Wang Yang, Wei Guo-Wei, Yang Siyang
Int J Numer Method Biomed Eng. 2011 Dec;27(12):1996-2020. doi: 10.1002/cnm.1452.
Nonlinear partial differential equation (PDE) models are established approaches for image/signal processing, data analysis and surface construction. Most previous geometric PDEs are utilized as low-pass filters which give rise to image trend information. In an earlier work, we introduced mode decomposition evolution equations (MoDEEs), which behave like high-pass filters and are able to systematically provide intrinsic mode functions (IMFs) of signals and images. Due to their tunable time-frequency localization and perfect reconstruction, the operation of MoDEEs is called a PDE transform. By appropriate selection of PDE transform parameters, we can tune IMFs into trends, edges, textures, noise etc., which can be further utilized in the secondary processing for various purposes. This work introduces the variational formulation, performs the Fourier analysis, and conducts biomedical and biological applications of the proposed PDE transform. The variational formulation offers an algorithm to incorporate two image functions and two sets of low-pass PDE operators in the total energy functional. Two low-pass PDE operators have different signs, leading to energy disparity, while a coupling term, acting as a relative fidelity of two image functions, is introduced to reduce the disparity of two energy components. We construct variational PDE transforms by using Euler-Lagrange equation and artificial time propagation. Fourier analysis of a simplified PDE transform is presented to shed light on the filter properties of high order PDE transforms. Such an analysis also offers insight on the parameter selection of the PDE transform. The proposed PDE transform algorithm is validated by numerous benchmark tests. In one selected challenging example, we illustrate the ability of PDE transform to separate two adjacent frequencies of sin(x) and sin(1.1x). Such an ability is due to PDE transform's controllable frequency localization obtained by adjusting the order of PDEs. The frequency selection is achieved either by diffusion coefficients or by propagation time. Finally, we explore a large number of practical applications to further demonstrate the utility of proposed PDE transform.
非线性偏微分方程（PDE）模型是图像/信号处理、数据分析和曲面构建的既定方法。以前的大多数几何偏微分方程都用作低通滤波器，可产生图像趋势信息。在早期工作中，我们引入了模式分解演化方程（MoDEEs），其作用类似于高通滤波器，能够系统地提供信号和图像的固有模式函数（IMFs）。由于其可调的时频定位和完美重构，MoDEEs的操作被称为PDE变换。通过适当选择PDE变换参数，我们可以将IMFs调整为趋势、边缘、纹理、噪声等，可在二次处理中进一步用于各种目的。本文介绍了变分公式，进行了傅里叶分析，并开展了所提出的PDE变换的生物医学和生物学应用。变分公式提供了一种算法，可将两个图像函数和两组低通PDE算子纳入总能量泛函。两个低通PDE算子具有不同的符号，导致能量差异，同时引入一个耦合项作为两个图像函数的相对保真度，以减少两个能量分量的差异。我们使用欧拉-拉格朗日方程和人工时间传播构建变分PDE变换。对简化的PDE变换进行傅里叶分析，以阐明高阶PDE变换的滤波器特性。这样的分析还为PDE变换的参数选择提供了见解。所提出的PDE变换算法通过大量基准测试得到验证。在一个选定挑战示例中，我们展示了PDE变换分离sin(x)和sin(1.1x)两个相邻频率的能力。这种能力归因于通过调整PDE的阶数获得的PDE变换的可控频率定位。频率选择可通过扩散系数或传播时间实现。最后，我们探索了大量实际应用，以进一步证明所提出的PDE变换的实用性。
