1. A method for reconstructing a vector x representing compressible signals of interest, based on a vector y comprising n<m measurements produced by a compressed sensing scheme, comprising the steps of: a processor taking n traditional measurements to obtain x; a processor applying a known transform B; a processor compressing a measured vector x to a nearly sparse vector by said matrix B, where Bx is a vector which can be well-approximated by a relatively small number of large-amplitude entries, with remaining entries relatively small in amplitude; a processor delivering among approximate solutions y=Ax+e, an approximate solution x # for which B x # is sparse or nearly sparse; and a processor applying post processing filtering to reduce a noise level.