Analytic reconstruction of magnetic resonance imaging signal obtained froma periodic encoding field

We have proposed a two-dimensional PERiodic-Linear (PERL) magnetic encoding field geometry B(x,y)=g(y)y cos(q(x)x) and a magnetic resonance imaging pu lse sequence which incorporates two fields to image a two-dimensional spin density: a standard linear gradient in the x dimension, and the PERL field. Because of its periodicity, the PERL field produces a signal where the pha se of the two dimensions is functionally different. The x dimension is enco ded linearly, but the y dimension appears as the argument of a sinusoidal p hase term. Thus, the time-domain signal and image spin density are not rela ted by a two-dimensional Fourier transform. They are related by a one-dimen sional Fourier transform in the x dimension and a new Bessel function integ ral transform (the PERL transform) in the y dimension. The inverse of the P ERL transform provides a reconstruction algorithm for the y dimension of th e spin density from the signal space. To date, the inverse transform has be en computed numerically by a Bessel function expansion over its basis funct ions. This numerical solution used a finite sum to approximate an infinite summation and thus introduced a truncation error. This work analytically de termines the basis functions for the PERL transform and incorporates them i nto the reconstruction algorithm. The improved algorithm is demonstrated by (1) direct comparison between the numerically and analytically computed ba sis functions, and (2) reconstruction of a known spin density. The new solu tion for the basis functions also lends proof of the system function for th e PERL transform under specific conditions. (C) 2000 American Association o f Physicists in Medicine. [S0094-2405(00)02009-5].