COMPARATIVE STUDY OF SOME ALGORITHMS FOR IMAGE RECONSTRUCTION

Abstract

Image reconstruction from projections is required in many practical problems, where the original object is either not visible or not measurable directly, but its projections from various directions are measurable. Such problems occur mainly in medical diagnostics imaging, radio and radar astronomy, electron microscopy, magnetic resonance imaging (MRI), non destructive testing and many more applications. More generally image reconstruction from projections deals with viewing the internal structure within an object in a non-invasive manner. In this process the plane sections of three dimensional objects are visualized using X-ray (in most applications) or γ- ray (some industrial applications) measurements. These measurements are taken as loss of energy and the internal structure is reconstructed as attenuation of these energies, which is closely related to the density of the object. Starting from 1970, the image reconstruction from projection earlier referred as computerized tomography (CT) or computer assisted tomography (CAT) has been initialized. In early development it is only getting two dimensional reconstruction from projections modelled as line or strip integrals along parallel lines or strips in different directions covering the whole objects (as 180 degrees). Further, with more technological advancement, the interest to get three dimensional reconstructions increased. These advancements mainly took place more recently in last decade. In present thesis the advancement of techniques from two dimensional image reconstruction to three dimensional image reconstructions from one dimensional and two dimensional projections are discussed. An image is a three dimensional object’s internal structure, the region of interest (ROI) for reconstruction will always be unit sphere in 3 for three dimensional reconstruction and unit circle in 2 in two dimensional reconstructions. ii In two dimensional image reconstruction problems, transform methods and algebraic methods have been analyzed. In transform method the analysis is focussed on convolution backprojection method based on Fourier slice theorem with parallel beam projection and convolution back projection method transformed for fan beam projections for both cases of detector arrangements, i.e. equiangular detector array and equispaced detector array. In algebraic reconstruction method the results have been compared with modified simultaneous algebraic reconstruction technique (MSART). In three dimensional image reconstruction methods, first, slice reconstruction has been discussed then direct three dimensional reconstruction from two dimensional projection data with circular source trajectory with parallel plane data has been discussed. Next, the reconstruction from cone beam projection data in helical or spiral source trajectory has been discussed for both types of detector arrangements viz. curved surface detector array and flat surface detector array. In both cases Katsevich PI line reconstruction method and the practical cone beam algorithm of Feldkemp Davis & Kress known as FDK method has been analyzed. All these methods of image reconstruction in two and three dimensions are converted to algorithms applicable to simulated Shepp-Logan phantoms and Jaw phantom. For converting the reconstruction methods to applicable algorithms all the variables have been standardized for two dimension case. Similarly the standardization is done for three dimensional and all algorithms are provided namely, convolution back projection (CBP) method for parallel beam projection, CBP method for equiangular detectors in fan beam X-ray and CBP method for equispaced detectors array in fan beam projections. All the algorithms are applied in two dimensional reconstruction as well as slices of three dimensional phantoms. The algorithms are compared on the basis of error analysis and time complexity. The errors reported are actual errors of the methods, they do not contain any measurement errors, since the projection data is simulated on the simulated image these errors off course contain approximation and interpolation errors.