PDE Techniques in Wavelet Based Image Processing

小波图像处理中的偏微分方程技术

• 批准号: 0410062
• 负责人: Haomin Zhou
• 金额: --
• 依托单位: Georgia Tech Research Corporation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-09-01 至 2008-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0410062&HistoricalAwards=false
• 关键词: PDE; Techniques; Wavelet; Based; Image

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.

本研究的目的是为数字图像处理中的问题和应用发展数学原理和新方法，并研究它们的数学基础，重点是将偏微分方程（PDE）技术与多分辨率小波表示相结合，特别是，我们关注两种不同的图像处理应用：小波修复（填充丢失或损坏的小波系数）和PDE图像压缩。为了实现这些任务，我们将把一些高级数学工具，包括几何偏微分方程，多分辨率谐波分析和变分框架与一些最先进的工程方法，如图像压缩的分组测试小波（GTW）算法结合在一起。其关键是利用小波的多分辨率特性，同时使用PDE技术来系统地和明确地控制图像中的几何信息，以便在重建中恢复显著特征，如边缘和角点。图像压缩和图像修复是图像处理的两个典型任务。由于数字图像的像素数很大，大多数图像都必须以压缩格式存储。目前国际上流行的图像压缩标准（JPEG2000）是基于小波变换的图像压缩标准之一，图像压缩在我们的日常生活中已经被广泛应用，例如互联网上传输的图像、无线通信、医学图像（如MRI）、卫星图像等。高效的压缩方法是非常可取的，通用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）。图像修复是指对图像中不完整、缺失或损坏的信息进行自动填充的过程。在无线传输、情报、国土安全（机场安检）、机器人在未知环境中的路径搜索、二维医学图像的三维重建等应用中，这种损失是不可避免的。由于许多数字图像都是以小波格式存储的，而这种格式的图像的损坏对应于小波系数的丢失，因此小波修复需要特别关注。 同样，填充缺失信息的一个关键目标是恢复缺失的几何属性。我们将研究新的小波修复模型和方法以及相关的数学理论。