Basis function cross‐correlations for Robust k‐space sample density compensation, with application to the design of radiofrequency excitations

Abstract

The problem of k -space sample density compensation is restated as the normalization of the independent information that can be expressed by the ensemble of Fourier basis functions corresponding to the trajectory. Specifically, multiple samples (complex exponential functions) may be contributing to each independent information element (independent basis function). Normalization can be accomplished by solving a linear system based on the cross-correlation matrix of the underlying Fourier basis functions. The solution to this system is straightforward and can be obtained without resorting to discretization since the cross-correlations of Fourier basis functions are analytically known. Furthermore, no restrictions are placed on the k -space trajectory and its point-spread function. Additionally, the linear system can be used to elucidate key trade-offs involved in k -space trajectory design. The approach can be used to compensate samples acquired for image reconstruction or designed for low flip angle radiofrequency (RF) excitation. Here it is demonstrated for the latter application, using reversed spiral trajectories. In this case the linear system approach enables one to easily incorporate additional constraints such as smoothness to the solution. For typical RF excitation durations (<20 ms) it is shown that density compensation can even be achieved without numerical iteration.Magn Reson Med 57:338-352, 2007.

Comparing MR Imaging Properties of Spiral Trajectories Using the Singular Spectrum of the Analytical Fourier Basis Cross-Correlation Matrix

“…The result is a matrix of cross-correlation values which determines a linear system whose solution gives the desired density compensation. As we describe in more detail below, this matrix gives the normal equations for the overdetermined linear system presented in [5], modulo the difference between direct sampling of the Fourier functions as in [5] compared to analytically computing the matrix elements as in [4]. More importantly, unlike the direct linear system in [5] where the final image must be known (as is the case for the problem of radio-frequency (RF) excitation design), the normal equations can also be solved for the problem of image reconstruction from acquired samples.…”

“…Our group recently presented a new algorithm [4] that has potential application for several aspects of imaging with spiral trajectories. This algorithm, presented in the context of density compensation, projects, or cross-correlates the spiral samples onto the underlying Fourier basis vectors.…”

“…More importantly, unlike the direct linear system in [5] where the final image must be known (as is the case for the problem of radio-frequency (RF) excitation design), the normal equations can also be solved for the problem of image reconstruction from acquired samples. Further, in [4] the singular value spectrum of the cross-correlation matrix was used to illustrate differences between trajectories in terms of the degree of independence of their components and their noise sensitivity.…”

Free‐breathing inner‐volume black‐blood imaging of the human heart using two‐dimensionally selective local excitation at 3 T

“…Since then, a steadily growing number of applications for 2D‐selective pulses have been reported, facilitated by advances in gradient hardware and the demand for fast imaging. Recently, the use of self‐refocused 2D‐selective RF pulses that account for T 2 *‐dependent relaxation during the excitation has been reported for sharper FOV localization and better outer‐volume suppression (2, 11–16). Although the theoretical aspects of these strategies show a potential for removing the need for refocusing gradients, and thereby shortening the minimum TE, practical implementations have only been demonstrated in phantoms (11, 15), at low field strength (1, 14, 17), and for imaging stationary or slow‐moving organs (1–3, 13, 14, 17, 18).…”

Fast, exact k‐space sample density compensation for trajectories composed of rotationally symmetric segments, and the SNR‐optimized image reconstruction from non‐Cartesian samples

“…The problem of MR image reconstruction from k-space samples arranged on non-Cartesian grids has been extensively studied in the past two decades (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20). The essence of this problem is that the samples collected do not provide independent information about the imaged object within the confines of the desired field-of-view (FOV) (20). That is, direct expansion of the collected coefficients using the encoding basis (i.e., complex exponential Fourier basis functions) produces a reconstructed object that possesses a different Fourier coefficient spectrum than that observed.…”