Sparse Shearlet Representation: Analysis, Implementation and Applications

在过去的几年里，在处理越来越大和更高维的数据集方面取得了惊人的成功。这些进展是基于多尺度技术的，并被应用于新的联邦调查局指纹数据库和新的图像压缩标准JPEG2000中。然而，最近的数学进步表明，这些技术远远不是最佳的：还有很大的改进空间。换句话说，可以构建新一代方法，从而显著扩展它们的应用范围。这满足了许多敏感应用程序对更强大的工具的日益增长的需求，以便以高效和准确的方式表示数据和提取相关信息。在遥感、医疗诊断、数据传输和分类、视频监控和数据存储等领域具有潜在的应用前景，提出了一种结合多尺度方法的能力和独特的捕捉多维数据几何特征的剪切波表示方法。这种方法打开了新一代方法的大门，这些方法在处理多维数据时是最有效的(没有改进的余地)，并且计算速度很快。研究人员提出了一个研究计划，涉及到剪切片表示的数学和数值两个方面的发展。这不仅将导致改进的数据压缩和分析算法，而且也是一种非常有前途的偏微分方程组的数值求解方法。这两种调查方式都将在该项目中进行。还将考虑生物医学和地质数据的具体应用。

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.