Signal Reconstruction Using Variational Methods Based on Lp Norms
Continuous-domain signals can be reconstructed from their discrete measurements using variational approaches. The reconstructed signal is then defined as a minimizer of a functional, which is composed of a convex combination of two terms, namely data-fidelity (quadratic) and regularization. This project aims at considering new regularizations characterized by the choice of the function norm (Lp norm) and the associated regularization operator. The classical L2 norm relies on the theory of reproducing kernel Hilbert space (RKHS), and we have recently obtained breakthrough results using L1 norm (total variation), yielding solutions that are sparse in the continuous-domain. Investigating the transition between p=1 and p=2 requires the development of new mathematical tools and will be at the heart of the project. The work also includes the design of new algorithms for signal reconstruction, with potential applications in machine learning and neural networks. The student should have strong mathematical interests and basic knowledge on optimization theory.
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