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

随着小波在信号和图像处理中的巨大成功，人们进行了几次尝试，以使它们的最佳效率从一维调整到多维设置。事实上，尽管小波具有显著的性质，但在捕捉多维现象的内在几何学方面并不是很有效。近年来，研究人员及其合作者提出的剪切法等定向多尺度方法已成为小波框架向多维环境最有效的扩展。事实上，Shearlet表示包含了仿射系统的数学理论，并且到目前为止，是唯一能够通过多分辨率分析的能力结合最佳稀疏性和快速变换的方法。所提出的研究集中于将剪切法应用于多维数据的分析和处理中的一些具有挑战性的问题。首先，应用Shearlet表示对多元函数和分布的不连续性进行精确的几何刻画。结合了调和分析和微分几何的技术，为改进的森林检测和特征提取算法的发展奠定了基础。其次，应用Shearlet框架发展了新一代不适定问题的正则化逆问题的方法。基于Searlet能够提供傅里叶积分算子的稀疏表示的能力，计算了Radon Andray变换的有效分解。这些算法被用于从局部和不完整数据中进行Radon反演和图像反褶积。第三，提出了一种新的视点不变纹理检索的数学和计算方法。这是通过在适当的统计设置下联合设计特征提取和相似性度量的框架来实现的，并且依赖于剪切器独特的捕捉局部几何信息的能力。在过去的几年里，人们面临着越来越大的压力，需要更有效地处理从电子监视、遥感和医学成像等广泛应用中产生的越来越大和更高维的数据集。挑战是快速、准确和可靠地提取相关信息，以便能够有效地处理、传输和存储这些信息。该项目侧重于Shearlet表示的应用--这是研究人员及其合作者提出的一种方法，在创新的数学和计算框架内提供最佳稀疏性和计算效率的独特组合。最优稀疏性的概念特别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。具体地说，该项目带来了从医学、工业和卫星图像中进行边缘检测、特征提取和纹理检索的先进技术。这导致了分析和处理高维数据的创新和改进的计算算法，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.