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

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.