Network Inpainting via Optimal Transport

The precise digital reconstruction of natural networks such as blood vessels or plant roots is crucial to ensure the quality of simulation-driven predictions. However, these structures can often be accessed only via noninvasive techniques, leading to artifacts that compromise the reliability of the data and the derived simulations. No technological solution is currently able to recover digital reconstructions of "real" networks from corrupted images. The NIOT (Network Inpainting via Optimal Transport) project aims to fill this technological gap by defining for the first time a robust mathematical formulation of the image network reconstruction problem. Thanks to the most recent advances of the optimal transport theory, we will finally encode into equations the well-known fact that several natural networks are designed to transport resources with the least effort possible. We will adopt a variational image processing method, where the reconstructed network is obtained as the density minimizing the sum of the discrepancy with the observed data and a branch inducing functional. As such, our proposed methodology builds a bridge between the image regularization and optimal transport communities. A major ambition of the project is to pair the theoretical analysis with robust simulation tools that are capable of handling real data arising from MRI acquisition techniques. This will require exploitation and development of dedicated components to handle large datasets, both from a data handling and a multiscale simulation perspective. Our algorithm will be tested on a sequence of increasingly channeling problems. We will start from simple synthetic networks, then we will use an high-quality map of the blood vessel network of a mouse brain. The final benchmark will be to reconstruct of corrupted vascular networks in MRI scans of human patients.