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

研究者为多维数据开发、实现并应用了一种新的多尺度表示方法。提出的shearlet方法包含了仿射系统的数学框架，并且是迄今为止唯一能够结合最优稀疏性（需要计算的系数很少）、通过多分辨率分析（快速计算）的能力进行快速变换和完整的数学证明和框架（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多维多尺度特征的直接结果，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的调查方向，有几个具体的目标。首先，研究支撑shearlet的数学框架，为最优稀疏多维表示的构建和分析奠定基础。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，shearlet被用作各向异性函数空间的构建块。这一步在近似理论、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，将shearlet应用于图像处理和图像分析问题。具体来说，改进的算法实现被开发并应用于图像去噪，边缘检测和形状识别。在生物医学数据上进行测试，以解决特定的应用驱动问题，包括从共聚焦图像和神经元分类中进行神经元形态的几何重建。在过去的二十年里，多尺度方法和小波已经发展了信号处理，并刺激了数学和工程领域令人印象深刻的研究。事实上，小波提供了一维数据的最优有效表示，并具有快速的数值实现。结果，小波被成功地应用于许多战略应用，包括新的FBI指纹数据库和JPEG-2000（图像压缩的新标准）。尽管小波取得了显著的成功，但总的来说，它还远不是最优的。尽管它们优于其他传统方法，但它们无法捕捉多维现象的内在几何特征。例如，它们在处理图像的边缘或固体物体的边界表面等特征方面表现不佳，因此，它们无法有效地处理许多现代应用程序所需要的更大的多维数据集。相比之下，这个项目所处理的方法是真正多维的，为新一代高效的数据存储、传输和处理方法打开了大门。这项研究的应用促进了遥感、医学诊断、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.