Compressed sensing technology for atmospheric turbulence wavefront slope measurement can greatly improve the wavefront signal measurement speed, while reducing the pressure of wavefront measurement system hardware. Different from the existing wavefront slope measurement method, the compressed sensing wavefront measurement increase a process which from sparse measurement of wavefront slope value to the reconstruction of the wavefront slope signal. Therefore, a fast and accurate wavefront slope reconstruction algorithm is needed if the compressed sensing technology is used for wavefront measurement. Smoothed L0 Norm (SL0) algorithm is an optimized iterative reconstruction algorithm with approximate L0 norm estimation, and compared with other algorithms, it is not necessary to know the sparsity of the signal in advance, and the calculation is low and the estimation accuracy is high. Based on the SL0 algorithm, this paper implements a subregion parallel algorithm- Block-Smoothed L0 Norm (B-SL0) which can quickly and accurately reconstruct the signal by measuring the wavefront slope signal in subarea and parallel operations through theoretical analysis and experiments. The experimental results show that B-SL0 is significantly better than other existing reconstruction algorithms in the calculation time and accuracy, and explore the feasibility of compressed sensing technology for measurement of atmospheric turbulence wavefront preliminarily.