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We show that a reciprocal space squared intensity map of a material can be recovered, for each characteristic length scale, from diffraction tomography data by a simple slice-by-slice reconstruction method. Moreover if the reciprocal space map can be represented by a finite sum of spherical harmonic components for each length scale then the coefficients of that expansion can be recovered from inverting the transverse ray transform (TRT), where the data are polynomial coefficients of the azimuthal diffraction pattern for each length scale.

Reconstructing a 2D slice or a 3D volume from a set of insufficient tomographic data is a difficult problem, and it is often tackled with analytical reconstruction algorithms. However, these types of methods fall short on delivering a quality image due to the severe artefacts introduced by the insufficiency of the data. The presented work shows the effects of taking tomographic scans with fewer radiographs on the quality of the reconstructed images. The aim here is to show the advantages of using iterative reconstruction methods over analytical methods, which are demonstrated by a quantitative comparison.

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 recon- struction 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 vol- ume. 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. Fur- thermore, 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 im- age 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 t

A nonlinear elastic microstructural model is used to investigate the relationship between structure and function in energy-storing and positional tendons. The model is used to fit mechanical tension test data from the equine common digital extensor tendon (CDET) and superficial digital flexor tendon (SDFT), which are used as archetypes of positional and energy-storing tendons, respectively. The fibril crimp and fascicle helix angles of the two tendon types are used as fitting parameters in the mathematical model to predict their values. The outer fibril crimp angles were predicted to be 15.1° ± 2.3° in the CDET and 15.8° ± 4.1° in the SDFT, and the average crimp angles were predicted to be 10.0° ± 1.5° in the CDET and 10.5° ± 2.7° in the SDFT. The crimp angles were not found to be statistically significantly different between the two tendon types (p = 0.572). By contrast, the fascicle helix angles were predicted to be 7.9° ± 9.3° in the CDET and 29.1° ± 10.3° in the SDFT and were found to be statistically highly significantly different between the two tendon types (p < 0.001). This supports previous qualitative observations that helical substructures are more likely to be found in energy-storing tendons than in positional tendons and suggests that the relative compliance of energy-storing tendons may be directly caused by these helical substructures.

Reconstructing a 2D slice or a 3D volume from a set of insufficient tomographic data is a difficult problem, and it is often tackled with analytical reconstruction algorithms. However, these types of methods fall short on delivering a quality image due to the severe artefacts introduced by the insufficiency of the data. The presented work shows the effects of taking tomographic scans with fewer radiographs on the quality of the reconstructed images. The aim here is to show the advantages of using iterative reconstruction methods over analytical methods, which are demonstrated by a quantitative comparison.

Log scanning to determine internal characteristics to maximize lumber value has been a long-term goal of the sawmilling industry. Juvenile wood is wood formed in southern pine trees during the first 10 years of growth. Juvenile wood is characterized by wood with high moisture content and that undergoes a high degree of longitudinal shrinkage during kiln drying. This shrinkage results in large amounts of warp that produce lumber degrade and very high value loss. Detection of juvenile wood prior to sawing logs will allow application of sawing patterns and drying procedures that result in reduced influence on the final lumber value. Our research tested EIT scanning with brush electrodes to determine the potential for detecting juvenile wood in green southern pine logs. EIT and computed tomography (CT) images were compared to determine that juvenile wood could be detected with acceptable.

We show that a reciprocal space squared intensity map of a material can be recovered, for each characteristic length scale, from diffraction tomography data by a simple slice-by-slice reconstruction method. Moreover if the reciprocal space map can be represented by a finite sum of spherical harmonic components for each length scale then the coefficients of that expansion can be recovered from inverting the transverse ray transform (TRT), where the data are polynomial coefficients of the azimuthal diffraction pattern for each length scale.