Applications of Singular Entropy to Signals and Singular Smoothness to Images

This paper explores signal and image analysis by using the Singular Value Decomposition (SVD) and its extension, the Generalized Singular Value Decomposition (GSVD). A key strength of SVD lies in its ability to separate information into orthogonal subspaces. While SVD is a well-established tool in ECG analysis, particularly for source separation, this work proposes a refined method for selecting a threshold to distinguish between maternal and fetal components more effectively. In the first part of the paper, the focus is onmedical signal analysis,where the concepts of Energy Gap Variation (EGV) and Singular Energy are introduced to isolate fetal and maternal ECG signals, improving the known ones. Furthermore, the approach is significantly enhanced by the application of GSVD, which provides additional discriminative power for more accurate signal separation. The second part introduces a novel technique called Singular Smoothness, developed for image analysis. This method incorporates Singular Entropy and the Frobenius normto evaluate information density, and is applied to the detection of natural anomalies such asmountain fractures and burned forest regions. Numerical experiments are presented to demonstrate the effectiveness of the proposed approaches.