1. A method for reconstructing a signal from a set of noisy measurements, the method comprising: receiving, by a receiving unit, the set of noisy measurements; identifying, by a reconstruction unit, a new element of support of the signal in a sparse domain based on a projection matrix, the set of noisy measurements and at least one postulated parameter, the projection matrix transforming the signal from a sparse domain to a compressed domain; and obtaining, by the reconstruction unit, an estimate of the signal based on at least the set of noisy measurements, the identified new element of support, the at least one postulated parameter and the projection matrix.