Generalized Two-Dimensional Orthogonal Spatial Encoding Fields

Introduction: The PatLoc concept has been introduced to perform imaging using non-bijective, curvilinear encoding fields [1]. Simulations have shown that a pair of mutually rotated multipolar encoding fields with cylindrical symmetry combined with a linear gradient for slice-selection have very good imaging properties especially in the outer half of the enclosed cylinder [2]. The gradients of these fields are mutually orthogonal as depicted in Fig. 1. Not least, this property is responsible for the good quality of the reconstructed images. We have addressed the problem of finding a simple method capable of generating all possible orthogonal fields for 2D encoding. Basic findings from complex analysis show that it is possible to describe the fields as real and imaginary parts of a complex-valued holomorphic function. Moreover, the set of holomorphic functions on a given simply connected region is equivalent to all mutually orthogonal source free fields on the same region.