Advanced Image Reconstruction Methods in MRI - Advance Image Processing in Magnetic Resonance Imaging - page 39

For finite sampling, there are not sufficient data to define this series. The
conventional Fourier reconstruction method treats the unknown coefficients as
zero and, as a result, we have

N / 21 −

∑

1

ˆ ()

ρ x = ∆ k Dn ( ∆ k e )

i 2 π n x ∆

, | | x < ,

(2.3)

2 ∆ k

nN =− / 2

which can be evaluated efficiently using the fast Fourier transform (FFT) algo-
rithm. Some basic properties of the Fourier reconstruction method are summa-
rized in the following remarks:

ˆ ()

Remark 1: Given D ( n ∆ k ) for n = − N /2, − N /2 + 1, … , N /2 − 1, any ρ x

given below with satisfies, Equation 2.1,

N −

∑

/ 21

/;

ˆ ()

ρ x = ∆ k Dn ( ∆ k e )

i 2 π n x ∆

+ c e

i 2 π n ∆ kx

, (2.4)

n

nN =− / 2 nNnN <− 2 ≥ / 2

which is often called a feasible reconstruction of ρ ( x ).

ˆ ()

Remark 2: The Fourier reconstruction, ρ x , given in Equation 2.3 is the

minimum-norm feasible solution because

1

∫

2 ∆ k

| ˆ ()|

c = arg min ρ x

2

dx = 0 . (2.5)

n

1

−

2 ∆ k

ˆ ()

Remark 3: The Fourier reconstruction, ρ x , is related to the true image

ρ ( x ) by

1

∫

2 ∆ k

ˆ ()

( ˆ )(

ˆ ) ˆ ,

ρ x = ρ x h x − x dx (2.6)

1

−

2 ∆ k

where h ( x ), known as the point spread function (PSF), is given by

sin( π

π

Nkx
kx

∆
∆

)

hx () = ∆ k e

− i x π ∆

. (2.7)

sin( )

Note that h ( x ) is a periodic function, and within each period it is similar to
a sinc function. The width of its main lobe, as measured by the interval between
the first two zero crossings, is 2/( N

∆ k ). The effective width W h of h ( x ) is often