Filtered backprojection (FBP) algorithm is a popular choice for complicated trajectory SAR image formation processing due to its inherent nonlinear motion compensation capability. However, how to efficiently autofocus the defocused FBP imagery when the motion measurement is not accurate enough is still a challenging problem. In this paper, a new interpretation of the FBP derivation is presented from the Fourier transform point of view. Based on this new viewpoint, the property of the residual 2-D phase error in FBP imagery is analyzed in detail. Then, by incorporating the derived a priori knowledge on the 2-D phase error, an accurate and efficient 2-D autofocus approach is proposed. The new approach performs the parameter estimation in a dimension-reduced parameter subspace by exploiting the a priori analytical structure of the 2-D phase error, therefore possesses much higher accuracy and efficiency than conventional blind methods. Finally, experimental results clearly demonstrate the effectiveness and robustness of the proposed method.