Radon Transformation - Digital Signal Processing for Medical Imaging Using Matlab - page 14

Fig. 1.13 Relationship between parallel beam and the fan-beam projection geometry-1

( 1.21 ) is rewritten using the new set of variables θ = β + δ and l = C sin (δ) .

f ( r ,φ)

δ 2 π + δ

max

= R ( C sin (δ), β − δ) u ( r cos (φ − β − δ) − C sin (δ)) | J | d β d δ,

− δ min 0 + δ

(1.22)

where | J | , is the jacobian of the transformation θ = β + δ and l = C sin (δ) (i.e)

⎡
⎣

⎤
⎦ ⇒|

∂β
∂θ

∂β
∂δ

J = J |=| C cos (δ) |

l
∂θ

∂

∂θ
∂δ

Hence, ( 1.22 ) is rewritten as

f ( r ,φ)

δ − δ + 2 π max

= R ( C sin (δ), β + δ) u ( r cos (φ − β − δ) − C sin (δ)) | C cos (δ) | d β d δ (1.23)

− δ max − δ + 0

With constant δ , R ( C sin (δ), β + δ) and u ( r cos (φ − β − δ) − C sin (δ)) are the

periodic functions of the variable β with period 2 π . Thus the final reconstruction