Over the last fifteen years, there has been a rapid growth in the use of high resolution X-ray computed tomography (HRXCT) imaging in material science applications. We use it at nanoscale resolutions up to 50 nm (nano-CT) for key research problems in large scale operation of polymer electrolyte membrane fuel cells (PEMFC) and lithium-ion (Li-ion) batteries in automotive applications. PEMFC are clean energy sources that electrochemically react with hydrogen gas to produce water and electricity. To reduce their costs, capturing their electrode nanostructure has become significant in modeling and optimizing their performance. For Li-ion batteries, a key challenge in increasing their scope for the automotive industry is Li metal dendrite growth. Li dendrites are structures of lithium with 100 nm features of interest that can grow chaotically within a battery and eventually lead to a short-circuit. HRXCT imaging is an effective diagnostics tool for such applications as it is a non-destructive method of capturing the 3D internal X-ray absorption coefficient of materials from a large series of 2D X-ray projections. Despite a recent push to use HRXCT for quantitative information on material samples, there is a relative dearth of computational tools in nano-CT image processing and analysis. Hence, we focus on developing computational methods for nano-CT image analysis of fuel cell and battery materials as required by the limitations in material samples and the imaging environment. The first problem we address is the segmentation of nano-CT Zernike phase contrast images. Nano-CT instruments are equipped with Zernike phase contrast optics to distinguish materials with a low difference in X-ray absorption coefficient by phase shifting the X-ray wave that is not diffracted by the sample. However, it creates image artifacts that hinder the use of traditional image segmentation techniques. To restore such images, we setup an inverse problem by modeling the X-ray phase contrast optics. We solve for the artifact-free images through an optimization function that uses novel edge detection and fast image interpolation methods. We use this optics-based segmentation method in two main research problems - 1) the characterization of a failure mechanism in the internal structure of Li-ion battery electrodes and 2) the measurement of Li metal dendrite morphology for different current and temperature parameters of Li-ion battery cell operation. The second problem we address is the development of a space+time (4D) reconstruction method for in-operando imaging of samples undergoing temporal change, particularly for X-ray sources with low throughput and nanoscale spatial resolutions. The challenge in using such systems is achieving a sufficient temporal resolution despite exposure times of a 2D projection on the order of 1 minute. We develop a 4D dynamic X-ray computed tomography (CT) reconstruction method, capable of reconstructing a temporal 3D image every 2 to 8 projections. Its novel properties are its projection angle sequence and the probabilistic detection of experimental change. We show its accuracy on phantom and experimental datasets to show its promise in temporally resolving Li metal dendrite growth and in elucidating mitigation strategies. Keywords: X-ray computed tomography, 4D X-ray computed tomography, phase contrast optics, fuel cells, Li-ion batteries, signal processing and optimization.