Improving the Solution of an Incomplete Problem in Computed Tomography
Computed Tomography (CT) is an imaging method that uses radiation, such as Xrays, to scan an object from di erent orientations to provide projection-measurements to reconstruct a pixelated image of the interior of the object using mathematical algorithms. To successfully recover an unambiguous image of the scanned object, the number of measurements should be equal or higher than the number of the pixels (unknowns). This requires relatively long data acquisition time and high radiation exposure. To reduce radiation exposure, particularly in medical applications, this thesis investigates whether credible CT images can be recovered from a lower number of radiation projections; particularly when the CT problem is incomplete, i.e. when the number of measurements is less than the number of unknowns. In practice images are typically piece-wise constant i.e. consist of a limited number of materials in a few regions and consequently have a small set of image attributes. Then, constrained optimization methods, such as total-variation (TV) regularization and compressed sensing, can produce credible images in incomplete (underdetermined) CT problems. However, the uniqueness of such solutions cannot always i be assured, since an incomplete problem can have multiple solutions that match the input measurements. Therefore, it is proposed in this work to produce a unique image reconstructed by formulating the problem as an overdetermined problem. This was done by reconstructing a coarse image from the same CT data so that the number of pixels (unknowns) is higher than the number of available measurements. The reliable coarse image was then fused with the ne image reconstructed from incomplete data, in an attempt to improve the quality of the latter image. Several common image fusion methods were applied, but were found not to be e ective due to the inevitable smoothing of the image attributes due to the larger size of the pixels of the coarse image. Therefore, a new fuzzy image re nement approach was developed. It employs a membership function produced using the histogram of the reliable coarse image to re ne the attributes of the ne image, except when the coarse image was highly blurred because of high noise accompanied by a low degree of determination. This technique was shown to improve the quality of ne images reconstructed from incomplete data. The e ect of various CT parameters, on the methods applied in this work, were studied. Image reconstruction methods, measurement noise, resolution, phantom's con guration, degree of incompleteness and coarseness were examined. The fuzzy re nement technique was shown to improve image quality except when the coarse image was highly blurred. In future work, techniques for image reconstruction in non-sparse phantom and more sophisticated image fusion methods should be studied.
Networked Digital Library of Theses and Dissertations
Faculty of Graduate Studies and Research, University of Regina