Analytic Reconstruction of the PERL Transform
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Analytic Reconstruction of the PERL Transform M.I. Hrovat and S. Patz Mirtech Inc. and Brigham & Women’s Hospital Purpose We have determined that the Ψm ( y ) are given by Previously, we proposed a two-dimensional m PERiodic-Linear (PERL) encoding field geometry of the Ψm ( y ) = J m ( k y y ) Sgn ( y )  y form Bx (x, y) = gy y cos(q x x) and an imaging sequence where Sgn(y) (1 for y>0 & –1 for y<0), is the sign function. which incorporates an additional standard linear gradient along x to image a 2D spin density.1,4 We have recently The PSF is then given by discovered an analytic procedure for the reconstruction ∞ J n (ky ) process in contrast to the numerical method2 we employed h ( x − y ) =δ ( y ) J 0 ( kx ) + ∑ n Sgn( y ) J n ( kx )  earlier and the semi-analytic procedure recently proposed.3 n =1 y Introduction We have been able to validate this expression. The In the PERL imaging technique, the phase of the [ ] summation term can be shown in the limit of large x,y or signal is given by Φ(t) = γ Gx xt − g y yT cos(q x x ) . where large k to be equivalent to sin(k(x-y))/(x-y). Gx is the amplitude of the linear gradient, gy is the Discussion amplitude of the PERL field, T is the time duration of the The analytic solution greatly simplifies the PERL field, and qx is the wavelength of the PERL field. reconstruction process and furthermore eliminates the The resulting exponential of the phase factor may be truncation error induced from our prior numerical expanded (Jacobi-Anger expansion) to reveal that the x - approach. Furthermore the sinc-like character of the PSF dimension is encoded in a Fourier fashion, but the y - which we had observed previously2 has been verified. dimension appears as the argument of a Bessel function. The PT is based upon expanding a function as Consequently, the time-domain signal and image spin series of Bessel functions with respect to order. This is a density are not related by a two-dimensional Fourier fundamentally different expansion then that associated with transform (2DFT). The x-dimension is treated by a 1D- a Fourier-Bessel series or a Hankel transform. The early FFT. However, the y-dimension transform features a new literature5-7 had concluded that this expansion was quite Bessel function integral transform, which we have termed limited and did not assign much importance to it. To our the PERL transform (PT). knowledge this view is still currently held. We have learned ∞ from our application and analysis that this expansion has ρ ’m ( x ) = ∫ ρ ( x, y ) J −∞ m ( k y y )dy  wider applicability. In particular the factor k, which is fixed in the PERL experiment, is intimately connected to the with k y = γgy T . The solution to the inverse of the PT limiting resolution, similar to the limiting resolution created provides a reconstruction algorithm for the y -dimension of by a truncated Fourier series. This in turn implies that with the spin density from the signal space. To date, the inverse a suitable scaling factor, the expansion is applicable to most PT has been computed numerically by a Bessel function cases of interest. expansion over its basis functions2 and a semi-analytic References procedure which involves a spherical Bessel function 1. S. Patz, M.I. Hrovat, Y.M. Pulyer, and F. J. Rybicki. expansion.3 “Novel encoding technology for ultrafast MRI in a Methods limited spatial region”. Int J of Imaging Syst and The inverse of Eq. requires finding the Technol (1999) 10: 216-224. complimentary functions, Ψm ( y ) , which satisfy the 2. F.J. Rybicki, S. Patz, and M.I. Hrovat. “Reconstruction algorithm for novel ultrafast MR imaging.” Int J criteria: Imaging Syst and Technol (1999) 10: 209-215. ∞ 3. F.J. Rybicki, S. Patz, M.I. Hrovat, Analytic ∫Ψ −∞ m ( y ) J m ’ (k y y )dy = δmm’  Reconstruction of MRI Signal Obtained from a ∞ Periodic Encoding Field.85th RSNA Proceedings, 1999, ∑J m =0 m (k y y )Ψm ( y ’) = h( y − y ’)  Chicago, Il. 4. S. Patz, M.I. Hrovat, F.J. Rybicki. Properties of a where h(y) is a point spread function (PSF) which for an Periodic and Linear Spatial Encoding Field for MR exact solution would behave as a delta function. Imaging. 85th RSNA Proceedings, 1999, Chicago, Il. Results 5. G.N. Watson, A Treatise on the Theory of Bessel Based on the following property5-7: Functions. University Press. Cambridge, 1948. ∞ 6. H.A. Webb, Messenger Math 33, 55 (1904). dk 2 sin (( m − n ) π 2) ∫J 0 m ( kx ) J n ( kx ) = k π ( m 2 −n 2 )  7. W. Kapteyn, Messenger Math 35, 122 (1906).