Mathematical analysis and applications of neural networks, with applications to image reconstruction

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dc.catalogadorpva
dc.contributor.advisorCourdurier, Matías
dc.contributor.authorMolina Mejía, Juan José
dc.contributor.otherPontificia Universidad Católica de Chile. Facultad de Matemáticas
dc.date.accessioned2025-06-30T21:35:21Z
dc.date.available2025-06-30T21:35:21Z
dc.date.issued2025
dc.date.updated2025-06-27T14:33:44Z
dc.descriptionTesis (Phd in Mathematics)--Pontificia Universidad Católica de Chile, 2025
dc.description.abstractThis thesis explores two fundamental aspects of neural networks: their frequency learning behavior and their application to quantitative Magnetic Resonance Imaging (MRI) reconstruction. The first part investigates the phenomenon of frequency bias, the empirical observation that neural networks tend to learn low-frequency components of a target function more rapidly than high-frequency ones. To provide a rigorous understanding of this behavior, we develop a theoretical framework based on Fourier analysis. Specifically, we derive a partial differential equation that governs the evolution of the error spectrum during training in the Neural Tangent Kernel regime, focusing on two-layer neural networks. Our analysis centers on Fourier Feature networks, a class of architectures where the first layer applies sine and cosine activations using pre-defined frequency distributions. We demonstrate that the network's initialization, particularly the initial density distribution of first-layer weights, plays a crucial role in shaping the frequency learning dynamics. This insight provides a principled way to control or even eliminate frequency bias during training. Theoretical predictions are validated through numerical experiments, which further illustrate the impact of initialization on the inductive biases of neural networks.The second part of the thesis applies neural network techniques to the reconstruction of quantitative MRI data. Quantitative MRI enables the estimation of tissue-specific parameters (e.g., T1, T2, and T2*) that are vital for clinical diagnosis and disease monitoring. However, these methods typically require long acquisition times, which are often mitigated through aggressive undersampling of k-space data. Undersampling, in turn, introduces reconstruction artifacts that must be addressed through regularization. To this end, we propose CConnect, a novel iterative reconstruction method that incorporates convolutional neural networks into the regularization term. CConnect connects multiple CNNs through a shared latent space, allowing the model to capture common structures across different image contrasts. This design enables the effective suppression of aliasing artifacts and improves image quality, even in highly undersampled scenarios. We evaluate CConnect on in-vivo brain T2*-weighted MRI data, demonstrating its superiority over classical low-rank and total variation methods, as well as standard deep learning baselines.
dc.description.funderANID
dc.fechaingreso.objetodigital2025-06-27
dc.format.extent101 páginas
dc.fuente.origenAutoarchivo
dc.identifier.urihttps://repositorio.uc.cl/handle/11534/104797
dc.information.autorucFacultad de Matemáticas; Courdurier, Matías; 0000-0002-2161-0356; 1007892
dc.information.autorucFacultad de Matemáticas; Molina Mejía, Juan José; S/I; 1193837
dc.language.isoen
dc.nota.accesocontenido completo
dc.rightsacceso abierto
dc.rights.licenseAtribución-CompartirIgual 4.0 Internacional (CC BY-SA 4.0)
dc.rights.urihttps://creativecommons.org/licenses/by-sa/4.0/deed.es
dc.subject.ddc510
dc.subject.deweyMatemática física y químicaes_ES
dc.subject.ods03 Good health and well-being
dc.subject.odspa03 Salud y bienestar
dc.titleMathematical analysis and applications of neural networks, with applications to image reconstruction
dc.typetesis doctoral
sipa.codpersvinculados1007892
sipa.codpersvinculados1193837
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