Computed tomography (CT) relies on computational algorithms to reconstruct images from CT projections. Current filtered backprojection reconstruction methods have inherent limitations in situations with sharp density gradients and limited beam views. In this thesis two novel reconstruction algorithms were introduced: the Algebraic Image Reconstruction (AIR) algorithm, and the Geometric Image Reconstruction Algorithm (GIRA). A CT simulation was developed to test these novel algorithms and compare their images to filtered backprojection images. AIR and GIRA each demonstrated their proof of principle in these preliminary tests. AIR and its extension, the Parsed AIR algorithm (PAIR), were able to reconstruct optimal images compared to filtered backprojection after empirically determining parameters relevant to the algorithms. While GIRA reconstructed optimal images in preliminary tests, reconstruction was complicated by error propagation for larger imaging domains. The initial success of these novel approaches justifies continued research and development to determine their feasibility for practical CT image reconstruction.