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Digital Tomographic Imaging with Time-Modulated Pseudorandom Coded Aperture and Anger Camera

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Digital Tomographic Imaging with Time-Modulated Pseudorandom Coded Aperture and Anger Camera DIGITAL TOMOGRAPHIC IMAGING WITH TIME-MODULATED PSEUDORANDOM CODED APERTURE AND ANGER CAMERA Kenneth F. Koral, W. Leslie Rogers, and Glenn F. Knoll The University of Michigan, Ann Arbor, Michigan The properties of a time-modulated pseudo gamma-ray imaging in nuclear medicine. In common random coded aperture with digital reconstruc with the pinhole and converging-hole collimator tion are compared with those of conventional they permit magnification of the object so that de collimators used in gamma-ray imaging. The tector resolution need not limit object resolution. theory of this coded aperture is given and the Contrary to the pinhole and converging collimator, signal-to-noise ratio in an element of the recon however, the field of view is not reduced to a point structed image is shown to depend on the entire as magnification is increased but rather approaches source distribution. Experimental results with a the dimensions of the coded aperture as a limit. preliminary 4 X 4-cm pseudorandom coded Another property peculiar to images formed by aperture and an Anger camera are presented. coded apertures is their insensitivity to detector non These results include phantom and human thy uniformity and noise. The gains with respect to roid images and tomographic images of a rat detector noise or indeed any noise component not bone scan. The experimental realization of the modulated by the aperture may be realized because theoretical advantages of the time-modulated coded apertures can subtend open solid angles that coded aperture gives reason for continuing the are orders of magnitude greater than conventional clinical implementation and further develop apertures. Detector uniformity is of less importance ment of the method. since each point in the source distribution is recorded as coded information over a large region of the de tector and nonuniform response is averaged out. Application of coded apertures to gamma-ray im These additional degrees of freedom pave the way aging in nuclear medicine is under active investi for innovations in detector design. gation by a number of research groups (1—14). In We will also show that a coded aperture is capable coded aperture imaging, the conventional pinhole or of giving improved statistical information for a num collimator is replaced by a lead plate with openings ber of nuclear medicine imaging situations. This arranged in a specific coded pattern. The image is gain can be thought of as arising from improved recorded in coded form and must be subsequently utilization of detector area and can be realized for decoded. The code is chosen to facilitate this recon any source region that is more intense than the struction process. To date with a few exceptions, average taken over the entire field of view. Tomog most investigators have employed some form of a raphy is an additional feature of coded apertures that Fresnel zone plate. Another type of coded aperture results from the fact that each source point is viewed in the form of a time-varying hole pattern based on from many angles. Tomographic information is si pseudorandom sequences of binary digits was first multaneously obtained for all possible source plane applied to gamma-ray imaging by Knoll (15,16). depths without the need to move the detector. Related work including that employing stochastic Time modulation of the aperture code yields fur codes has been reported (17—20) . This paper de ther advantages over stationary codes such as the scribes the theory of the time-modulated pseudo conventional Fresnel zone plate. These advantages random aperture and presents results obtained with an Anger camera. Received Oct. 26, 1974; original accepted Dec. 22, 1974. For reprintscontact: KennethF. Koral, R 3054KresgeII Coded apertures in general exhibit a number of University of Michigan Medical Center, Ann Arbor, Mich. unique features which make them attractive for 48104. 402 JOURNAL OF NUCLEAR MEDICINE stem from the fact that the point response function Source Point for a stationary code depends on a spatial integral over the recorded data. Both the scale factor and I Source the limits of the integration depend on the source position. For the time-modulated code the point response function depends on a time integration and spatial limits and scale factors do not enter into the problem. This difference manifests itself in the fol Aperture lowing ways: I. Source regions focus and defocus smoothly with no coherent artifacts. 2. The source distribution may be recon structed using a single detector element or widely spaced detectors. i—I—Detector 3. The resolution is uniform over the field of view. Radiation incident at other than 90 — CountRote deg on a zone plate, for instance, is prefer FIG. 1. Geometryinonedimensionfortime-modulatedcoded entially blocked by the finer rings because aperture. Counting rate in detector array from one point in source of the finite thickness of lead. For the time is shown for aperture in permutation depicted. modulated plate, the time average distribu dimensional aperture and detector. The source tion of the code over the plate is uniform aperture and aperture-detector spacing are repre and this problem is avoided. sented by A and B, respectively. The aperture is 4. The mean transmission of the code may be divided into L segments each of width w whose varied as a means for optimizing signal to gamma-ray transmittance is 1 or 0. A binary trans noise in the image. mittance function is not necessary but yields im 5. The time-modulated aperture offers one a! proved signal to noise over a continuous distribution ternative to the off-axis zone plate and half of transmittance. The detector counting rate due to tone screen. The off-axis zone plate requires a single point in the source is shown at the bottom about three times the detector resolution of the figure for the first aperture permutation. compared with the time-modulated aperture The total measurement time is divided into L equal for a given object resolution. This means intervals during which the counts in each detector the Anger camera cannot be used as a de are recorded. At the end of each interval the code tector but one must use instead a film pattern is changed and a new interval is begun. The screen combination with an attendant sacri raw data is composed of the counts in the jth detector fice in detection efficiency. during the @thtime interval, C1@,and the known trans In addition to the advantages offered by time mittance of the ith aperture element during the vth modulation are those that occur simply through time interval, T1+ digital acquisition and data processing. Among these Singledetector.In Appendix 1 it is shownthat advantages the following are significant: (A) no film the source strength viewed by the jt@@detector through grain noise, (B) increased dynamic range, (C) im the jth region of the aperture is given by the corre proved signal to noise in the reconstructed image lation coefficient compared with coherent optical processing, and (D) readily available quantitative results. L L @ oh = x T1+@—m Z (1) In the sections that follow, the theoretical devel v=1 v=1 opment of a time-modulated aperture for a single under the condition that the transmittance fiuctua detector is outlined and extended to a detector array. tions of different regions of the code pattern are out Expressions are derived for signal to noise and its of phase with each other. In this work, the sequence dependence on the mean transmission of the code. of code patterns fulfilling this requirement is ob The performance of a time-modulated aperture and tamed by cyclically permuting certain pseudorandom Anger camera-computer combination is illustrated binary sequences which are characterized by peaked and the results compared with an Anger camera autocorrelation functions with flat sidelobes (21,22). utilizing conventional apertures. The constant m in Eq. 1 is the sidelobe-to-peak ratio THEORY which is approximately equal to the mean transmis The essential features of the time-modulated coded sion for long code sequences (see Appendix 1). aperture geometry are portrayed in Fig. 1 for a one Figure 2 gives a pictorial rendition of the recon Volume 16, Number 5 403 KORAL, ROGERS, AND KNOLL CODE: llOI00000l000 L.13 r@4 q.I source activity within the field of view. This charac TERM I TERM 3 teristic can be seen qualitatively from Fig. 2 and Source Eq. 1; the single point source generates a uniform background in the image which can only be sub tracted on the average. The statistical noise in this background will add to the uncertainty for all other sources in the field of view. The magnitude of this uncertainty is derived in Appendix 2 where it is 0 Detector 0 a) b) shown that for the single detector case, the square TERM 4 TERM 8 of the relative standard deviation, S12,of the i@hsource element is given by 02_L0@ @B L 1 2 @ @,I r(L—r) ‘ The length of the code sequence is L, r is the number of open holes in the code sequence, and N1@@tis the @ total number of counts from the source the de SUM OF 4 TERMS tector would register through a completely open aperture during the entire measurement. The influ ence of all the other sources on the measurement of the jth source element is contained in B@which is the ratio of the total source strength in the field of view excluding the jth source to the strength of the itI@source: C) 0 I L B1@- @:N@. (3) FIG.2. Typicalpseudorandomcodeisshownat top.Contri .INj i/i bution to apparent source strength in each direction is shown by fan-shaped beams in A to D for time intervals 1, 3, 4, and 8.
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