Feature Extraction and Image Processing

Object description 273

That is, under dilation, all the coefficients are multiplied by the scale factor except a

x0

and

a

y0

which remain invariant.

Rotation can be defined in a similar way to Equation 7.59. If represents the rotation

angle, then we have that

xt

yt

a

a

ab

ab

kt

kt

x

y

k

xk xk

yk yk

()

()

=

1

2

+

cos( ) sin( )

– sin( ) cos( )

cos( )

sin( )

0

0

=1

(7.61)

This equation can be obtained by translating the curve to the origin, rotating it and then

returning it to its original location. By comparing Equation 7.55 and Equation 7.61, we

have that

aa a

xk xk yk

= cos( ) + sin( )

bb b

xk xk yk

= cos( ) + sin( )

aa a

yk xk yk

= – sin( ) + cos( )

bb b

yk x k yk

= – sin( ) +