High-field MRI systems significantly increase signal-to-noise ratio (SNR) (

1), but in vivo imaging at high field is impeded by the presence of severe

inhomogeneity (

2) arising due to wavelength interference effects (

3,

4) and tissue-conductive radio-frequency (RF) amplitude attenuation (

5). Inhomogeneity is also a concern at low field when structures such as the spine (

6) and body (

7) are imaged. When standard slice-selective RF excitation wave-forms are used for imaging,

inhomogeneity causes images to exhibit center brightening, spatial contrast variation, and SNR nonuniformity, despite the use of homogeneous volume RF excitation coils (

3,

4,

8 –

10).

The three-dimensional (3D) RF pulse designs proposed in (

11–

13) describe a class of slice-selective pulses capable of mitigating

inhomogeneity that offer improvements over high specific absorption rate (SAR) adiabatic pulses (

14) and image postprocessing methods (

15). These pulses are played in the presence of echo-volumnar-like 3D gradients. They consist of modulated sinc-like pulse segments (“spokes”) in the

*k*_{z} direction of excitation

*k*-space positioned at locations in (

*k*_{x},

*k*_{y}). Spoke-based pulses are used in the small-tip-angle regime (

16), in which the sinc-like RF depositions in

*k*_{z} produce slice-selectivity in

*z* and the amplitude and phase modulation of each spoke in (

*k*_{x},

*k*_{y}) spatially tailors the excitation in (

*x*,

*y*) to mitigate the in-plane inhomogeneity. An ideal

mitigation pulse excites the point-wise inverse of the inhomogeneity and yields a uniform magnetization; therefore, in practice, spoke modulation terms are chosen such that they produce an in-plane excitation that closely resembles the ideal one. Unlike a shimming approach, a spoke-based waveform does not flatten the

field; rather, the gradient modulation of the excitation process is used to produce a uniform magnetization. The “standard slice-selective” pulse that we refer to throughout this work is equivalent to a single-spoke pulse whose spoke is located at the

*k*-space origin.

In prior work, relatively few spokes have been used for inhomogeneity mitigation on single-channel (

11) and multichannel parallel transmission systems (

12,

13,

17–

20). In all cases, work is performed at field strengths below 7T, where

inhomogeneity in the brain is less severe, resembling a quadratic function in space (

11). In contrast,

inhomogeneity at 7T exhibits significant spatial variation and is not quadratic (

3,

4). This means that spoke designs that utilize single-channel transmit systems and rely on quadratic assumptions about

(

11) are unlikely to mitigate brain inhomogeneity at 7T. Parallel excitation systems, on the other hand, are indeed useful for

mitigation at high field, but are expensive in terms of hardware and complexity: each transmission channel requires an RF power amplifier as well as a SAR monitor. Based on the above, it is evident that a method is needed to design fast, slice-selective,

mitigation pulses for use on high-field single-channel systems.

Since

is highly nonuniform at 7T (

3,

4), one approach to mitigating it would be to extend prior spoke-based designs by placing a large number of modulated spokes throughout (

*k*_{x},

*k*_{y})-space, covering both low and high spatial frequencies. Unfortunately, placing many spokes leads to impracticably-long pulses. An alternate method is to compute the Fourier transform of the ideal in-plane excitation and place spokes in

*k*-space where Fourier coefficients are largest in magnitude (

21). Unfortunately, this tends to concentrate spokes around (

*k*_{x} = 0 ·

*k*_{y} = 0) DC, analogous to a low-pass filter. Further, it places unneeded constraints on the design outside the given field of excitation (FOX) (e.g., outside the brain) and fails to account for the influence of the transmission profile. These problems reduce the Fourier method’s ability to produce an excitation with enough spatial variation to mitigate the inhomogeneity. In response to this problem, we introduce here a novel method that determines a minimal number of spokes needed for

mitigation within a specified FOX, producing short pulses that mitigate inhomogeneity at 7T; this is an extension of prior work (

22,

23). The method provides designers with control parameters that trade off

mitigation with pulse duration. Given a

map of the head, the algorithm finds the minimal number of spokes necessary to mitigate the inhomogeneity along with their placement in (

*k*_{x},

*k*_{y}) and their proper modulations. The algorithm, based on sparse approximation (

24,

25), enforces sparsity on the number of spokes allowed while encouraging those that remain to be placed and modulated in a way that maximizes

mitigation in the least-squares sense. In this work, we demonstrate the capabilities of sparsity-enforced pulse design by performing mitigation experiments at 7T in a head-shaped phantom and the human brain.