Sparse Shearlet Representation: Analysis, Implementation and Applications

在过去的几年里，在处理更大和更高维的数据集方面取得了惊人的成功。 这些进展以多尺度技术为基础，并应用于例如联邦调查局新的指纹数据库和新的图像压缩标准JPEG2000，但是，数学方面的最新进展表明，这些技术远非最佳：有很大的改进余地。 换句话说，可以建立新一代的方法，这将大大扩展其应用范围。 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，这些工具可以以有效和准确的方式表示数据并提取相关信息。 潜在的应用包括遥感，医疗诊断，数据传输和分类，视频监控，和数据storage.It建议开发剪切波表示，它结合了多尺度方法的力量与独特的能力，以捕捉多维数据的几何形状。 这种方法为新一代的方法打开了大门，这些方法对于处理多维数据是最有效的（没有改进的余地），并且计算速度快。研究人员提出了一个研究计划，涉及剪切波表示的数学和数值方面的发展。 这不仅将导致改进的算法，数据压缩和分析，它也是一个非常有前途的方法，数值解偏微分方程。 这两种调查途径都将在项目中进行。 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.