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方面的巨大成功，人们已经进行了多次尝试，以使其从一维到多维设置的最佳效率。事实上，尽管小波具有非凡的特性，但在捕捉多维现象的固有几何特征方面效率并不高。近年来，研究者及其合作者引入的定向多尺度方法，如shearlet表示，已经成为小波框架在多维环境中最有效的扩展。事实上，shearlet表示包含了仿射系统的数学理论，并且是迄今为止唯一能够通过多分辨率分析的力量将最优稀疏性和快速变换结合起来的方法。提出的研究重点是将剪切波方法应用于多维数据分析和处理中的一些具有挑战性的问题。首先，应用剪切波表示对多元函数和分布的不连续点进行精确的几何表征。结合谐波分析和微分几何技术，这为改进边缘检测和特征提取算法的发展奠定了基础。其次，应用shearlet框架开发了新一代的不适定问题正则化反演方法。基于shearlet提供傅里叶积分算子的稀疏表示的能力，计算了Radon和ray变换的有效分解。这些用于开发从局部和不完整数据进行Radon反演和图像反卷积的算法。第三，提出了一种新的视点不变纹理检索的数学计算方法。这是通过在适当的统计设置中共同设计特征提取和相似性测量框架来实现的，并依赖于shearlet捕获局部几何信息的独特能力。在过去的几年中，人们面临着越来越大的压力，需要更有效地处理来自电子监控、遥感和医学成像等广泛应用的更大、更高维度的数据集。如何快速、准确、可靠地提取相关信息，并对其进行高效的处理、传输和存储，是当前面临的挑战。该项目侧重于shearlet表示的应用-由研究人员及其合作者介绍的方法，该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，最优稀疏性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。具体来说，该项目将为医学、工业和卫星图像的边缘检测、特征提取和纹理检索带来先进的技术。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、数据分类和电子监测等敏感应用的技术进步。

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.