Illustrations on MRI Techniques Using Matlab - Digital Signal Processing for Medical Imaging Using Matlab - page 62

Fig. 3.6 a Illustrating the order in which the K-space is scanned; b sample polar scanned K-space

9. Apply inverse fourier transformation in polar form to obtain f ( x , y ) (refer

Sect. 3.3.1 ).
10. For checking the validity of reconstruction from polar data, k is chosen as zero
matrix (No dephasing) for illustration. This helps to obtain proton-density image
(n-matrix) as the reconstructed image.

3.3.1 Reconstructing f ( x , y ) from G ( r ,θ)

There are two major techniques to obtain f namely back-

projection technique and the interpolation technique as described below. The
interpolation technique is used for illustration purpose.

( x , y ) from G ( r ,θ)

3.3.1.1 Back-Projection Technique

The generalized formula to reconstruct f ( x , y ) from G ( r ,θ) is as shown below.

π ∞

exp j 2 π( xcos θ + ysin θ) r f ( x , y ) = G ( r ,θ) | J | drd θ (3.2)

− π

0

1. It it noted G ( r ,θ) is the fourier transformation of the radon transformation R ( l ,θ)

for the specific constant θ .

2. It is possible to get back f using inverse fourier transfor-

mation (refer Sect. 1.2 ) , followed by back-projection technique.

( x , y ) by getting R ( l ,θ)