X-ray tomography is an imaging technique to inspect objects' internal structures with externally measured data by X-ray radiation non-destructively. However, there are concerns about X-ray radiation damage and tomographic acquisition speed in real-life applications. Strategies with insufficient measurements, such as measurements with insufficient dosage (low-dose) and measurements with insufficient projection angles (sparse-view), have been proposed to relieve these problems but are generally compromising imaging quality. Such a dilemma inspires the development of advanced tomographic imaging techniques, in particular, deep learning algorithms to improve reconstruction results with insufficient measurements. The overall aim of this thesis is to design efficient and robust data-driven algorithms with the help of prior knowledge from physics insights and measurement models. We first introduce a hierarchical synthesis CNN (HSCNN), which is a knowledge-incorporated data-driven tomographic reconstruction method for sparse-view and low-dose tomography with a split-and-synthesis approach. This proposed learning-based method informs the forward model biases based on data-driven learning but with reduced training data. The learning scheme is robust against sampling bias and aberrations introduced in the forward modeling. High-fidelity X-ray tomographic imaging reconstruction results are obtained with a very sparse number of projection angles for both numerical simulated and physics experiments. Comparison with both conventional non-learning-based algorithms and advanced learning-based approaches shows improved accuracy and reduced training data size. As a result of the split-and-synthesis strategy, the trained network could be transferable to new cases. We then present a deep learning-based enhancement method, HDrec (hybrid-dose reconstruction algorithm), for low-dose tomography reconstruction via a hybrid-dose acquisition strategy composed of textit{extremely sparse-view normal-dose measurements} and textit{full-view low-dose measurements}. The training is applied for each individual sample without the need of transferring the trained models for other samples. Evaluation of two experimental datasets under different hybrid-dose acquisition conditions shows significantly improved structural details and reduced noise levels compared to results with traditional analytical and regularization-based iterative reconstruction methods from uniform acquisitions under the same amount of total dosage. Our proposed approach is also more efficient in terms of single projection denoising and single image reconstruction. In addition, we provide a strategy to distribute dosage smartly with improved reconstruction quality. When the total dosage is limited, the strategy of combining a very few numbers of normal-dose projections and with not-too-low full-view low-dose measurements greatly outperforms the uniform distribution of the dosage throughout all projections. We finally apply the proposed data-driven X-ray tomographic imaging reconstruction techniques, HSCNN and HDrec, to the dynamic damage/defect characterization applications for the cellular materials and binder jetting additive manufacturing. These proposed algorithms improve data acquisition speeds to record internal dynamic structure changes. A quantitative comprehensive framework is proposed to study the dynamic internal behaviors of cellular structure, which contains four modules: (i) In-situ fast synchrotron X-ray tomography, which enables collection of 3D microstructure in a macroscopic volume; (ii) Automated 3D damage features detection to recognize damage behaviors in different scales; (iii) Quantitative 3D structural analysis of the cellular microstructure, by which key morphological descriptors of the structure are extracted and quantified; (iv) Automated multi-scale damage structure analysis, which provides a quantitative understanding of damage behaviors. In terms of binder jetting materials, we show a pathway toward the efficient acquisition of holistic defect information and robust morphological representation through the integration of (i) fast tomography algorithms, (ii) 3D morphological analysis, and (iii) machine learning-based big data analysis. The applications to two different 4D material characterization demonstrate the advantages of these proposed tomographic imaging techniques and provide quantitative insights into the global evolution of damage/defect beyond qualitative human observation. / Doctor of Philosophy / X-ray tomography is a nondestructive imaging technique to visualize interior structures of non-transparent objects, which has been widely applied to resolve implicit 3D structures, such as human organs and tissues for clinical diagnosis, contents of baggage for security check, internal defect evolution during additive manufacturing, observing fracturing accompanying mechanical tests, and etc. Multiple planar measurements with sufficient X-ray exposure time among different angles are desirable to reconstruct the unique high-quality 3D internal distribution. However, there are practical concerns about X-ray radiation damage to biology samples or long-time acquisition for dynamic experiments in real-life applications. Insufficient measurements by reducing the number of total measurements or the time for each measurement, are proposed to solve this problem but doing so usually leads to the sacrifice of the reconstruction quality. Computational algorithms are developed for tomographic imaging under these insufficient measurement conditions to obtain reconstructions with improved quality. Deep learning has been successfully applied to numerous areas, such as in recognizing speech, translating languages, detecting objects, and etc. It has also been applied to X-ray tomographic imaging to improve the reconstruction results by learning the features through thousands to millions of corrupted and ideal reconstruction pairs. The aim of this thesis to design efficient deep learning-based algorithms with the help of physical and measurement priors to reduce the number of training datasets. We propose two different deep learning-based tomographic imaging techniques to improve reconstruction results with reduced training data under different insufficient measurement conditions. One way is to incorporate prior knowledge of the physics models to reduce the required amount of ground truth data, from thousands to hundreds. The training data requirement is further simplified with another hybrid measurement strategy, which could be implemented on each individual sample with only several high-quality measurements. In the end, we apply these two proposed algorithms to different dynamic damage/defect behavior characterization applications. Our methods achieve improved reconstruction results with greatly enhanced experimental speeds, which become suitable for dynamic 3D recording. Final results demonstrate the advantages of the proposed tomographic imaging techniques and provide quantitative insights into the global dynamic evolution inside the material. This quantitative analysis also provides a much more comprehensive understanding than qualitative human observation.