Page 300 - Introduction to Information Optics

5.10. Geometric Transform 28!)

Hence, the inverse coordinate transform is

l
x = (p~ (u}.
The original input can be recovered from its geometrical transform as

»] =/(x). (5,42)

(a) Shift transformation. This transformation is the simplest one of the
1
geometric transformations. The mapping is u — cp(x) = x + a and x = <p~ (w) =
u — a. Hence, the geometrical transformed function becomes F(u) = f(u — a}.
The transformation shifts the input function to a distance, a, along the positive
x direction.

(b) Scaling transformation. When the mapping is in the form u = cp(x) — ax,
the geometric transform becomes the scaling transform. The input function is
magnified by a factor a, and the scaling transformed function becomes F(«) =
f(u/a).
(c) Logarithmic transformation. This transformation is used in scale-invari-
ant pattern recognition (see Exercise 5.12). The coordinate transform is u =
l
<p(x) = In x and the inverse transform is x = (p~ (u) = e". Hence, the geometric
transformed function becomes F(w) = f(e").

(d) Rotation transformation. This process is better described in 2D formula,
where we define two mappings as

u — (p^x, y) = x cos 6 + y sin 6

v — (p 2(x, y) — —x sin 6 + y cos 6.
The geometrical transform of the input pattern /(x, y) is

F(u, v) = /[(w cos 9 — v sin 0), (u sin 8 + v cos #)],

which is found to be the original pattern rotated by an angle — 8.

(e) Polar coordinate transformation. This transformation is defined by the
mappings from the Cartesian coordinate system to the polar coordinate system as