Mognerrc Re.,onanre /,,UZ@,ZR, Vol. 9, pp. 611-620, 1991 Printed in the USA. All rights reserved. 0730-725X/91 $3.M) + -00 (‘opyrighl @ 1991 Pergamon Pres\ plc zyxwvutsrqp BAYESIAN IMAGE PROCESSING IN MAGNETIC RESONANCE IMAGING zyxwvutsrqponm XIAOPING Hu,* VALEN JOHNSON,~ WING H. WONG,$ AND CHIN-TU CHEN* *Department of Radiology, University of Chicago Hospitals, Chicago, Illinois 60637, $.Department of Statistics, University of Chicago, Chicago, Illinois 60637, tInstitute of Decision and Statistics, Duke University, Durham, North Carolina 27706, USA In the past several years, image processing techniques based on Bayesian models have received considerable at- tention. In our earlier work, we developed a novel Bayesian approach which was primarily aimed at the’ processing and reconstruction of images in positron emission tomography. In this paper, we describe how the technique has been adopted to process magnetic resonance images in order to reduce noise and artifacts, thereby improving image quality. In this framework, the image is assumed to he a statistical variable whose posterior probability density conditional on the observed image is modeled by the product of the likelihood function of the observed data with a prior density based our prior knowledge. A Gibbs random field incorporating local continuity information and with edge-detection capability is used as the prior model. Based on the formalism of the posterior den&y, we can compute an estimate of the image using an iterative technique. We have implemented this technique and applied it to phantom and clinical images. Our results indicate that the approach works reasonably well for reducing noise, enhancing edges, and removing ringing artifact. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Keywords: Magnetic resonance imaging; Image processing; Bayesian image restoration. INTRODUCTION The past several years have seen a great deal of work in Bayesian approaches for image processing and re- construction. ‘-’ The general idea, was first formalized by Geman and Geman’ in 1984. In their model, the prior information employed was derived from the ob- servation that intensities of neighboring pixels of an image should be similar if the pixels belong to the same homogeneous region. This property of local con- tinuity was modeled by a Gibbs prior. In order to take into account the presence or absence of an edge be- tween pixels, they introduced horizontal and vertical line sites between pixels in addition to the conven- tional pixels (or intensity sites). In order to compute an appropriate estimate of the image under this model, they employed a simulated annealing ap- proach, namely the stochastic relaxation algorithm, to produce maximum aposteriori (MAP) estimates from the posterior density. Other workers have followed the same direction in developing new methods for process- ing images in various areas. ‘s3-’ Recently, we have extended the Geman and Ge- man’s approach for the processing and reconstruction of images in positron emission tomography (PET).7-9 Unique features included in our modified method are briefly described as follows. Firstly, line sites can take any value between 0 (no edge present) and 1 (edge present) and thus fuzzy edges are allowed. Secondly, we have expanded the neighborhood system so that it contains distant neighbors in order to account for the limited spatial resolution of the imaging system. Fi- nally, instead of using the time-consuming simulated annealing algorithm, we have adopted an approxima- tion algorithm similar to that proposed by Besag’ for computational efficiency. This approach has been suc- cessfully applied to PET. In this paper, we present our modification to the method and results from phantom zyxwvutsrq RECEIVED 9/4/90; ACCEPTED 2/6/91. Acknowledgment-This work was supported in part by Siemens medical systems (XH), Department of Energy (CTC, DE-FGO2-86ER60418) and National Science Founda- tion (WHW, NFS DMS 89-02667). We would like to thank Julian Besag for pointing out the necessity of a line site in- 611 hibiting clique. We would like to thank Drs. David N. Levin and Arthur E. Stillman for comments and discussions, and reviewers for helpful suggestions. Address correspondence to Xiaoping Hu, Ph.D, Dept. of Radiology, Box 292, UMHC, 420 Delaware Street, S.E., Minneapolis, MN 55455, USA.