用分数阶变分法滤除拉曼光谱噪声，保留关键化学特征。
A Fractional Variational Approach to Spectral Filtering Using the Fourier Transform
- 在频域通过傅里叶变换实现分数阶变分滤波，简化计算。
- 结合香农熵优化正则化参数与导数阶数，有效去噪并保峰。
- 适用于拉曼光谱和图像处理，易实现且鲁棒性强。
荧光信号与噪声的干扰仍是拉曼光谱分析中的重大挑战，常掩盖对准确分析至关重要的细微光谱特征。受图像去噪中变分方法的启发，本文提出一种最小化包含分数阶导数的泛函的方法，在抑制噪声的同时保留信号的关键化学特征，如峰位、强度和面积。原问题通过傅里叶变换在频域重新表述，实现简便且快速。本文讨论了该方法的理论框架、实际实现，以及在模拟拉曼数据和图像处理中的优缺点。主要贡献在于将频域变分方法、分数阶导数与基于香农熵的正则化参数及导数阶数优化相结合。研究揭示了分数阶与正则化参数如何共同影响去噪效果并保持光谱与图像的本质特征。结果表明，所提策略组合能生成高效、稳健且易于实现的滤波器。
原文摘要 · Abstract (English)
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.
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