Doctorado en Matemáticas · 2026

Variational Active Contour and Deep Learning Methods in Biomedical Image Segmentation

Resumen

This thesis deals with the applications of variational image processing methods, in particular image segmentation. These methods have become a cornerstone of image processing, offering mathematically rigorous frameworks for tasks like denoising, registration, and segmentation. Their ability to incorporate physical constraints, energy minimization principles, and prior knowledge makes them indispensable. In image segmentation, variational approaches provide robustness to noise and topological flexibility, addressing limitations of purely data-driven methods. Image segmentation is the task of dividing an image into different regions such that each region is homogeneous in color, intensity or texture i.e., the same in some sense. The aim is to select specific features out of an image from distinguishing them from the background. For image segmentation, and in general for image processing, the variational modeling is a well established technique. Variational methods are particularly valuable in biomedical imaging due to their ability to handle noise and low contrast (common in MRI/CT/X-ray), incorporate anatomical priors (e.g., smooth boundaries, shape constraints) and provide interpretable models (unlike purely data-driven deep learning). In this thesis we test the performance of the main variational models for image segmentation as well as the U-net model corresponding to deep learning methods. In some applications as magnetic resonance imaging (MRI), segmentation is challenged by an inherent artifact, called intensity inhomogeneity, which manifests itself as slow intensity variations in the same tissue over the image domain. Due to the intensity inhomogeneity, there are overlaps between the ranges of the intensities of different tissues, which often causes misclassification of tissues. Therefore, it is often a mandatory step to remove the intensity inhomogeneity through a procedure called bias field correction before performing quantitative analysis of MRI data. This procedure can be introduced in the level set method by using radial kernels. Given that traditionally the applied kernel has been Gaussian, in this thesis we introduce two families of well-known radial kernels into the level set method in order to improve its performance on biomedical image segmentation. In order to provide comparison with other methods the second part of this work deals with deep learning methods. Deep learning method conducts image segmentation by training a multi-layer artificial neural network to automatically extract image features. It has an amazing performance in image segmentation with higher accuracy. However, it still has the following drawbacks: (1) its performance significantly depends on the training samples, so it is usually limited for some specialized tasks, (2) it is a black box, which lacks the theoretical interpretability. In contrast, variational models uses image information such as intensity, texture, shape to build a mathematical model to realize image segmentation, which does not require much data and has a solid theoretical basis. Therefore, variational methods still play an irreplaceable role in image segmentation. As for deep learning methods, we propose some changes in the U-net architecture that may improve the results of biomedical image segmentation.