基于分数阶偏微分方程的信号保特征滤波研究

Fractional order partial differential equation is generated by replacing derivatives of the classical partial differential equation into fractional-order derivatives. Compared to the classical partial differential equation, fractional-order partial differential equation has a new advantage that is the adjustable differential order which will lead to greater flexibility for signal filtering. However, because fractional-order derivative is a nonlocal operator, computation of fractional-order partial differential equations are more complex than that of classical partial differential equations. Due to the advantage of the classical partial differential equations in edge-preserving denoising of images, in this project, we take the fractional-order differential equation as tool, some key issues of applying the fractional-order partial differential equation to signal preserving-feature filtering will be explored. This project focuses on the design of the diffusion functions which can preserve some important features of a signal and the development of fast numerical algorithms for fractional-order models as well as analysis of stability and convergence for these algorithms. Fast numerical algorithm will be presented in this project so that obstacles that delay the application of fractional-order partial differential equations in signal processing will be removed. In theory, this project will further expand application of partial differential equations in signal processing; in practice, some algorithms of preserving-peak smoothing will be given for a spectrum signal and some algorithms and models of preserving-edge denoising will be given for an image in this project. Applying fractional-order differential equations in signal processing, on the one hand, can maintain the advantages of partial differential equations on signal filtering, and on the other hand, can reflect some new advantages.

分数阶偏微分方程，是用分数阶导数替代经典偏微分方程中的导数产生的，相比于经典偏微分方程，分数阶偏微分方程具有新的优势——微分阶数是可调的，因此，在信号滤波过程中具有更大的灵活性。不过，分数阶导数是一个非局部算子，使得分数阶偏微分方程的计算较整数阶偏微分方程要复杂得多，鉴于经典偏微分方程在图像保边界滤波上的优势，本项目以分数阶偏微分方程为工具，对分数阶偏微分方程应用于信号保特征滤波的关键问题进行研究，重点解决保信号特征的扩散函数设计和求解分数阶模型的快速数值算法以及算法的稳定性、收敛性问题。给出求解分数阶模型的快速算法，扫除分数阶偏微分方程应用于信号滤波中的障碍。在理论上，进一步拓展偏微分方程在信号处理中的应用；在技术上，为谱信号提供保峰滤波算法，为图像提供保边界滤波模型和算法。一方面，保持偏微分方程在信号滤波上的优势，另一方面，又能发挥分数阶微积分的新优势。

分数阶偏微分方程是整数阶偏微分方程的推广，它是用分数阶导数替代整数阶导数而产生的，因为其阶数可变，在信号处理中有望得到更好的滤波性能。本项目对分数阶偏微分方程应用于信号处理的关键问题进行了研究。针对一维信号，提出了时间分数阶扩散模型和空间分数阶扩散模型。对每个模型进行了详细的讨论，给出了对应模型的数值算法，分析了模型参数对平滑性能的影响，并将所提模型与其他经典平滑方法进行了对比，时间分数阶扩散模型有更好的平滑性能，空间分数阶扩散模型有更好的保峰性能。在图像降噪方面，针对图像保边缘的问题，改进了各向异性扩散模型，将其推广到基于分数阶导数的各向异性扩散模型，讨论了参数的影响，并从图像去噪效果、算法收敛速度、图像视觉效果三个方面与其他同类方法进行了比较和分析，结果表明该模型对于边缘明显的图像相比其他方法具有很好的峰值信噪比提升；在分数阶扩散模型的基础上结合整数阶扩散模型提出了一种基于混合扩散模型的图像去噪方法，该模型对于不同的图像可以统一参数的选取，与同类方法相比在去噪效果方面具有一定的优势。分数阶偏微分方程相比于经典偏微分方程的优势在于其微分阶次是可调的，在信号滤波过程中能提升滤波性能，不过，分数阶导数是一个非局部算子，使得分数阶偏微分方程的计算较整数阶偏微分方程也要复杂得多，因此需要更多的处理时间。