Mathematical Sciences: Computational Mathematics Issues in Nonlinear Diffusion Modules in Image Processing

研究者和他的同事研究图像处理中非线性扩散模型的计算数学问题。这些问题计算量大，需要精确高效的方法。传统上，标准方法涉及频域计算，由高效的FFT和小波算法促进。最近，有一种新的运动是基于偏微分方程（PDE）的方法，这是一种更系统的方法来恢复具有尖锐边缘的图像，以及用于图像分割。根据非线性各向异性扩散偏微分方程对图像进行扩散（去噪），使图像在近边缘处的扩散较小。此外，偏微分方程被设计成具有某些理想的几何性质，如仿射不变性和因果性。从计算的角度来看，PDE公式需要不同于传统频域和代数方法的新计算技术。到目前为止，与传统方法相比，非线性扩散模型被认为有些昂贵，而本项目的目标之一是使它们在保持其理想的几何特性的同时更有效。此外，还尝试将这些方法改进和扩展到彩色和其他矢量值图像。正在研究的具体计算算法包括原始对偶最小化方法、预处理技术和小波算法。图像处理在物理科学和医学中都有许多重要的应用。在当前的通信革命和信息高速公路的出现中，越来越多的图像被传输，需要更好的数学算法来压缩和去除传输中出现的噪声和其他失真。在医学和生物技术领域的应用范围从计算机地形图到分子结构的显微图像处理。在环境领域，卫星成像已被用于绘制自然资源和环境污染。在制造领域，成像系统被用来自动检测缺陷。在所有这些应用中，一个关键的过程是图像恢复，即清理被噪声和模糊污染的图像。这是本课题的主要课题。由于大量的像素和图像序列（如视频）的可能性，这些问题的计算量非常大，解决它们需要聪明的数学算法以及高性能计算机。

Chan 9626755 The investigator and his colleagues study computational mathematics issues in nonlinear diffusion models in image processing. These problems are computationally intensive and accurate and efficient methods are needed. Traditionally, the standard methods involve computation in the frequency domain, facilitated by efficient FFT and wavelet algorithms. Recently, there has been a new movement towards a partial differential equation (PDE) based approach, which is motivated by a more systematic approach to restoring images with sharp edges, as well as for image segmentation. The image is diffused (denoised) according to a nonlinear anisotropic diffusion PDE, designed to diffuse less near edges. Moreover, the PDEs are designed to possess certain desirable geometrical properties such as affine invariance and causality. From a computational standpoint, the PDE formulations call for new computational techniques that are different from the traditional frequency domain and algebraic approaches. As yet, the nonlinear diffusion models are considered somewhat expensive compared to traditional methods, and it is one of the goals of this project to make them more efficient while retaining their desirable geometric properties. Additionally, attempts are made to improve and extend these methods to color and other vector-valued images. Specific computational algorithms being studied include primal-dual minimization methods, preconditioning techniques, and wavelet algorithms. Image processing has many important applications in both the physical and the medical sciences. In the current revolution in communication and the advent of the information highway, more and more images are being transmitted and better mathematical algorithms are needed to compress and remove noise and other distortions occurring in the transmission. Applications in the medical sciences and biotechnology field range from computer topography to processing of microscopic images of molecular structures. In the environmental area, satellite imaging has been used to map natural resources as well as environmental pollution. In the area of manufacturing, imaging systems are used to detect defects automatically. In all of these applications, a key process is that of image restoration, namely, cleaning up an image polluted by noise and blurring. This is the main subject of this project. These problems are very computationally intensive due to the large number of pixels and the possibility of sequences of images (e.g. videos), solving them requires clever mathematical algorithms as well as high performance computers.