PDE Techniques in Wavelet Based Image Processing

本研究的目标是发展数字图像处理问题和应用的数学原理和新方法，并研究其相关的数学基础。重点是将偏微分方程（PDE）技术与多分辨率小波表示相结合。我们特别关注两种不同的图像处理应用：小波插值（填充缺失或损坏的小波系数）和PDE图像压缩。为了完成这些任务，我们将整合几个高级数学工具，包括几何PDE，多分辨率谐波分析和变分框架，以及一些最先进的图像编码工程方法，如用于图像压缩的组测试小波（GTW）算法。关键是利用小波的多分辨率特性，同时利用PDE技术系统地、明确地控制图像中的几何信息，以便在重建中恢复边缘和角等显著特征。将设计和分析有效的计算算法。图像压缩和图像绘制是图像处理的两个典型任务。由于数字图像的像素数量很大，大多数图像都必须以压缩格式存储。目前的国际压缩标准（JPEG2000）主要基于小波表示，是最受欢迎的方案之一。图像压缩在我们的日常生活中无处不在，例如在互联网上传输的图像，无线通信，医学图像（如MRI），卫星图像。高效的压缩方法是非常可取的，普遍的目标是使用尽可能少的资源来表示最显著的特征（通常是几何结构）。这是这个项目的目标之一。图像补绘是指对不完整、缺失或损坏的图像信息进行自动补绘的程序。在无线传输、情报、国土安全（机场安检）、未知环境下的机器人寻径、二维医学图像重建三维物体等许多应用中，这种损失往往是不可避免的。由于许多数字图像以小波格式存储，并且对这种格式图像的损坏对应于小波系数的损失，因此小波图像处理需要特别注意。同样，填补缺失信息的一个关键目标是恢复缺失的几何属性。我们将研究小波绘画的新模型和新方法以及相关的数学理论。

The goal of this research is to develop mathematical principlesand new methodologies for problems in and applications of digitalimage processing, and study their associated mathematical foundations.The emphasis is on combining partial differential equation (PDE)techniques with multi-resolution wavelet representations.In particular, we focus on two different image processing applications:wavelet inpainting (filling in missing or damaged wavelet coefficients),and PDE image compression. To achieve the tasks, we will integrate severalhigh level mathematical tools including geometrical PDE's,multi-resolution harmonic analysis and variational frameworkstogether with some state-of-the-art engineering methods in imagecoding, such as group testing wavelet (GTW) algorithms for imagecompression. The key is to exploit the power of multi-resolutionproperties of wavelets while at the same time use PDE techniques tosystematically and explicitly control the geometrical informationin the image so that salient features, such as edges and corners,can be recovered in the reconstructions. Efficient computation algorithmswill be designed and analyzed.Image compression and inpainting are two typical tasks of imageprocessing. Due to the large number of pixels of digital images,most of them have to be stored in compressed format. The currentinternational compression standard (JPEG2000), which is largelybased on wavelet representation, is one of the most popular schemes.Image compression has been used everywhere in our daily life, examplesinclude images transmitted on the Internet, wireless communications,medical images (such as MRI), satellite images. Efficient compressionmethods are highly desirable and the universal goal is to represent themost salient features (typically geometrical structures) using minimalpossible resources. This is one of the objectives of this project.Image inpainting refers to automatic procedures to fillin incomplete, missing or damaged image information. Such lossis often unavoidable in many applications such as wirelesstransmission, intelligence, homeland security (airports' screening),robotic path finding in unknown environment, three-dimensionalobject reconstruction from two-dimensional medical images. Becausemany digital images are stored in wavelet formats, and damages to suchformatted images correspond to loss of wavelet coefficients,wavelet inpainting demands special attention. Again, a key objectivein filling in the missing information is to restore missinggeometrical properties. We will investigate new models and methods forwavelet inpainting and related mathematical theories.