Radon Transformation - Digital Signal Processing for Medical Imaging Using Matlab - page 3

∞ ∞

R ( l ,θ) = f ( x , y )δ( x cos θ + y sin θ − l ) dxdy (1.1)

−∞ −∞

For the fixed θ, R ( l ,θ) = R ( l ) . Taking fourier transformation of R ( l ) , we get

the following.

∞

exp − j 2 π Ul dl

G ( U ) = R ( l ) (1.2)

−∞

∞ ∞ ∞

dxdy exp − j 2 π Ul dl ⇒ G ( U ) = f ( x , y )δ( x cos θ + y sin θ − l ) (1.3)

−∞ −∞ −∞

∞ ∞

exp − j 2 π( x cos θ + y sin θ) U ⇒ G ( U ) = f ( x , y ) dxdy (1.4)

−∞ −∞

Let the 2D-Fourier transformation of the image matrix f ( x , y ) be represented as

U , V ) U , V ) , when U =

F ( . It is noted from the ( 1.4 ) that G ( U ) = F ( U cos θ

and V

= U sin θ . If the values are collected from the 2D-Fourier transformation

U , V ) of f(x,y), along the line U cos V sin

F ( (θ) + (θ) = U ((i.e) for various U ),

we get G ( U ) . This is called projection-slice theorem.

U , V ) f ( x , y ) is obtained from the F ( using inverse 2D-Fourier transformation

as mentioned below.

∞ ∞

exp j 2 π( xU + yV ) dU dV U , V ) f ( x , y ) = F ( (1.5)

−∞ −∞

Substituting U and V

= U cos θ = U sin θ in ( 1.5 ), we get the following.

F ( U cos θ, V sin θ) = G ( U ,θ) , which is the Fourier transformation of g ( l ,θ) for the

constant θ (refer ( 1.2 )). Changing the variables from ( x , y ) to ( U ,θ) ,( 1.5 ) becomes

π ∞

exp j 2 π( x cos θ + y sin θ) U

( , ) = ( ,θ) | | θ

f x y G U J dUd (1.6)

− π

0

√ U 2

tan − 1 V U . V 2

where | J | , is the jacobian of the transformation U = + ,θ =

∂ U
∂ U

U
∂ V

∂

(i.e) J = ⇒| J |=| U |

∂θ
∂ U ∂θ V

∂