Page 301 - Introduction to Information Optics

286 5. Transformation with Optics
After the polar coordinate transform, a pattern f(x, y) turns out to be

F(r, 0) = f(r cos 0, r sin 0).

This transformation has been used for rotational-invariant pattern recognition.
Example 5.1

Find the transformation between two functions:
2
2
2
2
/(x, y) = x + 4y and g(x, y) = 5x + 5y — 6xy.
Solution
From the contours determined by /(x, y) = 4 and g(x, y) = 4 in Fig. 5.4, we
can see that the transformation involves rotation and scaling. Assuming /(x, y)
is the rotation and scaling transformation of g(x, y), we have

u cos 0 — u sin 0\ (u sin 0 + u cos 0
/(«, i') = ,9

'u cos 9 — v sin 0V (u sin 0 + v cos 0\ 2
a / \ a
(u cos 0 — v sin 0\ (u sin 0 + i> cos 0
-6( —

2
2
= M + 4y .

2 3
(a) (b)

2
2
Fig. 5.4. (a) The contour of the functions /(x, y) = x + 4y = 4, (b) the contour of the functions
2
2
y(x, y) = 5x + 5y - 6xy = 4.