Image Processing Reference

In-Depth Information

Fig. 11.9
Representation of

a point
(P
x
,P
y
)
using polar

coordinates
(θ, r)

copy the corresponding input pixel to the output. In Figs.
11.8
(d) and
11.8
(e) the

effect of the algorithm is illustrated.

11.2 Visual Effects Based on Geometric Transformations

One could argue that the geometric transformations presented in Chap. 10 all cre-

ate visual effects. They are, however, not characterized as such, but rather as image

manipulation. Other geometric transformations exist that are aimed at creating vi-

sual effects. These transformations can be compared to the magic mirrors found in

entertainment parks, where for example the head of the person facing the mirror is

enlarged in a strange way while the legs are made smaller. Such transformations

are said to be
non-linear
as opposed to the transformations in Chap. 10, which are

linear. What is meant is that transformations which can be written as a product

between a vector and a matrix (for example as in Eq. 10.9) are said to be
linear
.

Transformations involving, for example, trigonometric operations, square roots, etc.

are said to be
non-linear
. Below four such transformations are presented followed

by a so-called local transformation.

11.2.1 Polar Transformation

A point P in 2D can be represented as
(P
x
,P
y
)
see Fig.
11.9
. But we can also

represent it by an angle
θ
and length
r
, see Fig.
11.9
. From the law of right-angled

triangles, see Sect. B.8, we can write

P
x
=
r
·

cos
(θ)

(11.1)

P
y
=
r
·

sin
(θ)

(11.2)

This is denoted a polar transformation. When using this to create a visual effect,

we use the image coordinates
(x, y)
as
θ
and
r
, respectively. That is, the forward

mapping is given as

x
=

y

·

cos
(x)

(11.3)

y
=

y

·

sin
(x)

(11.4)