New Approaches for Better Spatial Frequency Localization in Two- and Three-Dimensional Data Analysis

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This research project concerns mathematical and algorithmic developments for image analysis (image compression, and the like). To compress or analyze images, it is useful to decompose them into elementary building blocks that are especially effective - in other words, it pays to use mathematical decomposition tools that achieve high accuracy with relatively few coefficients. For instance, in the transition from the JPEG standard to JPEG2000, the wavelet transform replaced the discrete cosine transform because it provided a sparser representation for images and because its performance degrades more gracefully when bandwidth is variable. This project aims to develop methods for improved sparse representation of images.Curvelets and shearlets constitute a variant on the wavelet approach that can be shown, mathematically, to lead to even sparser representations for images. So far, they have not lived up to this promise in practice -- the only implementations in existence have a high overhead and a large redundancy factor. The Principal Investigator and her collaborators have identified several approaches that they expect will provide better basis constructions, which in turn will give better implementations, with great potential for applications. These will be developed in this project.