Career: Sparse directional multiscale representations: theory, implementation and applications

Labate 0746778研究者开发、实现并应用了一种新的多维数据多尺度表示方法。 所提出的剪切波方法包含了仿射系统的数学框架，迄今为止，是唯一能够结合最佳稀疏性（计算系数少），通过多分辨率分析（快速计算）和充分的数学理由和框架（极大的灵活性和多功能性）的快速变换的方法。 所提出的剪切波表示的稀疏性是其真正的多维多尺度特性的直接结果，其快速实现是其仿射几何结构的结果。 该项目分为三个主要的调查方向，有几个具体的目标。 首先，研究支撑剪切波的数学框架，为构造和分析最优稀疏多维表示奠定基础。 然后，将这些表示应用于函数、空间和算子的分解. 更具体地说，剪切波被用作各向异性函数空间的构建块。 这一步在逼近理论、傅立叶积分算子的研究以及与偏微分方程相关的各种非标准正则空间中具有重要意义。第三，剪切波被应用到图像处理和图像分析中。 具体而言，改进的算法实现，并应用于图像去噪，边缘检测和形状识别。 对生物医学数据进行测试，以解决特定的应用驱动的问题，包括神经元形态的几何重建共聚焦图像和神经元分类。在过去的20年里，多尺度方法和小波变换使信号处理发生了革命性的变化，并在数学和工程领域引起了大量的研究。 事实上，小波提供了一维数据的最佳有效表示，并具有快速的数值实现。 阿萨作为一个结果，被成功地应用在一些战略性的应用，包括新的FBI指纹数据库和JPEG-2000，新的标准，为图像压缩。尽管小波取得了显著的成功，但一般来说，小波远不是最优的。 尽管它们优于其他传统方法，但它们未能捕捉多维现象的内在几何特征。 例如，它们在处理诸如图像的边缘或固体物体的边界表面等特征方面做得很差，因此，它们无法有效地处理许多现代应用所需要的越来越大的多维数据集。 相比之下，本项目中提出的方法是真正多维的，并为新一代高效的数据存储、传输和处理方法打开了大门。 从这项研究中获得的应用促进了遥感，医疗诊断，数据传输和分类，视频监控和数据存储等敏感应用的技术进步。

Labate0746778The investigator develops, implements, and applies a newmultiscale representation method for multidimensional data. Theproposed shearlet approach encompasses the mathematical frameworkof affine systems and, to date, is the only method able tocombine optimal sparsity (few coefficients to compute), fasttransforms through the power of multiresolution analysis (fastcomputation) and full mathematical justification and framework(great flexibility and versatility). The sparsity of theproposed shearlet representation is a direct consequence of itsgenuinely multidimensional multiscale character, and its fastimplementation a consequence of its affine mathematicalstructure. The project is organized into three main directionsof investigation, with several specific goals. First, themathematical framework underpinning the shearlets is investigatedto set the foundation for the construction and analysis ofoptimally sparse multidimensional representations. Next theserepresentations are applied to the decomposition of functionsspaces and operators. More specifically, the shearlets are usedas building blocks of anisotropic function spaces. This step hassignificant implications in approximation theory, in the study ofFourier integral operators, and for various nonstandardregularity spaces associated to partial differential equations. Third, shearlets are applied to problems from image processingand image analysis. Specifically, improved algorithmicimplementations are developed and applied to image denoising,edge detection and shape recognition. Tests are conducted onbiomedical data to address specific application-driven problems,including geometric reconstruction of neuronal morphology fromconfocal images and neuronal classification. Over the past twenty years, multiscale methods and wavelets haverevolutionized signal processing and stimulated an impressiveamount of research in mathematics and engineering. In fact,wavelets provide optimally efficient representations ofone-dimensional data and have fast numerical implementations. Asa result, wavelets are successfully employed in a number ofstrategic applications, including the new FBI fingerprintdatabase and JPEG-2000, the new standard for image compression. In spite of their remarkable success, wavelets are far from beingoptimal in general. Even though they outperform othertraditional methods, they fail to capture intrinsic geometricalfeatures of multidimensional phenomena. For instance, they dopoorly at dealing with features such as the edges of an image orthe boundary surfaces of a solid object, and, as a consequence,they are unable to handle efficiently the ever largermultidimensional data sets which are required by many modernapplications. By contrast, the approach addressed in thisproject is truly multidimensional and opens the door to a newgeneration of highly efficient methods for the storage,transmission and processing of data. The applications arisingfrom this research facilitate technological advances in sensitiveapplications such as remote sensing, medical diagnostics, datatransmission and classifications, video surveillance, and storageof data.