This paper analyzes a recent sparse reconstruction algorithm applied to the nonlinear inverse scattering problem arising in microwave imaging to determine the dielectric contrast from measured fields. The sparsity of contrast profile with respect to a certain base is a priori assumed. The proposed iterative algorithm of soft shrinkage type enforces the sparsity constraint at each nonlinear iteration. The scheme produces sharp and good reconstruction of dielectric profiles in sparse domains by adapting Barzilai and Borwein (BB) step size selection criteria and projection constraint on contrast by maintaining its convergence during the reconstruction. Numerical results show the effectiveness and accuracy of the proposed method by considering randomly measured fields and emphasizing the sensitivity of the choice of regularization parameter.