The investigator develops, implements, and applies a new multiscale representation method for multidimensional data. The proposed shearlet approach encompasses the mathematical framework of affine systems and, to date, is the only method able to combine optimal sparsity (few coefficients to compute), fast transforms through the power of multiresolution analysis (fast computation) and full mathematical justification and framework (great flexibility and versatility). The sparsity of the proposed shearlet representation is a direct consequence of its genuinely multidimensional multiscale character, and its fast implementation a consequence of its affine mathematical structure. The project is organized into three main directions of investigation, with several specific goals. First, the mathematical framework underpinning the shearlets is investigated to set the foundation for the construction and analysis of optimally sparse multidimensional representations. Next these representations are applied to the decomposition of functions spaces and operators. More specifically, the shearlets are used as building blocks of anisotropic function spaces. This step has significant implications in approximation theory, in the study of Fourier integral operators, and for various nonstandard regularity spaces associated to partial differential equations. Third, shearlets are applied to problems from image processing and image analysis. Specifically, improved algorithmic implementations are developed and applied to image denoising, edge detection and shape recognition. Tests are conducted on biomedical data to address specific application-driven problems, including geometric reconstruction of neuronal morphology from confocal images and neuronal classification.
Over the past twenty years, multiscale methods and wavelets have revolutionized signal processing and stimulated an impressive amount of research in mathematics and engineering. In fact, wavelets provide optimally efficient representations of one-dimensional data and have fast numerical implementations. As a result, wavelets are successfully employed in a number of strategic applications, including the new FBI fingerprint database and JPEG-2000, the new standard for image compression. In spite of their remarkable success, wavelets are far from being optimal in general. Even though they outperform other traditional methods, they fail to capture intrinsic geometrical features of multidimensional phenomena. For instance, they do poorly at dealing with features such as the edges of an image or the boundary surfaces of a solid object, and, as a consequence, they are unable to handle efficiently the ever larger multidimensional data sets which are required by many modern applications. By contrast, the approach addressed in this project is truly multidimensional and opens the door to a new generation of highly efficient methods for the storage, transmission and processing of data. The applications arising from this research facilitate technological advances in sensitive applications such as remote sensing, medical diagnostics, data transmission and classifications, video surveillance, and storage of data.
