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The problem of reconstructing an image from a set of tomographic data is not new, nor is it lacking attention. However there is still a distinct gap between the mathematicians and the experimental scientists working in the computed tomography (CT) imaging community. One of the aims in this thesis is to bridge this gap with mathematical reconstruction algorithms and analysis approaches applied to practical CT problems. The thesis begins with an extensive analysis for assessing the suitability of reconstruction algorithms for a given problem. The paper presented examines the idea of extracting physical information from a reconstructed sample and comparing against the known sample characteristics to determine the accuracy of a reconstructed volume. Various test cases are studied, which are relevant to both mathematicians and experimental scientists. These include the variance in quality of reconstructed volume as the dose is reduced or the implementation of the level set evolution method, used as part of a simultaneous reconstruction and segmentation technique. The work shows that the assessment of physical attributes results in more accurate conclusions. Furthermore, this approach allows for further analysis into interesting questions in CT. This theme is continued throughout the thesis. Recent results in compressive sensing (CS) gained attention in the CT community as they indicate the possibility of obtaining an accurate reconstruction of a sparse image from severely limited or reduced amount of measured data. Literature produced so far has not shown that CS directly guarantees a successful recovery in X-ray CT, and it is still unclear under which conditions a successful sparsity regularized reconstruction can be achieved. The work presented in the thesis aims to answer this question in a practical setting, and seeks to establish a direct connection between the success of sparsity regularization methods and the sparsity level of the image, which is similar to CS. Using this connection, one can determine the sufficient amount of measurements to collect from just the sparsity of an image. A link was found in a previous study using simulated data, and the work is repeated here with experimental data, where the sparsity level of the scanned object varies. The preliminary work presented here verifies the results from simulated data, showing an "almost-linear" relationship between the sparsity of the image and the sufficient amount of data for a successful sparsity regularized reconstruction. Several unexplained artefacts are noted in the literature as the `partial volume', the 'exponential edge gradient' or the 'penumbra' effect, with no clear explanation for their cause, or established techniques to remove them. The work presented in this paper shows that these artefacts are due to a non-linearity in the measured data, which comes from either the set up of the system, the scattering of rays or the dependency of linear attenuation on wavelength in the polychromatic case. However, even in monochromatic CT systems, the non-linearity effect can be detected. The paper shows that in some cases, the non-linearity effect is too large to ignore, and the reconstruction problem should be adapted to solve a non-linear problem. We derive this non-linear problem and solve it using a numerical optimization technique for both simulatedand real, gamma-ray data. When compared to reconstructions obtained using the standard linear model, the non-linear reconstructed images show clear improvements in that the non-linear effect is largely eliminated. The thesis is finished with a highlight article in the special issue of Solid Earth, named "Pore-scale tomography & imaging - applications, techniques and recommended practice". The paper presents a major technical advancement in a dynamic 3D CT data acquisition, where the latest hardware and optimal data acquisition plan are applied and as a result, ultra fast 3D volume acquisition was made possible. The experiment comprised of fast, free-falling water-saline drops traveling through a pack of rock grains with varying porosities. The imaging work was enhanced by the use of iterative methods and physical quantification analysis performed. The data acquisition and imaging work is the first in the field to capture a free falling drop and the imaging work clearly shows the fluid interaction with speed, gravity and more importantly, the inter- and intra-grain fluid transfers.