Collaborative Research: Analysis and processing of multidimensional data using sparse directional multiscale representations

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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.