IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. MI-4, NO. 3, SEPTEMBER 1985 A Method for Reconstructing Images from Data Obtained with a Hexagonal Bar Positron Camera GERD MUEHLLEHNER, MEMBER, IEEE, JOEL S. KARP, MEMBER, IEEE, AND ALBERT GUVENIS Abstract-This paper describes the algorithms and procedures de- veloped for reconstructing images using the hexagonal bar positron camera. This camera has six continuous position-sensitive detectors which are completely stationary, and it has some special software re- quirements. In particular, spatial nonlinearities in the detectors must be removed in software, the large but sparsely populated data matrix must be reduced in size, and the gaps in the data from the intersections of the detectors must be compensated for. These problems have been investigated, and an appropriate algorithm for this system has been implemented. The effectiveness of this algorithm was evaluated by re- constructing both real and simulated data. I. INTRODUCTION T HE hexagonal bar positron camera comprises six po- sition-sensitive detectors arranged in a hexagon to form a single image plane. The detectors consist of 500- mm long continuous bars of NaI(T1) to which ten photo- multipliers are coupled. The position of a scintillation event along the bar is determined through Anger-type pro- cessing electronics [1], [2]. Since this design differs from other conventional posi- tron tomographs which use arrays of discrete detectors, it requires additional software which corrects for the various imperfections associated with continuous position-sensi- tive detectors, as well as an image reconstruction algo- rithm which compensates for the gaps that occur between neighboring detectors since the camera is completely sta- tionary. The imperfections include the nonuniformity of response along the detectors and positional distortions in the measurements. Rebinning the data into a sinogram prior to reconstruction introduces sampling patterns which must be corrected for as well. In this paper, we will first describe the methods used to correct the raw data. The overall image reconstruction software will then be described. Finally, the results ob- tained from real and simulated objects will be reported. II. DISTORTION REMOVAL Spatial distortions are systematic errors in the position- ing of scintillation events. Each event results in a distri- bution of light at the plane of the photocathodes. The cen- troid of this distribution, as measured by the phototubes, is used to define the position of the event. The measured position, however, is only an approximate linear function Manuscript received January 21, 1985; revised April 24, 1985. This work was supported by the Department of Energy under Contract DE-AC- 80EV 10402. The authors are with the Department of Radiology, Hospital of the Uni- versity of Pennsylvania, Philadelphia, PA 19104. of the true position because the light distribution is sam- pled only every 50 mm, and also because of effects near the edges of the crystal. In addition, the differential and integral nonlinearities of the analog-to-digital converters contribute to the spatial distortions. Since there is a one-to-one correspondence between the measured and true positions, the spatial distortions can be measured and compensated for [3]. This correction is achieved in practice by collimating a line source placed at the center of the patient aperture and using a lead ring which has 90 equidistant slits (see Fig. 1). This type of collimator allows narrow beams of gamma rays to hit the detectors at known positions. Since the actual positions of the coincident gamma rays on the detector surface are known, the distortion factors can be computed by taking the difference between the measured and the actual posi- tions. We obtain 360 factors (one distortion factor per de- gree) by rotating the lead ring four times in 10 increments. This allows us to calculate the distortion factors at ap- proximately 8-mm intervals along the detector. A com- plete set of distortion factors is formed by linearly inter- polating values between the measured points since the distortions change smoothly and linearly over short dis- tances. Fig. 2 shows the data accumulated for a detector pair. Projection of the data onto either axis gives a spec- trum of peaks for the corresponding detector. In subse- quent collections (when the lead ring is removed, of course), both position coordinates are digitized for each coincident event, and the "true" location is obtained by adding the distortion factor to the digitized coordinate. The accuracy of the distortion removal scheme was evaluated by placing a point source midway between two opposing detectors. Since annihilation events are colinear, the measured position of a coincident event on one detec- tor is linearly dependent on the measured position on the other detector. Without distortion removal, the average deviation on the detector from this linear dependence is 2.5 mm, whereas after distortion removal, the average de- viation is only 0.5 mm. This method of distortion removal can correct the position of a scintillation only to an aver- age position, independently of the depth of interaction in the crystal. Particularly for gamma rays entering the crys- tal obliquely, the uncertainty about the depth of interaction causes a lateral offset which results in some loss of spatial resolution. Fig. 3 shows data from a number of point sources which have been rebinned into polar coordinates (see Section III for a description of rebinning) both without [Fig. 3(a) and 0278-0062/85/0900-0134$01.00 © 1985 IEEE 134