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Compressive sensing (CS) is a theory that one may achieve an exact signal reconstruction from sufficient CS measurements taken from a sparse signal. However, in practical applications, the transform coefficients of SAR images usually have weak sparsity. Exactly reconstructing these images is very challenging. A new Bayesian evolutionary pursuit algorithm (BEPA) is proposed in this paper. A signal is represented as the sum of a main signal and some residual signals, and the generalized Gaussian distribution (GGD) is employed as the prior of the main signal and the residual signals. BEPA decomposes the residual iteratively and estimates the maximum a posteriori of the main signal and the residual signals by solving a sequence of subproblems to achieve the approximate CS reconstruction of the signal. Under the assumption of GGD with the parameter 0 <; p <; 1, the evolutionary algorithm (EA) is introduced to CS reconstruction for the first time. The better reconstruction performance can be achieved by searching the global optimal solutions of subproblems with EA. Numerical experiments demonstrate that the important features of SAR images (e.g., the point and line targets) can be well preserved by our algorithm, and the superior reconstruction performance can be obtained at the same time.