Hankel Transformation - Digital Signal Processing for Medical Imaging Using Matlab - page 103

Appendix C
Hankel Transformation

Hankel transformation is applicable for the image that is circular symmetric.
Hence assume f ð x ; y Þ is circular symmetric. The 2D-Fourier transformation in
polar form is given as follows.
F ð U ; V Þ¼ Z

Z

p 1

g ð r ; h Þ e j2prl rdrdh ð C : 1 Þ

p 0

where l ¼ Ucos ð h Þþ V sin ð h Þ Note that g ð r ; h Þ is the polar representation of the
original image f ð x ; y Þ (It is also the inverve fourier tranformation of the radon
tranformation of F ð U ; V Þ ), where U ¼ lcos ð h Þ and V ¼ lsin ð h Þ .Asf ð x ; y Þ is
circular symmetric, the corresponding radon transformation is identical and hence
g ð r ; h Þ is identical for all h. This implies that the radon transformation of F ð U ; V Þ
is independent of h. (i.e) F ð U ; V Þ is circular symmetric and hence it is enough to
compute the values for the co-ordinates on the y-axis. (i.e) U ¼ 0 and V ¼ 0to 1 .
Particularly for U ¼ 0 and V ¼ q, we get l ¼ q sin ð h Þ . Substituting back to the
Eq. ( C.1 ), we get

G ð q Þ¼ Z

Z

p 1

g ð r Þ e j2prqsin ð h Þ rdrdh ð C : 2 Þ

p 0

) G ð q Þ¼ Z

0

Z

1

g ð r Þ e j2prqsin ð h Þ rdrdh þ Z

p

Z

1

g ð r Þ e j2prqsin ð h Þ rdrdh

ð C : 3 Þ

p 0 0 0

) G ð q Þ¼ 2 Z

Z

p 1

g ð r Þ cos ð 2prqsin ð h ÞÞ rdrdh ð C : 4 Þ

0 0

G ð q Þ thus obtained is the hankel transformation of g ð r Þ :