When data collection is expensive, gathering fewer and strategically located points may reduce costs while maintaining important information. Inpainting then allows for the intelligent reconstruction of missing data from the sparse observations. In this thesis work, we propose the least squares differences algorithm, a new scheme for de-trending data with repeated observations based on classical least squares fitting along with au empirical guideline for constructing good sampling strategies. Furthermore, we develop and present a novel inpainting algorithm – penalized dictionary inpainting – that utilizes a variable penalty term and exhibits nonlocal sensitivity. Armed with these two innovations, we illustrate how current atomic force microscopy (AFM) imaging can be made more efficient. In particular, we apply least squares differences to the analysis of non-raster scanning methodologies, along with the inpainting of sparse or subsampled data.