Image Processing Reference
In-Depth Information
also constitutes a vector space,
E
N
. This is called the
N
-dimensional Euclidean
space (or room) and behaves very much like
E
3
as far as addition and scaling is
concerned, albeit it is not easy to visualize its members.
This concept can be easily extended to comprise arrays with more than one index,
e.g., with four indices:
A
=
{
A
(
k, l, m, n
)
}
(3.4)
where
A
(
k, l, m, n
) is a scalar (i.e., a real or a complex number). To fix the ideas,
the indicies
k, l, m, n
would be typically positive integers that can at most take the
values
K, L, M, N
, respectively. An addition and scaling rule that can make the set
of all
A
s having the same
K, L, M, N
, a vector space is as follows¡:
A
+
B
=
{
A
(
k, l, m, n
)+
B
(
k, l, m, n
)
}
(3.5)
α
A
=
{
α
A
(
k, l, m, n
)
}
(3.6)
Matrices, the double-indiced arrays studied in linear algebra courses, constitute a
special case of the above. Evidently, those having the same size are a vector space
too!
3.2 Discrete Image Types, Examples
There are many types of discrete images, and new types are being constructed as
the science studying their production from light, by sensors and other hardware ad-
vances. This science is also called
imaging
. The simplest image type is a 2D array,
where the array elements represent gray values
A
=
{
A
(
k, l
)
}
(3.7)
where
k, l
take
K, L
different values. The size of the image is
K
L
, and typically
k, l
corresponds to row (top-down) and column (left-right) counts. However, it is
also common that
k, l
represent the horizontal (left-right) and vertical (down-up)
counts. The pair
k, l
represents the coordinates of a point, also called
pixel
,onthe
discrete 2D grid of size
K
×
L
. The scalar
A
(
k, l
) is typically an integer ranging
0-255 and represents the
gray value
or the
pixel value
of the pixel. For convenience
we can assume that
A
(
k, l
) is a scalar (real or complex-valued), although it is not
possible to interpret negative values as gray values. Other names for gray value, are
intensity
,
brightness
, and
luminance
. Negative and complex pixel values are common
and can be obtained from computations in image processing, even from sensors. For
example, we can have an array with two indices representing a map of altitutes w.r.t.
the sea level. The points under the sea level will be negative, and those above will
be positive. There are examples of complex-valued images that can be obtained from
sensors as well as from computations. To limit the scope, we do not discuss the tax-
onomy of images at that level here. The space of discrete images of the same size
taking scalar values is thus a vector space. The necessary rules of vector addition
and scaling can be defined via the general elementwise addition and scaling rules
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