This paper introduces a new theory, called focusing computed tomography (CT) scanner theory based around the third generation, x-ray fan-beam, CT scanner principle. The theory suggests the design of the CT system along with the manner to acquire the data set. Once the requirements as suggested by the theory are met, a simple and cost-effective set-up and a fast sampling scheme can be performed. Using a simple image reconstruction algorithm, accurate reconstructed CT images, relative to the images of the first generation x-ray pencil-beam CT scanner, can be obtained. A set of examples supporting the principle is presented.
Keywords: CT, NDT, Focusing CT
Computed Tomography (CT) scanner is a desired device to create images of internal structure of unrecognized object [1, 2]. However, many criticisms have been addressed to the manner the CT technology developed. It is true that the development of CT devices have been extremely advanced. The system looks sophisticated, fast, but it tends to be more expensive, more complicated and lack of accuracy. Similar remark is also addressed by Carlsson .
The accuracy of CT image in representing a physical distribution of a cross-sectional 2-D structure is essential. Therefore, standard and reference for judging a CT performance is substantial. The work described in this paper explores the idea of focusing CT scanner in the manner to produce similar CT images to first, parallel-beam, CT images from a third, fan-beam, CT scanner. This can be accomplished by introducing a special arrangement of CT scanner called focusing design  based on third generation mode . This arrangement allows all physical parameters associated with the CT system are well defined. Therefore, the images resulted from this focusing set-up can be compared directly to the images from the first generation set-up. This set-up also allows a sampling scheme for sinogram following a typical rectangular or square pattern, in which all ray-sums in a projection are essentially parallel. This sinogram unnecessary requires pre-interpolation or rebinning  prior to the back-projection process. At last, as the pattern of data set is essentially similar to the first CT mode [7, 8], the image reconstruction can be performed in a similar way with the first generation . Therefore, the term of focusing is introduced to highlight the idea of developing a clear and well-defined first CT images from third fan-beam CT scanner.
The basic principle of focusing scanner  is described in Figure 1. The center of curved array of detectors is at the locus of radiation source that emits fan-beams with a fan angle of g. The center of rotation, O, is at the middle of the line from radiation source to the central detector in the array (s = 0). The radius of image circle is defined as R = S sin(g/2), where S is distance from source to center of rotation. The focusing scanning produces parallel projections f from fan projections q. The focusing principle requires a strict relation of f = q + s, where s indicates orientation of a detector element from the source. The focusing scanning also requires step sampling of fan-projection Dq = (S/R)Ds. This strict requirement ensures the sinogram forms a square pattern in (f, xs) space, where xs = Ss, in which a fan-projection, a set of ray-sum at certain object rotation, q, is along an inclined line (Figure 2). Each detector element associates with beam-width of w = S Ds.
Next section describes the experiment, followed by results and discussion. The concluding remarks are presented in the last section.
Fig 1: The basic principle of Focusing CT scanner set-up.|
Fig 2: Sinogram mapping for Focusing CT scanner. The fan-beam projection lies on the inclined line. The parallel projection consists of parallel ray-paths at angle of projection f|
Method and Experiments
The study has utilized data sets acquired in the Department of Physics, Monash University, Victoria Australia . The experiment was carried out using x-ray source from Cu anode using single NaI(Tl) detector, but the sinograms were sampled following focusing scheme for scanning interval of 360o. Two focusing systems, for fan angle of 45o and 60o were set-up. The object being inspected was a piece of pinewood, approximately 10 mm x 10 mm in size with a hole of 2 mm in diameter drilled perpendicular to the scan plane. After sinogram acquisition, the focusing sinograms were reconstructed using similar process to the first generation [4, 10]. As the focusing systems are well defined, the focusing images can be compared directly to the image acquire using conventional, first, translate-rotate, CT mode.
3. Results and Discussion
The images acquired using focusing CT principle for fan-beam 45o and 60o, in which a first CT image as reference are shown in Figure 3. The images have equal dimension of pixels matrix, 115 x 115 and equal size of pixel, 0.15 mm x 0.15 mm. Therefore, the focusing images can be compared directly with the associated first generation CT image. The image differences are shown in Figure 4 in which the image differences are not significant for both fan angles. Root mean square difference is up to 5.0% for fan angle of 60o.
Fig 3: CT images. From left to right: first generation CT scanner, Focusing CT scanner for fan-angle 45o and Focusing CT scanner for fan-angle 60o.|
Fig 4: CT images. From left to right: first generation CT scanner, Image difference of Focusing CT
scanner for fan-angle 45o and 60o relative to the first generation
Based on this study, it is essential to sample a sinogram data set in computed tomography considering the collimator or beam size, w. This parameter determines the physical spatial resolution of w/2 and should be conserved in any higher generation. The introduction of higher generations should consider both time constraint and physical representation associated with this beam-width. If this requirement is met, physical values of any CT images from higher generations can be compared directly to the associated first generation CT images.
The way to map the sinogram in (f, xs) space rather than in (q, s) is essential in which the fan-beam data set must along an inclined line. Since the ray-sums are parallel, the Radon property p(f, xs) = p(s+p, -xs) is preserved. As consequences, a full scanning of 360o in which M is even will provide redundant ray-sum and hexagonal sampling scheme can be performed. Significant saving on time and number of detectors in higher CT scanners can be achieved by implementing hexagonal sampling scheme into any CT scanners based around this focusing CT mode .
Standard, reference, purpose and specification are essential requirements. Based on this study, it is essential to adopt a first, translate-rotate, CT scanner as a standard CT system in order to justify a performance of higher generation CT scanners. Although, the first CT scanner is very slow, but the images resulted are considered to be accurate and precise according to the specifications characterized by the system. The focusing CT scanner provides a similar quality of CT images to the images from the first CT scanner in a similar and simple way, but in a fast manner.
Development of focusing design may re-direct and re-shape the development and designs of CT scanner technology appropriately based on its purpose and its specifications, along with its standard and reference, the first generation CT mode. Moreover, significant benefits may be achieved by a combination of focused CT scanner, hexagonal sampling scheme and apparently helical/spiral CT mode. Problem of scattering in the current CT scanners may be able to be minimized as the scanning interval for focused CT system is 360o, rather than p + g; in which the scattering profile is certain and a function of cattering angle.
The authors acknowledge the assistance of Associate Professor Peter Wells for his collaboration during finishing the thesis. The author also acknowledged the assistance from Dr. John Davis, Mr. Nino Benci, Mr. Cameron Kewish and other CT group members in Monash University, Melbourne, Victoria, Australia.
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