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

本研究项目涉及图像分析（图像压缩等）的数学和算法开发。 为了压缩或分析图像，将它们分解成特别有效的基本构建块是有用的-换句话说，使用数学分解工具以相对较少的系数实现高精度是值得的。 例如，在从JPEG标准到JPEG 2000的过渡中，小波变换取代了离散余弦变换，因为它为图像提供了更稀疏的表示，并且因为当带宽可变时，其性能会更温和地下降。 该项目旨在开发改进图像稀疏表示的方法。Curvelets和shearlets构成小波方法的变体，可以在数学上显示，以导致图像更稀疏的表示。 到目前为止，它们还没有在实践中实现这一承诺--现有的实现具有很高的开销和很大的冗余系数。 首席研究员和她的合作者已经确定了几种方法，他们希望这些方法能够提供更好的基础结构，从而提供更好的实现，具有巨大的应用潜力。 这些将在本项目中开发。

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.