Image Processing Reference

In-Depth Information

10

Geometric Transformations

Most people have tried to do a geometric transformation of an image when prepar-

ing a presentation or when manipulating an image. The two most well-known are

perhaps rotation and scaling, but others exist. In this chapter we will describe how

such transformations operate and discuss the issues that need to be considered when

doing such transformations.

The term “geometric” transformation refers to the class of image transformation

where the geometry of the image is changed but the actual pixel values remain

unchanged.
1

Let us recall from the previous chapters that an image is defined as
f(x,y)
,

where
f(

)
denotes the intensity or gray-level value and
(x, y)
defines the position

of the pixel. After a geometric transformation the image is transformed into a new

image denoted
g(x
,y
)
, where the tic (') means position in
g(x,y)
. This might

seem confusing, but we need some way of stating the position before the transfor-

mation
(x, y)
and after the transformation
(x
,y
)
.

As mentioned above the actual intensity values are not changed by the geometric

transformation, but the positions of the pixels are (from
(x, y)
to
(x
,y
)
). So if

f(
2
,
3
)

·

120. A geometric transformation basically

calculates where the pixel at position
(x, y)
in
f(x,y)
will be located in
g(x
,y
)
.

That is, a mapping from
(x, y)
to
(x
,y
)
. We denote this mapping as

=

120 then in general
g(
2
,
3
)

=

x
=
A
x
(x, y)

(10.1)

y
=

A
y
(x, y)

(10.2)

where
A
x
(x, y)
and
A
y
(x, y)
are both functions, which map from the position
(x, y)

to
x
and
y
, respectively.

1
For readers interested in a quick refreshment or introduction to linear algebra—in particular vec-

tors and matrices—please refer to Appendix B.