Breast ultrasound is an important technology for detecting breast lesions, but it faces a challenge in the form of speckle noise that negatively affects the quality of images. Effective methods are needed to eliminate this noise without compromising the image's fine edges and important features. In this work, we present a cutting-edge methodology based on a deep understanding of the dynamics of breast ultrasound images and the challenges of speckle noise. The method uses two separate stages of processing with two nonlinear filters. The first filter, the anisotropic diffusion filter, smooths out edges and boosts image contrast by lowering noise while keeping the tissue's structure. In the second stage, the SRAD filter is applied to remove residual noise and refine the image, increasing its clarity and improving the ability to visualize subtle lesions. This improved approach was evaluated using a comprehensive set of quantitative indices. The results confirmed the significant performance improvement provided, with the lowest MSE value of 0.2 and the highest PSNR of 59.3. Also, it reached the optimal value in the UQI and SSI, which indicates the robustness of the method in maintaining the quality of the structural image. The results confirm the value of the proposed approach and herald its promising potential for improving medical diagnostic results using breast ultrasound.