2134 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 30, NO. 12, DECEMBER 2011 Radial Imaging With Multipolar Magnetic Encoding Fields Gerrit Schultz*, Student Member, IEEE, Hans Weber, Daniel Gallichan, Walter R.T. Witschey, Member, IEEE, Anna M. Welz, Chris A. Cocosco, Jürgen Hennig, and Maxim Zaitsev Abstract—We present reconstruction methods for radial mag- netic resonance imaging (MRI) data which were spatially encoded using a pair of orthogonal multipolar magnetic fields for in-plane encoding and parallel imaging. It is shown that a direct method exists in addition to iterative reconstruction. Standard direct projection reconstruction algorithms can be combined with a previously developed direct reconstruction for multipolar en- coding fields acquired with Cartesian trajectories. The algorithm is simplified by recasting the reconstruction problem into polar coordinates. In this formulation distortion and aliasing become separate effects. Distortion occurs only along the radial direction and aliasing along the azimuthal direction. Moreover, aliased points are equidistantly distributed in this representation, and, consequently, Cartesian SENSE is directly applicable with highly effective unfolding properties for radio-frequency coils arranged with a radial symmetry. The direct and iterative methods are applied to simulated data to analyze basic properties of the al- gorithms and for the first time also measured in vivo data are presented. The results are compared to linear spatial encoding using a radial trajectory and quadrupolar encoding using a Cartesian trajectory. The direct reconstruction gives good results for fully sampled datasets. Undersampled datasets, however, show star-shaped artifacts, which are significantly reduced with the iterative reconstruction. Index Terms—Image reconstruction, nonlinear encoding fields, parallel imaging, radial imaging, spatial encoding. I. INTRODUCTION T HE first magnetic resonance imaging (MRI) data were ac- quired using a radial k-space trajectory [1]. Nowadays, radial imaging plays an important role in MRI, especially in clinical research settings. It offers unique and fast encoding op- tions [2], [3] and there are ongoing efforts to develop advanced imaging techniques [4], [5] and investigate interesting applica- tions [6], [7]. Recent developments in improving acquisition and reconstruction methods add to the importance of radial imaging for MRI [8], [9]. Manuscript received June 15, 2011; revised July 22, 2011; accepted July 28, 2011. Date of publication August 12, 2011; date of current version December 02, 2011. This work was supported by the INUMAC project funded by the German Federal Ministry of Education and Research under Grant 13N9208. Asterisk indicates corresponding author. *G. Schultz is with the Department of Radiology, Medical Physics, University Medical Center Freiburg, 79106 Freiburg, Germany (e-mail: gerrit.schultz@uniklinik-freiburg.de). H. Weber, D. Gallichan, W. R. T. Witschey, A. M. Welz, C. A. Cocosco, J. Hennig, and M. Zaitsev are with the Department of Radiology, Medical Physics, University Medical Center Freiburg, 79106 Freiburg, Germany. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMI.2011.2164262 New fields of application have emerged with the invention of PatLoc [10], an imaging concept where the requirement of gradient linearity for signal localization is relaxed in favor of performance resulting in non-bijective and curvilinear spa- tial encoding magnetic fields (SEMs). It has been proposed that depth-encoding with curvilinear SEMs, aligned with the anatomical geometry of the brain, would be more suitable than the conventionally applied linear field geometries. Moreover, it has been hypothesized that non-bijective SEMs will reduce peripheral nerve stimulation compared to linear gradient fields. Initial investigations [11], [12] were primarily based on a proto- type hardware configuration with two quadrupolar SEMs [13] for in-plane encoding in addition to the conventional -gradient for slice selection. Recently, four [14] and five [15] channel systems with simultaneous switching of linear and nonlinear SEMs have been implemented and investigated. This paper focuses on a second-generation three-channel configuration suitable for in vivo measurements [16], [17]. The PatLoc approach has already been studied in detail with respect to Cartesian sampling trajectories [11]. As a conse- quence of the nonlinearity of the SEMs, reconstructed images result with nonhomogeneous resolution. However, image con- trast is not influenced by the field geometry of the SEMs per se. Therefore, apart from spatial resolution, conventional imaging methods show a similar behavior when adopted to arbitrary field encoding. This is in particular true for radial imaging schemes, which can show a high tolerance to undersampling in situations with high imaging contrast and which have inherent self-navigating properties resulting from oversampling of the k-space center [2]. These and other properties make radial imaging trajecto- ries useful for medical imaging also in the context of PatLoc imaging. However, care must be taken whenever the gradient fields become very flat; nonlocal image artifacts might result, which could deteriorate the diagnostic usability of the acquired images without further artifact reduction [11]. In this paper, these issues are treated from an image recon- struction perspective with the purpose to develop, analyze, and compare different reconstruction methods for radial encoding. A focus lies on a fast and easy-to-interprete direct reconstruc- tion method. The basic observation is that, whereas in conventional imaging a projection is taken along one straight line, in PatLoc imaging projections are also taken along two or more curved lines. Standard back-projection of the data can, therefore, not directly result in a reconstructed image. From [11] and [15] it is known that iterative reconstruction methods can be used 0278-0062/$26.00 © 2011 IEEE