The study of mathematical problems related to image processing is common ground in many applications, in industry as in medical imaging, at the scale of microscopy, in nondestructive materials investigation and landslide thickness estimation, or in astronomical imaging. The interest of the group is related to problems in which it is necessary to reconstruct images from indirect measurements, which are often subject to degradation processes: no matter how well the physical models describing each application are known, the inversion process is ill-posed, that is, the existence, uniqueness, and stability of the solution are not guaranteed.
The most effective mathematical techniques integrate the inversion process with a priori information about the desired solution: this can be done by incorporating this knowledge into appropriate theoretical models (model-based approach) or by using large amounts of images to extract information and correlations present in the data (data-driven approach). Model-based approaches lead, on the one hand, to regularization techniques formulated as optimization problems (in convex and nonconvex domains and in the presence of constraints) and, on the other hand, to solving partial differential equations using numerical methods. Data-based methodologies exploit tools from statistical learning, developing reconstruction methods with neural networks. Given the ill-posedness of the problems addressed, it is crucial to formulate rigorous results for such techniques, in terms of existence and stability of solutions, interpretability and dependence on the data used for learning.
The group faces these challenges by developing transdisciplinary mathematical skills for the imaging sciences. The main problems addressed include:
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inversion of relaxometry and NMR spectroscopy data;
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tomography reconstructions in presence of severely under-sampled or randomly sampled data;
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parameter identification in differential diffusion-transport and nonlinear reaction models;
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deblurring/denoising of 2D/3D images;
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Image inpainting via semi-Lagrangian schemes for the game p-Laplacian operator;
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Compressed sensing and sparsity promotion techniques for signal, image and video reconstruction;
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segmentation of structures in 2D images and scalar fields on manifolds;
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3D reconstruction of surfaces from 2D images (problems belonging to the Shape-from-X class with non-Lambertian reflectance models, Photometric Stereo Shape-from-X, Multi-View Shape-from-Shading);
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reconstruction of 2D/3D images from electrical impedance tomography (EIT) with model-based and data-driven techniques;
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theoretical guarantees for regularization techniques based on statistical learning: convergence, stability, generalization;
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processing and inversion of geologic and topographic data from remote sensing aimed at landslide thickness estimation.