Persistence optimization for data visualization

Overview & Synopsis

This thesis addresses the challenge of managing the growing complexity of scalar data in both space and time, by leveraging topological methods for data reduction and reconstruction. As datasets become increasingly large and detailed, topological descriptors such as persistence diagrams offer compact and robust summaries that support efficient storage and analysis. However, these abstract representations often lack the necessary information for detailed visualization or interpretation, highlighting the need for methods capable of reconstructing scalar data from them. To this end, the thesis introduces two complementary contributions. First, it proposes an optimization framework for topological simplification that preserves meaningful features while removing noise, extending prior approaches by handling complex structures such as saddle pairs in three-dimensional data. Second, it presents a neural interpolation scheme that reconstructs intermediate scalar fields from sparse keyframes, guided by topological constraints. By incorporating topology-aware loss functions, the model improves both geometric and topological accuracy, enabling the faithful reconstruction of time-varying scalar data. These contributions are validated on synthetic and real-world datasets, including applications in medical imaging and meteorology, and are supported by open-source implementations to ensure reproducibility. Together, they provide practical tools for simplifying and reconstructing complex scalar data through a topological lens.