Simple
F.T.A.W.
Equation
for Optimal
Gridding
Parameters
Wajer, E. Woudenberg, R. de Beer, M. Fuderer”, A.F. Mehlkopf, D. van Ormondt
Spin Imaging
Group, Faculty of Applied Physics, Delft University of Technology, De&,
*Philips Medical Systems, Best, The Netherlands
Introduction
The Netherlands
loo
Gridding is a well established method for image reconstruction from a non-uniformly
sampled k-space. An essential part
of gridding is a convolution.
The convolution kernel usually
used is the Kaiser-Bessel window [l]. This window has two parameters, its width L, and a shape parameter B. The accuracy
of the gridding algorithm depends on B and L. In addition, it
depends on how fine a grid one convolves to, which is governed
by a factor f. On account of the FFT, f = 2 is usually chosen
to keep the gridsize a power of 2.
The calculation time of the convolution is determined by
the product of L and f. Fixing the calculation time leaves
only B as a free parameter to optimise the accuracy. Optimal
values for B, given L, for f = 1 and 2 can be found in [2].
In principle f may be any number greater or equal to 1.
This can be of use whenever the original gridsize is not a power
of 2. For example, if the original gridsize is 96, setting f = $
may be sufficiently accurate. Moreover, the resulting gridsize
is 128, instead of 192, which results in a faster FFT. Any loss
of accuracy due to the smaller f can be overcome by a small
increase in L.
In this abstract we present an equation for the optimal value
of B given L and f. The values for B in case f = 2 agree very
well with the values found in [2].
Method
The criterion we used for selecting the optimal value of B is the
relative amount of aliased energy (including roll-off correction)
of the Kaiser-Bessel window. This is the same criterion as used
in [2], and it is valid as long as the signal, which would alias
into the FOV without the use of the convolution, is sufficiently
uniformly distributed.
Otherwise, dependency on both the
sampling distribution
and the image is to be expected [3].
10 -30
4
6
8
10
12
14
fL 0; conv. talc. time
Figure 2: Aliased energy of the Kaiser-Bessel window against fL,
which is proportional to the calculation time of the convolution, for
f = 1.0,1.25,. . . ,2.0,2.5,3.0.
increases, because of the finer grid that needs to be FFT’d.
The best compromise between accuracy and speed is to find a
value for f which makes the gridsize FFT-friendly
and adapt
L to attain the required accuracy.
Validation
To determine if Eq. 1 does indeed give the optimal value of B
for an actual reconstruction,
a circular object with radius R
was reconstructed.
Fig. 3 shows the true optimal value of B
for this case. Eq. 1 fits very nicely, except for small values of
f, where the used criterion breaks down due to dependency on
the image and the sampling distribution
[3].
Figure 3: Optimal value for B/nL by reconstructing a circular object with radius R for different values of f and L = 3.0.
Figure 1: Optimal value for B against f for L = 3,4,. . . ,7, found
by minimizing the relative amount of energy of the Kaiser-Bessel
window, which will alias into the FOV.
Fig. 1
values of
rion. The
accurately
shows the optimal value for B against f for several
L found by minimizing the above mentioned critevalues clearly lie on straight lines, which can be very
described by:
B opt = (f - $rL
Fig. 2 shows the relative amount of aliased energy of the
Kaiser-Bessel window against f L, which is proportional to the
calculation time of the convolution.
Keeping it constant, increasing f beyond 2.0 does not significantly
increase the accuracy. The total calculation time, however, increases as f
Conclusion
An equation for obtaining an optimal value B of the KaiserBessel window used in gridding has been found. This equation
enables quick adaptation to the convolution gridsize.
Acknowledgements
This work is supported by the Dutch Technology Foundation (STW), the Advanced School for Computing and Imaging
(ASCI) and Philips Medical Systems.
References
O’Sullivan, J.D., IEEE Trans. Med. Imag,, 4, 200-207, 1985.
2. Jackson, J.I., et al., IEEE Trans. Med. Imag., 10, 473-478,
1991.
3. Zwaga, J.H., et al., Proc. ISMRM, 6th Ann. Mtg., 669, 1998.
1.