METHOD AND APPARATUS FOR MRT IMAGING WITH MAGNETIC FIELD MODULATION

Abstract

A method of magnetic resonance (MR) tomography imaging an object (1) comprises arranging the object in a static magnetic field, subjecting the object to at least one radiofrequency pulse and magnetic field gradients for creating spatially encoded MR signals, acquiring MR signals, and reconstructing an object image utilizing the spatial encoding of the MR signals, wherein, during the acquiring step, the MR signals are subjected to a locally specific frequency modulation by means of at least one spatially restricted, time-varying magnetic modulation field with a component parallel to the static magnetic field, and the step of reconstructing the object image further utilizes the frequency modulation for obtaining spatial information from the spatially encoded MR signals. An MR imaging device (100) includes an MR scanner (110) with a magnetic field modulation source device (114) for creating a spatially restricted, time-varying magnetic modulation field, a control device (120) and a reconstruction device (130) for reconstructing the object image by utilizing a frequency modulation of collected MR signals for obtaining spatial image information.

Claims

1. A method of magnetic resonance tomography (MRT) imaging an object, comprising the steps of: arranging the object in a static magnetic field, subjecting the object to at least one radiofrequency pulse and magnetic field gradients for creating spatially encoded magnetic resonance signals, acquiring magnetic resonance signals, and reconstructing an object image, wherein the spatially encoded magnetic resonance signals are utilized, wherein during the acquiring step, the magnetic resonance signals are subjected to a locally specific frequency modulation by use of at least one spatially restricted, time-varying magnetic modulation field, which has a component parallel to the static magnetic field, and the step of reconstructing the object image further utilizes the locally specific frequency modulation for obtaining spatial information from the spatially encoded magnetic resonance signals.

2. The method according to claim 1, wherein the at least one spatially restricted, time-varying magnetic modulation field has a modulation frequency being selected such that a temporal frequency modulation of local Larmor frequencies of the magnetization is obtained.

3. The method according to claim 1, wherein the at least one spatially restricted, time-varying magnetic modulation field has a modulation frequency of at least one of at least 100 Hz and at most 10 MHz.

4. The method according to claim 1, wherein the at least one spatially restricted, time-varying magnetic modulation field has a sinusoidal or triangular modulation shape.

5. The method according to claim 1, wherein the at least one spatially restricted, time-varying magnetic modulation field is created with at least one of at least one local magnetic field coil being arranged adjacent to the object, and a shimming device comprising spherical harmonic shim coils being arranged for shimming a magnetic field distribution in the object.

6. The method according to claim 1, wherein the static magnetic field is superimposed with at least two spatially restricted, time-varying magnetic modulation fields being localized in different spatial sections of the object.

7. The method according to claim 6, wherein the spatially restricted, time-varying magnetic modulation fields in different spatial sections of the object have at least one of different amplitudes, frequencies, phases and modulation shapes.

8. The method according to claim 1, including a step of selecting at least one of a number and an extension of spatial sections of the object being subjected to different spatially restricted, time-varying magnetic modulation fields in dependency on operation instructions provided by at least one of an operator and a preliminary object imaging process.

9. The method according to claim 1, wherein the step of acquiring the magnetic resonance signals includes at least one of parallel sensing the magnetic resonance signals with a plurality of RF coils, and changing the at least one spatially restricted, time-varying magnetic modulation field for each phase encoding step included in the step of acquiring the magnetic resonance signals.

10. The method according to claim 1, wherein the step of reconstructing the object image m includes solving a linear equation system s=E m by a regularized optimization, wherein s includes the magnetic resonance signals and E is an encoding matrix being determined by the spatial encoding of the magnetic resonance signals and depending on time-varying modulation components.

11. The method according to claim 1, wherein the static magnetic field is further superimposed with at least one spatially restricted, time-varying magnetic modulation field during the step of subjecting said object to the at least one radiofrequency pulse and magnetic field gradients.

12. A magnetic resonance imaging (MRI) device, a magnetic resonance scanner being configured for accommodating an object to be imaged, creating a static magnetic field, at least one radiofrequency pulse and magnetic field gradients, and collecting magnetic resonance signals, a control device being configured for controlling the magnetic resonance scanner, and a reconstruction device being configured for reconstructing an object image based on the magnetic resonance signals, wherein the magnetic resonance scanner includes a magnetic field modulation source device, which is further configured for superimposing the static magnetic field with at least one spatially restricted, time-varying magnetic modulation field, which has a component parallel to the static magnetic field, so that the magnetic resonance scanner is configured for subjecting the magnetic resonance signals to a locally specific frequency modulation during the collecting of the magnetic resonance signals, the control device is configured for setting the at least one spatially restricted, time-varying magnetic modulation field, and the reconstruction device is configured for reconstructing the object image by utilizing the locally specific frequency modulation of the magnetic resonance signals for obtaining spatial image information from the spatially encoded magnetic resonance signals.

13. A magnetic resonance imaging (MRI) device, which is configured for conducting the method according to claim 1, said MRI device comprising: a magnetic resonance scanner being configured for accommodating an object to be imaged, creating a static magnetic field, at least one radiofrequency pulse and magnetic field gradients, and collecting magnetic resonance signals, a control device being configured for controlling the magnetic resonance scanner, and a reconstruction device being configured for reconstructing an object image based on the magnetic resonance signals, wherein the magnetic resonance scanner includes a magnetic field modulation source device, which is further configured for superimposing the static magnetic field with at least one spatially restricted, time-varying magnetic modulation field, which has a component parallel to the static magnetic field, so that the magnetic resonance scanner is configured for subjecting the magnetic resonance signals to a locally specific frequency modulation during the collecting of the magnetic resonance signals, the control device is configured for setting the at least one spatially restricted, time-varying magnetic modulation field, and the reconstruction device is configured for reconstructing the object image by utilizing the locally specific frequency modulation of the magnetic resonance signals for obtaining spatial image information from the spatially encoded magnetic resonance signals.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0040] Further advantages and details of the invention are described in the following with reference to the attached drawings, which schematically show in:

[0041] FIG. 1: an MRI device according to an embodiment of the invention;

[0042] FIG. 2: a magnetic field modulation source device included in an MRI device according to an embodiment of the invention;

[0043] FIG. 3: an overview of magnetic field patterns created with the magnetic field modulation source device of FIG. 2;

[0044] FIG. 4: an illustration of free induction decay acquired at a constant main magnetic field and with a varying magnetic modulation field;

[0045] FIG. 5: an illustration of effects of the varying magnetic modulation field on the spatial encoding; and

[0046] FIGS. 6 and 7: illustrations of the invention based on the concept of generating virtual phase-shifted images by varying magnetic modulation fields.

[0047] FIG. 8: an illustration of effects of the varying magnetic modulation field from each individual coil on the spatial encoding;

[0048] FIG. 9: an illustration of effects of the varying magnetic modulation field from 8 coils on the spatial encoding without acceleration, and the result of the reconstruction;

[0049] FIG. 10: an illustration of effects of the varying magnetic modulation field from 8 coils on the spatial encoding with 2× acceleration, and the result of the reconstruction; and

[0050] FIG. 11: an illustration of the singular value spectrum of the encoding matrix in case no locally specific frequency modulation is applied versus the modulation according to the invention.

DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS OF THE INVENTION

[0051] Embodiments of the invention are described in the following with particular reference to the inventive locally specific frequency modulation of MR signals by means of at least one spatially restricted, time-varying magnetic modulation field. The invention preferably is implemented with an MR scanner as it is known per se. Accordingly, details of the MR scanner, the available control schemes thereof and available schemes of MR signal acquisition are not described as they are known from prior art. Exemplary reference is made to applications of the invention, wherein a magnetic field modulation source device comprises an arrangement of local magnetic field coils. The invention is not restricted to this embodiment, but correspondingly can be implemented with one single local magnetic field coil and/or with a shimming device comprising at least one local spherical harmonic shim coil. In particular, one single local magnetic field coil or one local spherical harmonic shim coil, e.g. covering a half of the object under investigation, is sufficient for providing additional spatial information for image reconstruction.

Embodiments of the MRI Device and Method

[0052] FIG. 1 schematically illustrates an embodiment of an MRI device 100 including an MR scanner 110, a control device 120 and a reconstruction device 130, which are configured for implementing the invention. The MR scanner 110 includes a main magnetic field device 111, a magnetic gradient device 112, an excitation/acquisition coil device 113 and a magnetic field modulation source device 114. Furthermore, a schematically illustrated holding device 116, like a supporting table, can be provided for supporting an object 1 to be investigated. The components 111 to 113 and 116 are configured as it is known from conventional MR scanners. For example, the excitation/acquisition coil device 113 can comprise a single RF coil or an array of RF coils arranged for parallel MR imaging.

[0053] The magnetic field modulation source device 114 is adapted for creating preferably less than 130 e.g. 10 to 30, spatially restricted, time-varying magnetic modulation fields, each covering a spatial section of the object 1. Magnetic field coils 115 of the magnetic field modulation source device 114, e.g. a set of surface coil loop elements, are designed such each magnetic field sufficiently covers the extension of the spatial section of the object to which the modulation is to be applied. Preferably, magnetic field coils 115 are arranged on at least two different sides of the object 1. Examples of the magnetic field modulation source device 114 are described below with reference to FIGS. 2 and 3.

[0054] The control device 120 includes a main field and gradient control unit 121, an RF pulse control unit 122 and a modulation source control unit 123, each including driving circuits, like excitation and modulation current sources, amplifiers and/or pulse modulators, and at least one computer unit. The components 121 to 123 can be provided with a common computer unit or with separate computer units, coupled with the driving circuits for driving the components 111 to 114. In particular, the modulation source control unit 123 is connected with the magnetic field modulation source device 114 for creating the inventive locally specific frequency modulation of the MR signals. The reconstruction device 130 includes a signal acquisition device 131 coupled with the excitation/acquisition coil device 113 and a calculation device 132. Optionally, the calculation device 132 can be coupled, e.g. via direct connection or any other type of data transmission, with the modulation source control unit 123, so that information on the modulation pattern applied to the magnetic field modulation source device 114 can be introduced in the MR image reconstruction. The components 131 and 132 can be provided with a common computer unit or with separate computer units. The computer units of the control device 120 and the reconstruction device 130 are adapted for running software controlling the setting of the components 111 to 114, for collecting and processing MR signals and for MR image reconstruction, resp.

[0055] FIG. 2 shows an example of a magnetic field modulation source device 114, which comprises 8 independent local magnetic field coils 115 (schematically shown) that are placed in one circular row perpendicular to the z-direction on a hollow cylinder shaped non-conductive carrier (not shown). With a preferred application of the invention, the object 1 to be investigated, for example a head of a human subject, is placed inside this arrangement (as mentioned with reference to the experimental tests below). Each local magnetic field coil 115 is made of e.g. 25 turns of copper wire with a coil diameter of 5 cm to 10 cm. Each coil is independently connected to an individually controllable modulation current source of the modulation source control unit 123 (see FIG. 1) that allows to drive the modulation current in each coil with preselected and independent, different temporal patterns during MR signal collection, e.g. during read-out of a FLASH sequence or another sequence.

[0056] The arrangement of FIG. 2 can be modified, e.g. by providing two circular rows of local magnetic field coils 115 perpendicular to the z-axis surrounding a hollow cylinder as schematically shown in FIG. 3A. With this embodiment, the magnetic field modulation source device 114 comprises 16 independent local magnetic field coils 115.

[0057] The inventive spatial encoding relies on injecting oscillating currents into the local magnetic field coils 115. For example, a sine-form modulation can be used in each coil loop element, where all coil elements share the same modulation frequency but have distinct phases. The technique allows to use characteristic spatiotemporal patterns of field variations due to oscillating currents during image reconstruction in order to improve SNR or accelerate the acquisition. An extra information required to solve a potentially under-determined underlying system of linear equations can be obtained by increasing the number of samples acquired in each readout.

[0058] With more details, according to the waveform of the modulation current, the local magnetic field coils 115 induce local and varying magnetic fields in the object 1, each having a field component parallel to the main magnetic field (z-direction) of the MR scanner. Each of the local magnetic field coils 115 produces a local field having an amplitude and local and temporal distribution according to Biot-Savart law. The local field deviates from the originally homogeneous main magnetic field into positive or negative direction. An example is shown in FIG. 3B, which shows an overview of the magnetic field patterns (in units of Hz) generated by a constant current through coils 1 to 16 in a cylindrical phantom object placed inside the cylinder shown in FIG. 3A. FIG. 3B can be considered as a snap-shot of a time-varying specific spatially confined magnetic field pattern in the object 1, which is illustrated along three Cartesian spatial directions for each individual coil.

[0059] The distinct magnetic field pattern generated by each individual coil is used for local spatial encoding. The principle of the invention is to apply temporal varying currents independently to each of the local magnetic field coils 115 during the acquisition of the MR signal. A temporal current variation induces a local varying magnetic field that modifies the local Larmor frequency of the magnetization which is sensed with the excitation/acquisition coil device 113. FIG. 4 shows an example where a sinusoidal current is applied to one coil element 115 (1 kHz at 1 A) during the acquisition of a free induction decay. In this example, this current produces a 1 kHz varying local magnetic field that generates a locally confined oscillation of the free induction decay (scale of decay signal axes: a.u.). The smooth curve a of FIG. 4A shows a free induction decay acquired at a constant magnetic field without applying any currents to the local magnetic field coils 115.

[0060] FIG. 4B shows the corresponding spectrum with a peak b around 0 Hz. Application of the varying magnetic modulation field of 1 kHz via the local coil 115 superimposes an oscillation onto the free induction decay (curve c in FIG. 4A). This produces 1 kHz separated side lobes d in the corresponding spectra shown in FIG. 4B, thus providing the additional spatial image information for the reconstruction of the MR signals.

[0061] Preferably, currents with different frequencies and/or phases are applied to each coil separately during the acquisition of the MR signal. Based on these locally different modulations introduced by the magnetic modulation field, the spatial origin of the acquired MR signal can be localized by the mathematical reconstruction procedures. With this additional spatial image information, the MR imaging process is accelerated.

[0062] The mathematical reconstruction procedure may comprise the reconstruction of local images associated to the side lobes shown in FIG. 4B and the subsequent superposition of the local images. Alternatively, the reconstruction may comprise a numerical optimization as outlined below.

Image Reconstruction by Numerical Optimization

[0063] In the following, the model of the image acquisition is specified, which is preferably used for image reconstruction. With this example, the reconstruction employs the assumption that an MR signal acquisition is performed using a single receive coil element of the excitation/acquisition coil device 113, which has a complex-valued sensitivity profile B.sub.1(r). If the MR signal acquisition is performed using multiple receive coil element, the reconstruction is extended as outlined below. Without loss of generality, relaxation effects are ignored in the present example model, and spatial encoding terms are considered.

[0064] Signals S acquired at time t and providing the spectrum s can be obtained via the integration over the excited volume.

[00001]$S\left(k,t\right)=\int \int \int m\left(r\right)B_1\left(r\right)\exp \left(-i\left(k\left(r,t\right)+B_c\left(r\right)\underset{t_1}{\stackrel{t}{\int }}{f}_c\left(r,\tau \right)d\tau \right)\right)\mathrm{dxdydz}\mathrm{where}k\left(r,f\right)=\underset{0}{\overset{t}{\int }}G\left(r,\tau \right)d\tau .$

[0065] The phase of the exponential term is composed of two parts. The term k(r,t) describes spatial linear gradients that are used to perform frequency and phase encoding (G: linear gradient vector, r: spatial vector, τ: time, and t: time point of signal acquisition). Finally, the second term B.sub.c(r)∫.sub.t.sub.1.sup.tf.sub.c(r,τ)dτ corresponds to a field induced by the local magnetic field coils 115 subject to an arbitrary waveform f.sub.c. B.sub.0 field profile of the coil element c in z-direction is indicated by B.sub.c, m(r) is the object image to be reconstructed. Furthermore, t.sub.1 indicates the beginning of the modulation. With the second term, the additional spatial image information of the spatially encoded magnetic resonance signals is provided. In general, other temporally varying patters can be applied.

[0066] The reconstruction results shown in FIG. 10 and FIG. 11 were obtained by discretizing the continuous model and assuming a sinusoidal waveform. With 2D acquisition protocol, K.sub.x, K.sub.y are the number of acquired k-space lines in readout and phase-encode directions, respectively, and N.sub.x, N.sub.y the number of pixels in spatial domain in readout/phase-encode directions, the complex-valued spectrum s∈.sup.K.sup.x.sup.×K.sup.y is acquired. From the spectrum s, the complex-valued image m∈.sup.N.sup.x.sup.×N.sup.y is to be reconstructed in the spatial domain. The image m is reconstructed by a numerical optimization procedure. The image acquisition process employed with the inventive imaging method can be described by a discrete linear operator E∈.sup.(K.sup.x.sup.K.sup.y.sup.)×(N.sup.x.sup.N.sup.y.sup.). The acquisition process is thus given by a linear equation s=Em. In case the currents injected into the local magnetic field coils are zero, the encoding matrix is an orthonormal Fourier transform matrix E=F. For ease of indexing the encoding matrix can be reshaped as E∈.sup.K.sup.x.sup.×K.sup.y.sup.×N.sup.x.sup.×N.sup.y. With injecting currents into the local magnetic field coils and the inventive encoding, the elements of the encoding matrix E are given by:

[00002]$E_{i,j,l,m}=F_{i,j,l,m}\exp \left(-\sqrt{-1}\frac{\underset{c}{.Math.}\left(a_c{B}_{c,l,m}\cos \left(2\pi w{\tau }_i+{\theta }_c\right)\right)}{w}\right)$

Here, i, j are the indices of the acquired k-space lines in readout and phase encode direction. Indices l,m are the spatial coordinates. F.sub.i,j,l,m are the elements of the Fourier transform matrix, B.sub.c∈.sup.C×N.sup.x.sup.×N.sup.y is the B.sub.0 field profile in z-direction of a local magnetic field from coil c, ω is the frequency of the inventive local modulation, a.sub.c is the current amplitude, θ.sub.c is the phase offset of the modulation, and τ.sub.i is the time vector.

[0067] It is assumed that there are no off-resonance field components and B.sub.0=0. In case the injected currents a.sub.c are non-zero, in order to reconstruct the image m the linear system s=Em is solved by computing a pseudoinverse of the encoding matrix E or by a numerical optimization. Here, E is the above linear operator that aggregates the exponential encoding terms and performs the summation (integration) over the spatial domain.

[0068] In MR signal acquisitions with geometry factors (g factor) close to unity, computing the pseudoinverse of the encoding matrix E and applying it to the measured k-space allows for a simple one-shot reconstruction. Otherwise, in case the g factor is greater than unity, inversion of the system can be unstable and results noisy. In this case, a modified least absolute shrinkage and selection operator (LASSO) optimization can be employed. The vanilla LASSO is modified such that in the regularizer term a total variation (L1 norm of the voxel differences in X/Y in spatial domain) is used instead of the plain L1 norm (as in usual LASSO). Total variation loss puts a high penalty on residual artifacts from the inventive encoding, which appear as blur and readout direction-smeared-out image content. The following regularized optimization problem is solved:

{circumflex over (m)}=argmin.sub.m(∥Em−s∥.sub.2.sup.2+λ|Dm|.sub.1)

[0069] The regularization coefficient λ sets the weight of the total variation term that penalizes high-frequency artifacts in the reconstruction. With the matrix D, finite pixel difference of the reconstructed image m is computed in spatial domain. A computationally intensive part in the optimization loop is the repetitive multiplications with the encoding matrix E, which can either be precomputed and stored in memory, or generated online. In the latter case, the operation can be efficiently performed on GPUs, since it relies on computing a massive number of independent complex-valued weighting coefficients subject to spatial location.

[0070] An extension of the model to the case of accelerated acquisition is straightforward and involves decreasing the number of rows in the matrix E subject to the k-space undersampling pattern.

Simulation Results

[0071] FIG. 5 (A, B, C—scale axes: pixel number) illustrates effects of the oscillating modulation current on the spatial encoding. For the reconstruction of simulated data, C=16 coil loops, a modulation frequency of 17 kHz, a current amplitude of 10 A are employed. FIG. 5A shows the simulated object to be imaged. An accelerated acquisition by keeping every 2.sup.nd k-space line and additionally keeping 4 central k-space lines around the DC component is assumed for the simulation. The sine form in each of the coil loops has a phase equal to

[00003]$\frac{2\pi c}{C},$

where c is the coil loop index. FIG. 5B shows the image as reconstructed with an inverse Fourier transform without taking into account additional field terms of inventive modulation. FIG. 5C shows the inventive reconstruction of an object image obtained by solving the above linear system. FIG. 5D shows a spectrum of singular values of the encoding matrix E.

[0072] In case multiple receive elements V of the excitation/acquisition coil device 113 are employed the model is extended to:

[00004]$S\left(k,t,\nu \right)=\int \int \int m\left(r\right)B_{1,\nu }\left(r\right)\exp \left(-i\left(k\left(r,t\right)+B_c\left(r\right)\underset{t_1}{\stackrel{t}{\int }}{f}_{c}t\left(r,\tau \right)d\tau \right)\right)\mathrm{dxdydz}$

[0073] The reconstruction problem is still formulated in terms of solution of s=Em, system, where both s and E now have additional rows originating from respective receive coil elements.

[0074] The image formation and reconstruction alternatively can be described with the concept of generating virtual phase-shifted images by the varying magnetic modulation fields as illustrated in FIGS. 6 and 7 (scale axes: pixel number). Each of the local magnetic field coils 115 modulates the MR signals in a restricted spatial section of the object, e.g. in 16 sections of the object 1 penetrated by the magnetic fields of the local magnetic field coils 115 (see FIG. 2). Accordingly, in each spatial section of the object, specific MR signals are locally collected. The MR signals are shifted in frequency space relative to the radiofrequency range of the unmodulated MR signals by the local modulation frequency as shown in FIG. 4B. MR signals from the different spatial sections provide virtual MR images with equal amplitudes but different phases. Accordingly, one or several sets of virtual images from the object are produced by the local frequency modulated magnetic fields. Based on a simulated object as shown in FIG. 6A, those virtual images are obtained as shown in FIGS. 6B and 7. In the step of image reconstruction, the virtual images and the information on the spatial origin thereof are used as additional local information.

[0075] Experimental Test Results

[0076] FIGS. 8 to 11 illustrated results of practical experiments performed on a whole body 9.4T MR scanner on a cylinder-shaped phantom filled with silicon oil. The experimental setup employed eight local magnetic field coils for the inventive magnetic field modulations. The eight local coils were arranged like one of the circular coil rows as shown in FIG. 2. For excitation and reception of the MR signal a patch antenna (see excitation/acquisition coil device 113 in FIG. 1) placed about 10 cm away from the phantom was used. The oil phantom was placed inside a larger hollow cylinder carrying the eight local magnetic field coils.

[0077] In a first test step, no currents have been applied to the eight local coils. A gradient echo sequence was used to image one slice of the oil phantom as shown in FIG. 8A (reference). Next, an alternating current of 2 A (zero-peak) and 1 kHz has been applied for only one of the local magnetic field coils (one single channel), and repeated for each local magnetic field coil separately. The alternating current was applied only during the readout period of each k-space line acquisition during the gradient echo sequence. FIGS. 8B to 81 (channels 1 to 8) show the resulting local image distortions and modulations. No reconstruction algorithm was applied except of a plain Fourier transformation of the k-space data.

[0078] Next, an alternating current of 2 A and 5 kHz was applied to all channels with a phase shift of 45° between all channels during each readout period of the gradient echo sequence. FIG. 9A (reference) shows the reference image without application of currents. FIG. 9B shows the resulting image modulations if only a plain Fourier transformation has been applied to the k-space data. FIG. 9C shows the object image reconstructed via the described method by inversion of the corresponding linear imaging equation. The spatial distribution of local magnetic fields produced by each single magnetic field coil has been measured before, and have been used to setup the matrix E in the linear imaging equation. This example shows that a complete reconstruction is possible using the described approach. However, in this example the local field modulations have not been used to accelerate the imaging process.

[0079] FIG. 10 shows an example where the local field modulations have been used to accelerate image acquisition with twofold undersampling. FIG. 10A (reference) represents the reference image. In FIG. 10B, again alternating currents of 2 A and 5 kHz were applied to all channels with a phase shift of 45° between all channels during each readout period of the gradient echo sequence. In contrast to FIG. 9B, in FIG. 10B only every other k-space line was acquired and thus image acquisition time was reduced by a factor of two. FIG. 10C shows the object image reconstructed via the described method by inversion of the corresponding linear imaging equation, but only using half of the sampled k-space lines as compared to FIG. 9. This example proves the basic acceleration advantage of the invention.

[0080] FIG. 11 shows an example of the singular values of the imaging encoding matrix E for the experimental test in FIG. 10 with two-fold undersampling in phase-encode direction, in case no locally specific frequency modulation is applied versus the modulation with 8 coils at 5 kHz and 2 A. FIG. 11A is with the application of alternating currents. Without application of alternating currents to the magnetic field coils (FIG. 11B) the upper half of singular values is zero, the matrix is rank-deficient and thus no reconstruction would be possible for two-fold k-space undersampling.

[0081] The features of the invention disclosed in the above description, the drawings and the claims can be of significance both individually as well as in combination or sub-combination for the realization of the invention in its various embodiments.