Abstract: Smoothed l0 norm algorithm (SL0) is a reconstruction algorithm in compressive sensing based on approximate l0 norm. It approches the solution using the specific iteration process with the steepest desent method and gradient projection principle. It has many advantages, such as the high matching degree, the short reconstruction time, the low computation complexity, and no need for the sparsity of a signal. However, it has “notched effect” due to the negative iterative gradient direction. Moreover, the property of continuous function in SL0 and its improved algorithm (NSL0) is not steep enough，which results in the estimations are not accurate and the convergence is slow. In this paper, we use hyperbolic tangent function as the approximation to the big “steep nature” in l0 norm. Based on it, we propose a novel reconstruction algorithm named ANSL0 with the Damped Newton method and the Revised Newton method. The numerical simulation results show that the ANSL0 algorithm has great improvement in both matching degree，the peak value signal-to-noise ratio and the signal-to-noise ratio, comparing with the SL0 algorithm and NSL0 algorithm under the same conditions.