Simple algorithm for L1-norm regularisation-based compressed sensing and image restoration

L1-norm regularisation plays an important role in compressed sensing reconstruction and image restoration. However, the discontinuity of L1-norm function makes solving the involved optimisation problem very challenging with traditional optimisation methods. In this article, a simple but efficient algorithm is proposed for the L1-norm regularised compressed sensing and image restoration problem. In the proposed algorithm, the L1-norm regularised optimisation problem is converted to a non-linear optimisation problem with L1-norm approximation by a smoothening function, which then can be solved by existing powerful non-linear optimisation methods. The simulation results show that the proposed algorithm is more efficient and results in a higher accurate solution. Compared to existing methods, the proposed algorithm is very easy to implement and promising for applications in medical and biological imaging.