Fast Fourier transform-based susceptibility-to-B0 calculation without aliasing artifacts
Lee Seungkyun1,2

1Center for Neuroscience Imaging Research (CNIR), Institute for Basic Science (IBS), Suwon, Korea, Republic of, 2Department of Biomedical Engineering, Sungkyunkwan University (SKKU), Suwon, Korea, Republic of


In the Fourier transform-based susceptibility-to-B0 calculation, the dipolar field kernel (1/3-kz2/k2) is discretely sampled in the k-space, which leads to aliasing artifacts in the spatial domain. We show that calculating and discretizing the dipolar field kernel in the spatial domain, before the Fourier transform, can effectively reduce the aliasing effect without resorting to large zero-filled buffers. In particular, aliasing is eliminated if the spatial-domain grid size is larger than the combined dimensions of the susceptibility source and the B0 target regions. The new method can accelerate repeated calculations of susceptibility-induced B0 fields.


Tissue magnetic susceptibility (χ)-induced B0 disturbances can be rapidly calculated by carrying out convolution of the susceptibility map with the dipolar field kernel in the Fourier domain [1,2]. Conventionally, the continuous k-space dipolar field kernel ($$$1/3-k_{z}^2/k^2$$$) is discretized in the k-space, causing Fourier aliasing artifacts in the spatial domain. This effect decreases, but is not eliminated, with zero-filled buffers [2] added in the spatial domain. Whereas a moderate buffer size is often sufficient for brain applications, disproportionately larger buffer sizes may be required in other body parts where the susceptibility source has a non-spherical, flat or elongated shape. Large buffers burden the computational memory and can slow down repeated χ-to-B0 calculations as in certain quantitative susceptibility mapping (QSM) algorithms [3]. Here we show that discretization of the dipolar field kernel in the spatial domain provides an effective way to avoid aliasing. The process utilizes closed-form integration of the dipolar field over a rectangular volume. It can be shown that for any bounded object, buffering by a factor of two in each Cartesian direction guarantees lack of aliasing.


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 Lobj, 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 ith 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 ith (target) voxel per unit susceptibility of the jth (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), δB0 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.

Method and Results

Numerical simulation was conducted in Matlab (Mathworks, MA, USA) on an ellipsoid with axis lengths =(0.4, 0.2, 0.8)[m] in a Cartesian grid with voxel size 2x2x4 [mm3]. The ellipsoid was uniformly magnetized at 3T with χ = 1 ppm. δB0 was calculated using both the proposed and the conventional methods with 3 different buffer dimensions in the shortest-axis (y) direction to assess the aliasing artifacts. Figure 3 shows the simulation models and results. For the two small buffers (a,b), the conventional method exhibited a large mismatch between the simulated and the analytical [4] δB0 profiles. The proposed method, on the contrary, showed excellent agreement in all cases. In particular, the proposed method was free from artifacts on the coronal mid-plane (y=0) without any buffer in the y-direction (case (a)). Figure 4 compares the computation times. The proposed method is slower in calculation of the k-space dipolar kernel (Step 1). However, this is compensated by faster subsequent calculations (Step 2) resulting from the smaller domain size afforded by the new method.


The proposed method adds one more operation in computing the Fourier-domain dipolar field kernel compared to the conventional method, but this is needed only once for a given computational grid. After that, reduction of the computational memory and guaranteed elimination of the Fourier aliasing effect can improve the speed and accuracy of repeated susceptibility-to-B0 calculations. Potential applications include respiration-induced B0 changes in the brain [2], susceptibility-induced off-resonance calculation in the body, including breast [5], and iterative high-resolution QSM outside the brain.


No acknowledgement found.


[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 Lobj is the object’s size, a buffered domain size L=2×Lobj 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 δB0 profiles along the major axis of the ellipsoid (d-f). In the proposed method, the δB0 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-B0 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 B0 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)