The performance of linear programming techniques that are applied in the signal identification and reconstruction process in compressed sensing (CS) is governed by both the number of measurements taken and the number of nonzero coefficients in the discrete basis used to represent the signal. To enhance the capabilities of CS, we have developed a technique called Variable Projection and Unfolding (VPU). VPU extends the identification and reconstruction capability of linear programming techniques to signals with a much greater number of nonzero coefficients in the basis in which the signals are compressible with significantly better reconstruction error.