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Compressed Sensing (CS), a popular technique which seeks to capture a discrete signal with a small number of linear measurements, could be used to compress a signal during the process of sampling. As an iterative greedy reconstruction algorithm for practical CS, sparsity adaptive matching pursuit (SAMP) takes advantage of the capability of signal reconstruction without prior information of the sparsity in the process of resuming the original high-dimension-data from low-dimension measurement. This paper presents a backward and adaptive matching pursuit reconstruction algorithm with fixed step sizes to avoid the overestimation phenomena of SAMP by using a standard regularized approach. Firstly, a fixed and biggish step size is set to make sure the size of support set of the signal to be reconstructed increasing stably. The energy difference between adjacent reconstructed signals is then taken as the halting condition of iteration. A standard regularized approach is employed to post-dispose the final iteration results, which backward eliminates superfluous atoms to acquire exact reconstruction. Experimental results show that such an improvement of SAMP is feasible in technology and effective in acquiring quick and exact reconstruction with sufficient measurement.