Fast Fourier transform-based susceptibility-to-B0 calculation without aliasing artifacts

Lee Seungkyun^{1,2}

Figure 1 illustrates the concept
of our method. The true, continuous dipolar field kernel (Fig. 1a)
is contaminated when the kernel is discretized in the Fourier domain by the
replicas of the dipole sources appearing at integer multiples of the domain
size (*L*) (Fig. 1b). The effect is generally non-zero everywhere in the spatial domain
due to the long range of the aliased dipolar fields. In contrast, if the true
dipolar field is first discretized and truncated in the spatial domain, the subsequent
(discrete) Fourier transform leads to aliasing which does not affect the true
dipolar field in the original domain (*L*) (Fig. 1c). If this kernel is
convoluted with the susceptibility of a source object with length *L*_{obj}, the
aliased portion of the kernel does not affect the true field inside the object when
$$$L\geq 2L_{obj}$$$.
Now consider a 3D Cartesian grid
where each voxel is a rectangular cuboid of volume $$${\Delta}v={\Delta}x{\times}{\Delta}y{\times}{\Delta}z$$$.
The susceptibility-induced, *z*-directional magnetic field in the *i*th voxel is given by the discrete convolution:

$$\delta B_0(\vec{r}_i)=\sum_{\vec{r}_j}K(\vec{r}_i-\vec{r}_j)\cdot\chi(\vec{r}_j)$$

where *K* is the *average*
dipolar field in the *i*th (target) voxel per unit susceptibility of the *j*th
(source) voxel. In computing *K* for anisotropic voxels, it is
important that the dipolar field be averaged over the target
voxel rather than merely sampled at its center. The voxel-averaged dipolar field
kernel *K* can be analytically calculated using the formula in Fig. 2. This
kernel is then Fourier-transformed discretely, multiplied by the Fourier
transform of *χ*, and inverse-transformed to give the off-resonance field,

$$\delta B_0 = FT^{-1}(FT(K)\cdot FT(\chi)).$$

As long as the spatial domain is
at least twice as large as the region of interest (in all Cartesian
directions), *δB*_{0} is free from aliasing artifacts. If the source and
target regions are different, a domain size that equals the sum of the sizes of
the source and the target regions (in all directions) is sufficient to avoid artifacts.

[1] Salomir R. el al, Concepts Magn Reson 19B:26 (2003). [2] Marques J.P. and Bowtell R., Concepts Magn Reson 25B:65 (2005). [3] de Rochefort L. et al, Magn Reson Med 63:194 (2010). [4] Osborn, J.A., Phys Rev 67:351 (1945). [5] Lee S-K. and Hancu I., J Magn Reson Imaging 36:873 (2012).

Figure 1.
(a) One-dimensional illustration of the continuous dipolar field convolution
kernel (*L* = infinity). (b) Kernel obtained by discretization in the Fourier
domain (heavy line). It is altered
everywhere
by aliased dipolar sources at integer multiples of *L*. (c) Proposed kernel
obtained by discretization and
truncation in the
spatial domain. If *L*_{obj}
is
the object’s size, a buffered domain size* L*=2×*L*_{obj}
ensures elimination
of the
Fourier aliasing
artifact in the object.

Figure 2. Analytical expression for the dipolar
field
averaged over a rectangular cuboid.

Figure
3. Uniformly magnetized ellipsoid in a buffered rectangular domain of different
sizes (a-c), and the calculated *δB*_{0} profiles along the major axis of the
ellipsoid (d-f). In the proposed method, the* δB*_{0} inside the ellipsoid closely
matches the analytical result (28.23 Hz, dashed line) in all cases (a-c). In
the conventional method, the Fourier aliasing effect is strong for the two
small domain sizes (a,b)
and is still appreciable in case (c).

Figure
4. Susceptibility-to-*B*_{0} calculation times for the proposed and the conventional
methods.
The
proposed method is
faster in Step
2 of the workflow since smaller buffer sizes (a,b)
can be used while avoiding the Fourier aliasing effect, compared to the
conventional method requiring a large buffer (c). This puts the proposed
method at advantage in applications requiring repeated *B*_{0} calculations such as
iterative QSM. A
laptop with
Intel
Core i5 at 1.7 GHz and 8
GB RAM
was used.

Proc. Intl. Soc. Mag. Reson. Med. 24 (2016)

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