Collaborative Research: Analysis and processing of multidimensional data using sparse directional multiscale representations

Labate，DMS-1008900 Guo，DMS-1008907 随着小波在信号和图像处理中的巨大成功，人们已经进行了几次尝试，以使其从一维到多维的最佳效率。 事实上，尽管小波具有显著的性质，但它在捕捉多维现象的内在几何结构方面并不是很有效。 近年来，方向多尺度方法，如研究人员及其合作者介绍的剪切表示，已经成为小波框架到多维设置的最有效的扩展。 事实上，剪切波表示包含了仿射系统的数学理论，迄今为止，是唯一能够通过多分辨率分析的力量将最佳稀疏性和快速变换结合起来的方法。 拟议的研究重点放在应用的剪切波方法的一些challengingproblems的分析和处理多维数据。首先，剪切波表示被应用于提供多元函数和分布的不连续性的精确几何表征。 结合调和分析和微分几何的方法，为改进边缘检测和特征提取算法奠定了基础。 第二，应用剪切波框架发展了新一代不适定问题的正则化反演方法。 剪切波的能力，提供稀疏表示的傅里叶积分算子的基础上，有效的分解Radon和射线变换计算。 这些被用来开发算法的氡反演从当地和不完整的数据和图像反卷积。 第三，介绍了一种新的视点不变纹理检索的数学和计算方法。 这是通过联合设计一个框架，在一个适当的统计设置的特征提取和相似性测量，并依赖于独特的capabilityofshearlets捕捉局部几何信息。 在过去的几年里，有一个不断增加的压力，以更有效地处理越来越大和更高维的数据集产生的广泛的应用，如电子监控，遥感和医学成像。 如何快速、准确、可靠地提取相关信息，以便有效地处理、传输和存储，是一个挑战。 该项目的重点是剪切波表示的应用-研究人员及其合作者介绍的方法，在创新的数学和计算框架内提供了最佳稀疏性和计算效率的独特组合。 特别是最佳稀疏度的概念意味着这种方法能够非常有效和可靠地识别数据中包含的最相关的特征。 具体而言，该项目导致先进的技术，边缘检测，特征提取，和纹理检索从医疗，工业和卫星图像。这导致了创新和改进的计算算法，用于分析和处理高维数据，并促进了遥感、医疗诊断、数据分类和电子监视等敏感应用领域的技术进步。

Labate, DMS-1008900Guo, DMS-1008907 Following the spectacular success of wavelets in signal andimage processing, several attempts have been made to adapt theiroptimal efficiency from the one- to the multi-dimensionalsetting. In fact, in spite of their remarkable properties,wavelets are not very efficient in capturing the intrinsicgeometry of multidimensional phenomena. In recent years,directional multiscale methods such as the shearletrepresentation, introduced by the investigators and theircollaborators, have emerged as the most effective extension ofthe wavelet framework to the multidimensional setting. Indeed,the shearlet representation encompasses the mathematical theoryof affine systems and, to date, is the only method able tocombine optimal sparsity and fast transforms through the power ofmultiresolution analysis. The proposed research focuses onapplications of the shearlet approach to a number of challengingproblems of analysis and processing of multidimensional data. First, the shearlet representation is applied to provide aprecise geometric characterization of the discontinuities ofmultivariate functions and distributions. Combining techniquesfrom harmonic analysis and differential geometry, this providesthe groundwork for the development of improved algorithms foredge detection and feature extraction. Second, the shearletframework is applied to develop a new generation of methods forthe regularized inversion of ill-posed problems. Building on theability of shearlets to provide sparse representations of Fourierintegral operators, efficient decompositions for the Radon andRay transforms are computed. These are used to developalgorithms for the Radon inversion from local and incomplete dataand for image deconvolution. Third, a novel mathematical andcomputational approach for viewpoint-invariant texture retrievalis introduced. This is achieved by jointly designing a frameworkfor feature extraction and similarity measurement in anappropriate statistical setting, and relies on the unique abilityof shearlets to capture local geometric information. Over the past several years, there has been a continuouslyincreasing pressure to handle more efficiently the ever largerand higher dimensional data sets generated from a wide rangeapplications such as electronic surveillance, remote sensing, andmedical imaging. The challenge is to rapidly, accurately andreliably extract the relevant information, so that it can beefficiently processed, transmitted and stored. The projectfocuses on the applications of the shearlet representation -- amethod introduced by the investigators and their collaboratorsthat provides a unique combination of optimal sparsity andcomputational efficiency, within an innovative mathematical andcomputational framework. The notion of optimal sparsity, inparticular, implies that this approach has the ability to veryeffectively and reliably identify the most relevant featurescontained in the data. Specifically, this project leads toadvanced techniques for edge detection, feature extraction, andtexture retrieval from medical, industrial and satellite imagery. This results in innovative and improved computational algorithmsfor the analysis and processing of high-dimensional data andfacilitates technological advances in sensitive applications suchas remote sensing, medical diagnostics, data classification andelectronic surveillance.