A Fractional Variational Approach to Spectral Filtering Using the Fourier Transform

The interference of fluorescence signals and noise remains a significant challenge in Raman spectrum analysis, often obscuring subtle spectral features that are critical for accurate analysis. Inspired by variational methods similar to those used in image denoising, our approach minimizes a functional involving fractional derivatives to balance noise suppression with the preservation of essential chemical features of the signal, such as peak position, intensity, and area. The original problem is reformulated in the frequency domain through the Fourier transform, making the implementation simple and fast. In this work, we discuss the theoretical framework, practical implementation, and the advantages and limitations of this method in the context of {simulated} Raman data, as well as in image processing. The main contribution of this article is the combination of a variational approach in the frequency domain, the use of fractional derivatives, and the optimization of the {regularization parameter and} derivative order through the concept of Shannon entropy. This work explores how the fractional order, combined with the regularization parameter, affects noise removal and preserves the essential features of the spectrum {and image}. Finally, the study shows that the combination of the proposed strategies produces an efficient, robust, and easily implementable filter.