Page 307 - Introduction to Information Optics

292 5. Transformation with Optics

Comparing these two integrals of the above equations, we get

271
(f)(x. y) — — (x In .x — x + y In y — y).
4f

The amplitude transmittance for logarithmic transformation is
r 9 ~i
t(x, y) = exp j — (x In x — x + y In y — y) .

5.11. HOUGH TRANSFORM

The Hough transform is a specific geometrical transform (see Sec. 5.10) and
is a special case of the Radon transform (see Sec. 5.9). Although the definition
of the Hough transform can be derived from the Radon transform, the basic
distinction between them is that the Radon transform deals with the projection
of an image into various multidimensional subspaces, while the Hough
transform deals with the coordinate transform of an image. The Hough
transform is useful for detecting curves in an image and tracking moving
targets, and for many image-processing and pattern-recognition applications
[33].

5.11.1. DEFINITION

A straight line in the Cartesian coordinate system can be Hough-trans-
formed as given by [34]

H(8, r) = I | /(x, y)d(r - x cos 0-y sin 0) dx dy
(5.49)
1,
0, otherwise,

where

1, (x, y)er l = xcosO^ + y sin
f(x, v) = ,
[0, otherwise

is a straight line, as shown in Fig. 5.7. Thus, we see that a straight line is
transformed into a point (6 l,r l) in the parameter space (6, r).