Nonconvex lp minimization for compressed sensing under general perturbations

Compressed sensing (CS) provides a framework for acquisition of signals far below the Nyquist rate if it is represented as sparse or compressible on an orthonormal basis. Recovering sparse and compressible signals using lp minimization with p<1 when some part of the support of the signal is known a priori is studied. A sparse reconstruction method based on lp minimization with partially known set is proposed and recovery conditions are given. Error bound noise constant and error bound compressibility constants are obtained for sparse and compressible signal cases. Theoretical results show that lp minimization with partially known support is stable and robust. Experimental results are presented to expose the modification of lp minimization improves performance and need fewer samples to reconstruct the signal. Also lp minimization with p<1 under both additive and multiplicative noise in compressed sensing is studied. The results are based on the restricted isometry constant and relative perturbations. The exact reconstruction of a signal is not possible under additive and multiplicative noise. However simulation results show that under multiplicative noise, lp minimization performs better than l1 minimization for average reconstruction error with varying parameters such as noise level, sparsity and measurement level.