Numerical approximation of continuous traffic congestion equilibria
Fethallah Benmansour Guillaume Carlier Gabriel Peyré Filippo Santambrogio
Networks & Heterogeneous Media 2009, 4(3): 605-623 doi: 10.3934/nhm.2009.4.605
Starting from a continuous congested traffic framework recently introduced in [8], we present a consistent numerical scheme to compute equilibrium metrics. We show that equilibrium metric is the solution of a variational problem involving geodesic distances. Our discretization scheme is based on the Fast Marching Method. Convergence is proved via a $\Gamma$-convergence result and numerical results are given.
keywords: Fast Marching Method. traffic congestion eikonal equation subgradient descent Wardrop equilibria
Non-local regularization of inverse problems
Gabriel Peyré Sébastien Bougleux Laurent Cohen
Inverse Problems & Imaging 2011, 5(2): 511-530 doi: 10.3934/ipi.2011.5.511
This article proposes a new framework to regularize imaging linear inverse problems using an adaptive non-local energy. A non-local graph is optimized to match the structures of the image to recover. This allows a better reconstruction of geometric edges and textures present in natural images. A fast algorithm computes iteratively both the solution of the regularization process and the non-local graph adapted to this solution. The graph adaptation is efficient to solve inverse problems with randomized measurements such as inpainting random pixels or compressive sensing recovery. Our non-local regularization gives state-of-the-art results for this class of inverse problems. On more challenging problems such as image super-resolution, our method gives results comparable to sparse regularization in a translation invariant wavelet frame.
keywords: Non-local regularization inpainting compressive sensing. super-resolution

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