Robust regression analysis using the weighted median model for improved denoising of MR data in image processing

This paper presents a comprehensive analysis of a multidimensional regression model using the weighted median as the regression function. The model is formulated as an optimization problem within the framework of the $L_1$-norm error fitting approach, exhibiting robustness to outliers, a critical advantage in various applications where data might be contaminated by extreme values. The core of the investigation focuses on the regression and objective functions of the proposed model. A detailed mathematical study reveals that the optimization problem inherent in the model can be effectively discretized, leading to computationally tractable solutions. The study's findings are further validated through a rigorous exploration of the model's application in the context of image denoising, a significant problem in image processing. Specifically, the model addresses the challenging task of impulse noise removal in Magnetic Resonance images. By integrating the proposed model into well-established adaptive denoising methods, this work demonstrates that significant improvements in image quality reconstruction and noise suppression are easily achievable. The results highlight the model's efficacy in balancing the competing demands of preserving essential image features while effectively reducing noise artifacts. This research offers a novel approach for robust regression analysis and provides a robust tool for image denoising, particularly in scenarios involving impulse noise. The mathematical underpinnings, along with the demonstrated practical application, contribute significantly to the field of robust statistical modeling and image processing.