MPI signal encoding can provide a system function that forms a well-behaved basis set capable of representing highly resolved image information.

For 1D harmonic excitation of ideal particles, the system function corresponds to a series of Chebyshev polynomials of the second kind. Therefore, a fast and exact reconstruction is provided by the Chebyshev transform.

The properties of realistic particles are introduced into the system function by a convolution-type operation leading to a blurring of the high-resolution components. This introduces a resolution limit which is determined by the steepness of the particle magnetization curve. While in principle, a higher resolution image can be regained by deconvolution, resolution provided by realistic particles without deconvolution is already in the sub-millimeter range [3].

The system function for 2D imaging is determined by the trajectory taken by the FFP and a kernel representing the region around the FFP which contributes to the signal. The shape of this FFP kernel is determined by the topology of the selection field. The simple case of constant selection field gradients in all spatial directions has been demonstrated. For ideal particles, the kernel has sharp singularities which provide high spatial resolution. However, regions around these sharp peaks also contribute to the signal. This is probably the reason for the observation that in 2D encoding using a 2D Lissajous pattern, the system function is not exactly represented by 2D Chebyshev functions. Therefore, reconstruction cannot be done by using the Chebyshev transform as in 1D, but requires the inversion of the system function matrix. However, a close relationship between the 2D Lissajous system function and the 2D Chebyshev polynomials is obvious. This may be used to transform the system function into a sparser representation using a Chebyshev or cosine-type transformation, resulting in lower memory requirements and faster reconstruction.

The 2D Lissajous system function does not form a fully orthogonal set, since it contains redundant components. Nonetheless, it is capable of encoding highly resolved image information as shown in figure 11. Possible mismatches between the information content of the acquired data and the desired pixel resolution can be mediated by using regularization schemes in the reconstruction [10]. In experimentally acquired data, the necessary degree of regularization will also depend on the SNR. Optimally, to take into account noise in the system function as well as in the measured object data, image reconstruction should be based on the total least squares approach [11].

To speed up the tedious experimental acquisition of the system function, one can use the parity rules derived for the 2D system function. In theory, these allow to construct the complete system function from measuring only one quadrant of the rectangle of the Lissajous figure. For a 3D Lissajous figure, one octant would suffice, accelerating the system function acquisition by a factor of eight. Experimentally, the symmetry can be disturbed by non-perfect alignment of coils. Nonetheless, knowledge of the underlying theoretical functions and their parity can help to model the system function from only a few measured samples.

In a real MPI experiment, one usually acquires many more frequency components than the desired number of image pixels. Therefore, one has the freedom to make a selection of system function components to constitute a more compact basis set providing better orthogonality. For instance, duplicate system function components can be removed after acquisition to arrive at a smaller system function matrix to speed up image reconstruction. Furthermore, a selection of harmonics according to their weight can help to reduce matrix size. It is also conceivable to modify the weight of certain components to influence image resolution and SNR.

2D imaging of realistic particles has not been simulated in this work, but from the 1D derivations, one can infer that a blurring of the FFP kernel depending on the steepness of the particle magnetization curve will occur. This would remove the kernel singularities, but would also result in a slight loss of resolution, as discussed for the 1D case.

3D imaging has not been shown, but 2D results can be directly extrapolated to 3D by introducing an additional orthogonal drive field enabling 3D FFP trajectories. For a 3D Lissajous trajectory, close resemblance of the system function to third order tensor products of Chebyshev polynomials can be expected.

The selection field topology and the FFP trajectories used in this work have been chosen for their simple experimental feasibility. However, many alternative field configurations are conceivable. For the FFP trajectory, one can as well use radial or spiral patterns [12], or even patterns that are tailored to the anatomy to be imaged. Trajectories can be adapted to deliver varying resolution over the image. For the selection field, a topology creating a field-free line instead of the FFP promises more efficient scanning [8]. More research is required to identify field configurations optimal for specific applications.