A maximum a posteriori (MAP) algorithm is presented for the estimation of spin-density and spin-spin decay distributions from frequency and phase-encoded magnetic resonance imaging data. Linear spatial localization gradients are assumed: the y-encode gradient applied during the phase preparation time of duration tau before measurement collection, and the x-encode gradient applied during the full data collection time t>/=0. The MRI signal model developed in M.I. Miller et al., J. Magn. Reson., ser. B (Apr. 1995) is used in which a signal resulting from M phase encodes (rows) and N frequency encode dimensions (columns) is modeled as a superposition of MN sinc-modulated exponentially decaying sinusoids with unknown spin-density and spin-spin decay parameters. The nonlinear least-squares MAP estimate of the spin density and spin-spin decay distributions solves for the 2MN spin-density and decay parameters minimizing the squared-error between the measured data and the sine-modulated exponentially decay signal model using an iterative expectation-maximization algorithm. A covariance diagonalizing transformation is derived which decouples the joint estimation of MN sinusoids into M separate N sinusoid optimizations, yielding an order of magnitude speed up in convergence. The MAP solutions are demonstrated to deliver a decrease in standard deviation of image parameter estimates on brain phantom data of greater than a factor of two over Fourier-based estimators of the spin density and spin-spin decay distributions. A parallel processor implementation is demonstrated which maps the N sinusoid coupled minimization to separate individual simple minimizations, one for each processor.
Model-based image reconstruction mcthods cuch as maximum-likelihood (ML) or maximum a-poytcriori (MAP) estimation rcquirc a signal modcl describing thc rclationship between the imagc paramctcrs to be eytimatcd and the measurement data. Two of the paramctcrc of intcrest in magnetic resonancc imaging (MRI) are thc tissuc spin dcnsity, A , and spin-spin dccay timc constant, Tz. This papcr presents a mathematical model for the signals obscrvcd in standard two-dimcnsional MRI expcriments and discucscs how this model is incorporated into a maximum a-posteriori parametcr estimation algorithm to compute imagc estimatcs of spin density and spin-spin dccay time. A detailcd dcwription of this work can be found in [I].The basic response of a magnetically scnsitivc population of nuclei to excitation in a magnetic rcsonancc cxpcrimcnt wac described by Bloch [2] as an exponcntially dccaying sinusoid whose frequcncy is proportional to the strength of thc static magnetic field to which the nuclei are exposed. In an MRI experiment, magnetic fields which vary with spatial position are employed to create a relationship in the observcci signal betwcen the frequency and phase of a sinusoidal signal component and the spatial position from which thc signal originated.The parameterized signal modcl which forms tlic ba+ of our M A P image reconstruction algorithm is bascd upon thrcc assumptions: 1) The frcqucncy and phasc encoding magnetic fields used for spatial localization vary lincarly with spatial position. 2) Voxels of dimcnsion D, x Dy cm2 arc small cnough that the spin density and spin spin dccay timc conytant within a single voxcl are constant.3) The loss of cignal cohcrcnce due to static magnetic field inhomogcncity rcwlts in signal attenuation which can bc rcprcscntcd as an exponcntial tlccay with time constant TM . Undcr thcsc assumption.; tlic signal emitted from a singlc voxcl a t position (xJ.') takcc thc formThe frequencies of oscillation ,fx(x) = CJ and f,b) = cyy arc linear functions of the encoding gradient strcngths c, and cy and position (xJ). The sinc-function paramctcrs (*xDx and cyDy are equal to the frequency bandwidths a c i w the (XJ) dimensions of the voxel. The full two-dimcnsional MRI signal is represented as a superposition of cinc-modukitcd, exponentially decaying sinusoids of the typc a bow, onc from each of the M x N voxels which form the imagc ficld.
Michael 1. Miller Washington University St. Louis, MO 63130-4899Under the assumption of additive, white, Gaussian noisc in the MRI mcasurcment data, thc maximum likelihood estimates of the spin dcnsity and spin-spin dccay image parameters are those paramctcrs which minimizc thc squarcd error between the measuremcnt data and a \ignal estimate computed from thc image paramctcrs ucing thc modcl described above.To compute thcsc imagc paramctcr estimates, we havc implemcntcd a form of tlic itcrativc expectation maximization (EM) algorithm of Dcmpctcr, Laird, and Rubin [ 3 ] .This algorithm has thc propcrty of decomposing the 2 x M x N-dimcnsional lcast-squarcs...
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.