Sparse Shearlet Representation: Analysis, Implementation and Applications

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The past few years have seen spectacular successes in the handling of ever larger- and higher-dimensional data sets. Those advances are based on multiscale techniques and are applied, for instance, in the new FBI fingerprint database and in JPEG2000, the new standard for image compression.However, recent advances in mathematics have shown these techniques to be far from optimal: there is ample room for improvement. In other words, a new generation of methods can be built that will significantly expand their range of applications. This answers a growing need in many sensitive applications for more powerful tools to represent data and extract relevant information in an efficient and accurate way. Potential applications include remote sensing, medical diagnosis, data transmission and classification, video surveillance, and data storage.It is proposed to develop the shearlet representation, which combines the power of multiscale methods with a unique ability to capture the geometry of multidimensional data. This approach opens the door to a new generation of methods that are optimally efficient (no room for improvement) for handling multidimensional data and are fast to compute.The investigators propose a program of research that involves the development of both mathematical and numerical aspects of the shearlet representation. Not only will this lead to improved algorithms for data compression and analysis, it is also a very promising approach for the numerical solution of partial differential equations. Both avenues of investigation will be pursued in the project. Specific applications to biomedical and geological data will also be considered.