Image Processing Reference
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 ·
P y = r ·
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