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Digital geometric models are fundamental to modern engineering, media, and manufacturing. However, models created by artists in-the-wild often contain ambiguities that precludes their use in simulation and manufacturing, while complex designs may need to be simplified for efficiency or functionally optimized to meet competing aesthetic and performance goals. This necessity for robust, useful, and high-performing geometry creates a critical need for advanced computational techniques that can automatically repair, simplify, and optimize digital shapes. Our research addresses these challenges by developing a suite of shape processing and optimization methods designed to enhance the quality and functionality of geometric models for a range of applications.
This thesis delivers solutions across three key areas. First, we present a Bézier curve simplification framework that simplifies complex vector graphics while preserving visual fidelity by defining a curve-to-curve distance metric and repeatedly conducting local segment removal operations. Second, we propose a solid or shell labeling technique for artist-created surface meshes that lack a well-defined interior, guided by a sparse set of user inputs. These labels reduce ambiguity and enable the construction of valid volumetric meshes for downstream applications. Finally, we introduce two powerful shape optimization frameworks: one that leverages neural network-based models to independently control the tactile properties and visual appearance of a texture, and another that optimizes the geometry and position of radiofrequency (RF) receive coil arrays to increase signal-to-noise ratio (SNR) in magnetic resonance imaging (MRI).