Partial differential equation (PDE) based methods have become some of the most powerful tools for exploring the fundamental problems in signal processing, image processing, computer vision, machine vision and artificial intelligence in the past two decades. The advantages of PDE based approaches are that they can be made fully automatic, robust for the analysis of images, videos and high dimensional data. A fundamental question is whether one can use PDEs to perform all the basic tasks in the image processing. If one can devise PDEs to perform full-scale mode decomposition for signals and images, the modes thus generated would be very useful for secondary processing to meet the needs in various types of signal and image processing. Despite of great progress in PDE based image analysis in the past two decades, the basic roles of PDEs in image/signal analysis are only limited to PDE based low-pass filters, and their applications to noise removal, edge detection, segmentation, etc. At present, it is not clear how to construct PDE based methods for full-scale mode decomposition. The above-mentioned limitation of most current PDE based image/signal processing methods is addressed in the proposed work, in which we introduce a family of mode decomposition evolution equations (MoDEEs) for a vast variety of applications. The MoDEEs are constructed as an extension of a PDE based high-pass filter (Europhys. Lett., 59(6): 814, 2002) by using arbitrarily high order PDE based low-pass filters introduced by Wei (IEEE Signal Process. Lett., 6(7): 165, 1999). The use of arbitrarily high order PDEs is essential to the frequency localization in the mode decomposition. Similar to the wavelet transform, the present MoDEEs have a controllable time-frequency localization and allow a perfect reconstruction of the original function. Therefore, the MoDEE operation is also called a PDE transform. However, modes generated from the present approach are in the spatial or time domain and can be easily used for secondary processing. Various simplifications of the proposed MoDEEs, including a linearized version, and an algebraic version, are discussed for computational convenience. The Fourier pseudospectral method, which is unconditionally stable for linearized the high order MoDEEs, is utilized in our computation. Validation is carried out to mode separation of high frequency adjacent modes. Applications are considered to signal and image denoising, image edge detection, feature extraction, enhancement etc. It is hoped that this work enhances the understanding of high order PDEs and yields robust and useful tools for image and signal analysis.

中文翻译：

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模式分解演化方程。

在过去的 20 年里，基于偏微分方程 (PDE) 的方法已成为探索信号处理、图像处理、计算机视觉、机器视觉和人工智能等基本问题的最强大工具。基于 PDE 的方法的优点是它们可以完全自动化，对图像、视频和高维数据的分析具有鲁棒性。一个基本问题是是否可以使用偏微分方程来执行图像处理中的所有基本任务。如果可以设计偏微分方程对信号和图像进行全尺度模式分解，那么由此产生的模式对于二次处理非常有用，可以满足各种类型的信号和图像处理的需要。尽管过去 20 年基于 PDE 的图像分析取得了很大进展，PDE 在图像/信号分析中的基本作用仅限于基于 PDE 的低通滤波器，以及它们在噪声去除、边缘检测、分割等方面的应用。 -尺度模式分解。大多数当前基于 PDE 的图像/信号处理方法的上述局限性在所提出的工作中得到解决，其中我们引入了一系列用于各种应用的模式分解演化方程 (MoDEE)。通过使用 Wei（IEEE Signal Process. Lett ., 6(7): 165, 1999)。使用任意高阶偏微分方程对于模式分解中的频率定位是必不可少的。与小波变换类似，现有的 MoDEE 具有可控的时频定位，可以完美地重建原始函数。因此，MoDEE 操作也称为 PDE 变换。然而，从本方法生成的模式在空间或时域中，可以很容易地用于二次处理。为了计算方便，讨论了所提出的 MoDEE 的各种简化，包括线性化版本和代数版本。在我们的计算中使用了傅立叶伪谱方法，它对于线性化的高阶 MoDEE 是无条件稳定的。对高频相邻模式的模式分离进行验证。应用被认为是信号和图像去噪、图像边缘检测、特征提取、增强等。