The coherent properties of radar give rise to speckle noise in synthetic aperture radar (SAR) images. Speckle noise, mixed with valid information, directly affects information extraction in SAR images, especially the accuracy of persistent scatter point selection. Based on a detailed analysis of speckle noise characteristics, an innovative speckle noise reduction method combining independent vector analysis and an improved fast independent component analysis (FastICA) is proposed in this study. First, the principle of independent vector separation is followed to retain the maximum correlation of internal information in each channel of the SAR image. Then, a high-order Newton iterative scheme is constructed and added to the traditional FastICA algorithm to improve the speed and stability of iteration processing. Finally, the relaxation factor is introduced to relax the initial value requirement to minimize image distortion during speckle noise reduction. To verify the proposed algorithm, two groups of SAR images are selected from “Sandia National Laboratories” and Sentinel-1A. The proposed algorithm is compared with several other algorithms on speckle noise reduction efficiency. The experimental results showed that the proposed method could more effectively reduce speckle noise and retain edge features of SAR images, indicating that it had a potential to enhance image quality for the subsequent interpretation of SAR images.