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

实验室0746778研究人员开发、实现并应用了一种新的多维数据的多尺度表示方法。所提出的Shearlet方法包含了仿射系统的数学框架，是迄今为止唯一能够结合最佳稀疏性(需要计算的系数很少)、通过多分辨率分析的能力(快速计算)快速变换以及充分的数学证明和框架(极大的灵活性和通用性)的方法。所提出的Shearlet表示的稀疏性是其真正的多维多尺度特性的直接结果，而其快速实现是其仿射数学结构的结果。该项目分为三个主要调查方向，并有几个具体目标。首先，研究了支持剪切片的数学框架，为构造和分析最优稀疏多维表示奠定了基础。接下来，将这些表示应用于函数空间和算子的分解。更具体地说，剪切片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有关的各种非标准正则性空间中都有重要的意义。第三，将剪切片应用于图像处理和图像分析中的问题。具体地说，提出了改进的算法实现并将其应用于图像去噪、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问题，包括从共焦图像进行神经元形态的几何重建和神经元分类。在过去的二十年里，多尺度方法和小波发展了信号处理，并在数学和工程中激发了大量的研究。事实上，小波提供了一维数据的最佳有效表示，并且具有快速的数值实现。因此，小波被成功地应用于许多战略应用中，包括新的联邦调查局指纹数据库和新的图像压缩标准JPEG-2000。尽管小波取得了显著的成功，但总的来说，它们远不是最优的。尽管它们的性能优于其他传统方法，但它们无法捕捉到多维现象的内在几何特征。例如，它们更擅长处理诸如图像的边缘或固体物体的边界表面之类的特征，因此，它们无法有效地处理许多现代应用所需的越来越大的多维数据集。相比之下，该项目涉及的方法是真正的多维方法，并为新一代高效的数据存储、传输和处理方法打开了大门。这项研究产生的应用促进了敏感应用领域的技术进步，如遥感、医疗诊断、数据传输和分类、视频监控和数据存储。

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.