Page 308 - Introduction to Information Optics

5,11. Hough Transform

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Fig. 5.7. A straight line in (x, >') plane is Hough-transformed into a point in the parameter space

On the other hand, an arbitrary point <5(x — x 0, 3; — y 0) in the (x, y)
coordinate space can be Hough-transformed as given by

H(d, r) — \\ S(x — x 0, y — y 0)<5(r — x cos 6 — y sin 6) dx dy
J J
— d(r — x 0 cos 0 — y 0 sin 8),

which describes a pseudo-sinusoidal function in the (9, r) plane, as shown in
Fig. 5.8. In fact, a point in the (x, y) plane can be represented as an intersection
of many straight lines. Each line is mapped into a point in the ($, r) plane
according to its orientation and its distance to the origin of the (x, y)
coordinate system. All those points constitute a pseudo-sinusoidal curve.

5.11.2. OPTICAL HOUGH TRANSFORM

The optical system performing the Hough transform as suggested by Gindi
and Gmitro is shown in Fig. 5.9. The optical system images an input object

Fig. 5.8. A point in the (x, y) plane is Hough transformed into a pseudo-sinusoidal curve in the
(0, r) parameter space, where the points numbered by 1, 2, 3, 4, 5 correspond to the straight lines
in the (x, y) plane.