From pixels to topological signals: scalable persistent homology for image-based learning

Images are usually processed as arrays of intensities, yet many relevant visual phenomena are structural: objects and textures are often defined by how regions, boundaries, and voids are organized across the image. Persistent homology (PH) provides a principled language for measuring this structure, but its computational cost and its abstract output have often limited its use in image-based learning pipelines. To address this gap, this thesis studies how pixels can be transformed into scalable, interpretable topological signals. The first part develops the computational foundation required for this transformation. We introduce PixHomology, an algorithm for computing PH on images by exploiting the regularity of pixel grids and intensity-induced filtrations. Starting from a distributed prototype for large collections of images, the work progresses toward single-machine implementations optimized for memory and runtime, including variants that handle both connected components and loops. The resulting algorithms reduce the practical barrier to repeated persistence computation and make topological descriptors usable in workflows where scalability is essential. The second part shows how these topological signals can support learning and image analysis. Spatial Persistent Images (SPI) and Topological Attention (TA) convert persistence information into spatial priors that can be injected into neural networks for classification, denoising, and segmentation. Persistent Trees (PTs) provide a complementary, fully interpretable representation for unsupervised denoising, where anisotropic diffusion is guided by the hierarchy of connected components and can be stopped automatically. Additional applications include topology-driven object proposals for H&E histological fiber segmentation and persistence-based methods for detecting and explaining Trojan behavior in neural networks. Taken together, these contributions show that scalable PH can move topology from a descriptor to an operational signal for image-based learning. The proposed methods demonstrate that topological information can be computed efficiently, localized spatially, integrated into neural and non-neural pipelines, and inspected in a way that preserves interpretability.