Generalized variational models for data processing

• Autores: Vicent Pallardó Julià
• Directores de la Tesis: Salvador Moll (dir. tes.)
• Lectura: En la Universitat de València ( España ) en 2025
• Idioma: inglés
• Número de páginas: 144
• Tribunal Calificador de la Tesis: Lorenzo Giacomelli (presid.) , Juan Calvo Yagüe (secret.) , Marta Latorre Balado (voc.)
• Programa de doctorado: Programa de Doctorado en Matemáticas por la Universitat de València (Estudi General) y la Universitat Politècnica de València
• Calificación: Sobresaliente Cum Laude
• Enlaces
  • Tesis en acceso abierto en:  webges.uv.es (pdf)  RODERIC 
• Resumen

  • Over the past forty years, image processing has led to the development of a wide variety of problems aimed at modelling technical processes. Within this field, one can encounter tasks such as noise removal, resolution enhancement, compression, reconstruction, optical flow detection, or the segmentation of relevant information, among many others. Mathema- tically, these problems constitute a framework in which a diverse range of strategies can be applied to address them: Fourier analysis, inverse problem theory, statistical methods, opti- mal transport techniques, or partial differential equations, to name a few examples. Alongside this academic interest, image processing proves essential in numerous real-world situations, including medicine, agriculture, surveillance, entertainment, robotics, and environmental sciences, among others. However, it is necessary to clarify the concept of an image. Although we commonly asso- ciate an image with a photograph, this definition may be overly restrictive when addressing situations where images represent diverse data types, such as those obtained through X-rays, ultrasounds, radar, telescopes, or electron microscopes. In these cases, the data often repre- sent complex structures-such as the human body, the Earth, or specific materials-captured at a given moment in time. Therefore, in this work, we adopt the concept of an image as akin to a snapshot, understanding it as any representation of an object or system at a par- ticular instant. This perspective allows us to regard image processing as a subgenre within the broader framework of data processing. Throughout this thesis, we focus on two classi- cal variational models originally developed for processing photographs, which we reinterpret from a broader perspective aligned with the concept of snapshot. Specifically, we study the Rudin-Osher-Fatemi (ROF) denoising model and the Chan-Vese (CV) segmentation model, and we extend them to more general and complex settings. In the first part of the thesis, we analyze and generalize the relationship between the ROF and CV models. In the classical isotropic setting, a partial minimizer of the CV model can be constructed from the minimizer of the ROF model via this connection. Our analysis extends this result to broader contexts, identifying conditions under which the correspondence persists when incorporating anisotropies, fractional perime- ters, random walks, or metric graph formulations, among others. This unified framework clarifies how variations in the regularization terms or the domain influence the interaction between both models. In particular, the anisotropic case is addressed in detail, offering an enhanced understanding of the linkage between the models compared to the isotropic case. In constrast, the rest of this work is devoted to the generalization of the ROF model, par- ticularly to a vector-valued variant constrained to Riemannian manifolds, where the image domain is represented by a surface. This manifold-based formulation introduces new the- oretical challenges motivated by practical applications, and although notable progress has been made in the applied literature, rigorous theoretical developments remain limited. In this thesis, we contribute to bridging this gap by proving the existence of minimizers and establishing their regularity properties, emphasizing the crucial role of the interplay between extrinsic and intrinsic viewpoints.