Browsing by Author "Molina Mejía, Juan José"

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    Mathematical analysis and applications of neural networks, with applications to image reconstruction
    (2025) Molina Mejía, Juan José; Courdurier, Matías; Pontificia Universidad Católica de Chile. Facultad de Matemáticas
    This 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.