A Novel RPCA Method Using Log-Weighted Nuclear and L_(2,1) Norms Combined with Contrast-Limited Adaptive Histogram Equalization (CLAHE) for High Dimensional Natural and Medical Image Data

Estimating the true underlying images from distorted high-dimensional data is crucial for applications in high-profile fields such as crime detection in security, clinical settings and medical diagnosis in healthcare, and radar imaging in signal processing. Existing statistical methods often struggle with robustness and image reconstruction quality when processing high-dimensional image data. While Robust Principal Component Analysis (RPCA) is widely used for image recovery, its reliance on uniform weights with singular value decomposition (SVD) weakens performance, especially in noisy environments. The L1 norm also fails to capture image details and recovery under high noise levels, a critical limitation for applications like medical diagnoses, where detail is essential. These challenges emphasize the need for improved methods to handle noise and enhance image quality in sensitive fields. Therefore, this paper proposes a novel RPCA method that integrates CLAHE with Log weighted nuclear norm (LWNN) and the L2,1 norm for high-dimensional natural and medical imaging. To reduce the computational load, our novel method is formulated into a new optimization problem and solved using the Alternating Direction Method of Multipliers (ADMM). This method leverages LWNN for enhanced low-rank approximation to drastically prune out the anomalies in images and the norm for improved sparse component recovery. Our approach has superior performance in image reconstruction compared to other state-of-the-art methods (SOTAs), showing significant advancements with real-world datasets. An interesting finding of this research is that combining the LWNN with the L2,1 norm is highly effective at removing noise from images. Furthermore, when the CLAHE technique is combined with LWNN and the L2,1 norm, it significantly enhances the extraction of previously unseen features, making blood vessels in medical images much clearer and more distinguishable. This combination proves to be a powerful approach for medical image analysis, revealing details that are otherwise difficult to detect. This method will be used for crime detection in security intelligence, and clinical settings and medical diagnosis in human retinal eyes.