Noise Removal Using Singular Value Decomposition (SVD) Method

In signal processing, noise represents a common challenge that can be effectively addressed using Singular Value Decomposition (SVD). SVD is a linear algebra technique that decomposes a matrix into the product of three distinct matrices. This method factorizes a matrix into an orthogonal matrix, a diagonal matrix, and the transpose of another orthogonal matrix. In implementation, the numpy.linalg.svd() function in Python or svd() in MATLAB can be used to perform this decomposition, where the diagonal matrix contains singular values that represent the signal's energy distribution. The key algorithm involves setting a threshold to truncate smaller singular values that typically correspond to noise components while preserving larger singular values representing the actual signal. By reconstructing the matrix using only the significant singular values, we can effectively separate the clean signal from noisy components, achieving noise removal. This technique is particularly effective for processing matrices representing time-series data or images where noise appears as low-energy components in the singular value spectrum. Therefore, utilizing Singular Value Decomposition for noise removal proves to be a highly efficient technique for signal denoising applications, with practical implementation involving appropriate threshold selection strategies such as percentage-based truncation or adaptive thresholding methods.