular region of unit radius we have (Figure 2.3(c))
circ(x, y)=
{
1, if x
2
+ y
2
≤1;
0, otherwise.
(2.3)
ese definitions can be extended to the three-dimensional case (where the region
is a cube or a sphere) or to higher dimensions in a straightforward fashion, and the
single notation rect(x) or circ(x) canbeusedtocoverallcases.Wewillseelaterhow
these basic signals can be shifted, scaled, rotated or otherwise transformed to generate a
much richer set of zero–one functions. Other zero–one functions that we will encounter
correspond to various polygonal regions such as triangles, hexagons, octagons, etc.
2.2.2 Sinusoidal Signals
As in one-dimensional signals and systems, real and complex-exponential sinusoidal sig-
nals play an important role in the analysis of image processing systems. ere are several
reasons for this but a principal one is that if a complex-exponential sinusoidal signal is
applied as input to a linear shift-invariant system (to be introduced shortly), the output